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Calculus

Power rule

We learn this this high school: a function xⁿ has the derivative nx^n-1.

Proof example

Let's take the square root function x^1/2. The power rule says this should result in 1/2 * x^-1/2.

We'll prove it using the limit "as delta x goes to zero". Some people like to call that variable h or d, but we'll go with delta x.

We cannot set delta x to zero, because it's sitting in the denominator. Let's change the expression. Using the difference of squares (a + b)(a - b) = a² - b², we multiply numerator and denominator by (a + b).

Normally, when we cancel something from the denominator, we add a clause that it must be nonzero. Irrelevant/implicit when you're working with limits.

We're left with the expression node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , and can now pretend that delta x is zero without breaking the equation. That's just node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , or node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , which fits the power rule.

Chain rule

A function f(x) that composes two functions g(x) and h(x) such that one is inside the other like: g(h(x)), will have the following derivative: g'(h(x)) * h'(x).

Note that in the first factor you neglect modifying the inner function.

Example: the derivative of sin(2x) is 2cos(2x), because you are deriving two functions: sin(something), and 2x. The derivative of 2x is just 2.

Aside: As you know, the sine of x derives to the cosine of x, but there are two functions in sine of x: the sine, and the x itself. What gives? Indeed you should derive both. But x derives to 1, which you just don't need to write.

Product rule

A product of two functions f(x) g(x) has the derivative f'(x) g(x) + f(x) g'(x).

For a product of three functions, derive "one at a time", leaving the rest untouched, for each term.

Function inputs

Take f(x).

Reflect across the x-axis: -f(x). This should be intuitive.
Reflect across the y-axis: f(-x)
Reflect across both axes: -f(-x)

Example: Periodic functions

sin(2x) är dubbelt så “snabb” som sin(x). sin(x/2) hälften så snabb. 2sin(x) gör kurvan dubbelt så hög, men ändrar inte svängningens frekvens. Dvs den når fortfarande 0 efter 1pi tid (alltså vid x=1pi). sin(x) + 2 förflyttar kurvan 2 steg uppåt, så att dess “noll-axel” är en horisontell linje som skär y vid 2.

TODO Plot a third-degree polynomial and paste images for the difference between f(x), -f(x), f(-x).

Odd and even functions

An even function is where f(-x) = f(x), i.e. symmetric, reflected across y-axis. An odd function is where -f(-x) = f(x). i.e. that if you reflect it across both axes, you end up with the same graph.

If there is a constant — an y-intersect offset from origin — it will fail to be odd.

A function is neither even nor odd when f(x) doesn't tell you anything about f(-x), i.e. when it isn't perfectly reflected across axes.

The terminology came from exponents of some functions, but the exponents don't necessarily determine it.

When the highest-degree exponent is odd, the function may be odd. When the highest-degree exponent is even, the function may be even.

What's a function

All these are functions:

sin x
x²
2x
x
ln x
e^x
2^x

A function evaluates to only one value for a given x.

Elementary function

Euclid's similarity cases

△ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.

Compare two triangles:

Two pairs of sides being proportional and the angle between them the same => similar triangles.
Any pairs of sides being proportional => similar. (??)
Any pair of angles the same => similar. (??)

Partial fraction expansion

A way to change a rational expression into multiple terms, which is then easier for some purposes – like taking the derivative or integral, or doing a Laplace transform.

Part of elementary school in some countries…

Take an example. node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

First check to see what degree these polynomials are in. If the denominator has a lower degree, then you do a polynomial division, and then do the partial fraction expansion, okay? The technique is for stuff that you can't divide, like the remainder left by a polynomial division.

Now observe the denominator. What x makes it reach zero? That's 2. So you can factor out (x - 2), right? (Scribble a polynomial division of node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 with node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 )

We get an altered denominator: node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

You could also try to factor the second-degree expression we just got, but doing it in our head tells us it will not be possible in the reals. Skip it.

Now the aim of the technique is to set variables A and B, sometimes C etcetera:

A is a constant because the denominator is first-degree. You perceive the rule: the numerator should be one degree less.

To put the right-hand side over the same denominator, expand the fractions. You know how…

Now you can just remove the denominator from both sides. Unfactor the RHS.

Logarithms

Addition: lg 5 + lg 5 = lg 25 Subtraction: lg 10 - lg 5 = lg 10/5 = lg 2

Exponents: lg 2³ = 3 lg 2

Change of base formula

Limits

"What is the limit of node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 when n goes to infinity?"

Solution: we want to do a variable substitution on the function inputs. We'll have to change the n out front so that it is something something 1/n. The expression above is equal to node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 .

Thereafter, subbing 1/n for x, we ask the limit of x as it goes to zero – because 1/infty goes to zero. We get the expression node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

What happens here? The solution lies in Standard limits. The limit goes to 1 * 1 = 1.

Standard limits🔗

node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1	x → 0		x → ∞	x → + ∞
	node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1		node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

	node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1
	node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1
	node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

Differential equation

Differential equations involve a function and its derivative, sort of. In a "first-order" differential equation, the highest order of derivative is one. In an "ordinary" differential equation, the function is of one variable.

If there is a requirement that the solution take on a particular value at a particular input, the exercise is called a `begynnelsevärdesproblem'.

You can plot the solutions to differential equations, in the form of a grid of 'directions' in a coordinate system: the tangent that a solution function would have at a given (x,y). `Riktningsfält'.

Now, by a linear differential equation, this is meant:

Take a generic equation node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , and name the LHS node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 . Linearity will imply that node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 and that node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 . This is similar to linearity in other fields of math, no?

Solving a first-order differential equation

The simplest form of a first-order equation is node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , which I will name A. A more general form is node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 To solve that one, turn it into A.

Choose a primitive node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 to the coefficient node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 for y. Multiply both sides of the equation with the so-called integrerande faktorn node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 . (Because node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 is nonzero for all x, the resulting equation is strictly equivalent!)

Notice, that the LHS is the derivative of node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 ! That is neat since we are working on differential equations. Rewrite the equation as

Now you can treat this as you would treat A. Namely, turn node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 into node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 (see my seminar writeups for another example):

node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

Applying this to the equation we got, we get

Now you can divide both sides by node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 and get an expression for y. Insert whatever required values you were given for y and x to find out C, etc.

Fundamental Theorem of Calculus

Proves the useful formula of Insattningsformeln (PB 298), which is just the basic rule of how to integrate a defined interval.

Can also be used for (PB 298)

Optimization

Finding the "best" solution to a problem, like the maximum payoff or minimal material use. Involves finding asymptotes, zeros of derivatives, etc.

Continuity

A function is continuous around a point if it isn't undefined nor jumps to some unexpected value at exactly that point. Mathematically, the limit from both sides of that point is equal to the value of the function at that point.

Differentiability

Standard derivatives

The derivatives of common functions are as follows

f(x)	f'(x)
sin x	cos x
cos x	-sin x
tan x	node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1
cot x	node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1
arcsin x	node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1
arccos x	node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1
arctan x	node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1
arccot x	node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

Standard integrals

Implicit differentiation

When you work with unknown functions and use their derivatives, it's called implicit differentiation. Nothing to be scared of.

Absolute value

|xy| = |x||y| Proof:
For |x + y|, see Triangle inequality

Taking the absolute value of a complex number lets you get rid of the "i".

Absolute equation

Consider the graph of an absolute value linear function like |x - 1|. It has two distinct segments, and no other notation would give it that shape. It's not a smooth function, it's two functions, depending on which x you're at.

So to rewrite |x - 1| without bar notation, make cases for different x. The quick way is to see where it becomes zero, which is at x=1. For the case of x>1, you can just remove the bars. For the case of x<1, also flip the signs.

It's easier with complex numbers. Consider the case of having a complex number inside bar notation, like |a + bi|. Rewriting it as a vector gives us a circle around origin, any point on which may be touched by the vector, but it's irrelevant - we want only the radius r in re^iθ{}.

Now consider the case of some unknown complex number z, plus other stuff: |z + 1 + i|. The origin of the "vector" z will now be at -1, -i, so this circle is centred off-origin.

Example solving the above: Set z := x + yi. Say its absolute value is 2. Write the equation.

The last form can be recognized as Pythagoras.

After squaring, the equation resembles that of a circle:

[2024-04-22 Mon] Created Maclaurin/Taylor series

Standardutvecklingar

In these cases, θ is a number between 0 and 1 that depends on x and n.

Proofs for the above (PB )

Mean value theorem

Partial integration

What links here

Math

Created 2016-Nov-17 (8 years ago)

Trigonometry

Reciprocal identities

Secant and cosecant not taught in Sweden.

If you prefer expressing them in sine and cosine:

Interestingly, note that tan is 1/cotan, but sine is not 1/cosine.

Offset identities

Imagine the unit circle. Instead of starting at zero, you start at pi/2. Then, due to subtraction, you walk clockwise, not counterclockwise.

Unit circle

Inverse trig functions

There are different definitions of the domain of inverse trig functions. Multi-valued arcsin is a thing. Mathematica in particular might define the cotangent differently from Sweden. But here are the rules for basic-uni Swedish math.

TODO Explain how to think, don't just make a reference table

Law of sines

We can use it to find stuff about nonright triangles.

Angle A, opposing side a;

Num and denom can be swapped.

Law of cosines

Can tell you stuff about nonright triangles. Like with the Law of Sines, the capital letters denote angle opposing a side denoted in lower case.

The Pythagorean Theorem is a special case of this Law.

Pythagorean identity

In Swedish, it's called trigonometriska ettan. cos²x + sin²x = 1²

Should be obvious if you look at the unit circle and remember the Pythagorean Theorem.

Half-angle formula

Double-angle formula

sin(2x) = 2 sin(x) cos(x)

Proof?

Example: Find the sine of π/12

Using the double angle formula, we know that

sin(π/6) = 2 sin(π/12) cos(π/12).

We also know from our standard angles that sin(π/6) is 1/2. So

The reason I left cosine by itself is that we'll exploit the Pythagorean identity. First, let's square this thing.

Now, look at a different equation, the Pythagorean identity:

We have an expression for node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , so replace it.

Multiply the equation so we're rid of the fraction.

Now, we can do variable substitution. Say node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 .

Solve like any second-degree equation.

So, given that we made y the squared sine of π/12, we square root the equation and get our answer.

To figure out whether the plus or minus applies, test them. The result node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 is definitely larger than sin(π/12) ought to be, so the plus-minus sign should be a minus.

Euler's formula🔗

General form: e^iθ = cos(θ) + isin(θ)

Insert π and check the value. It gives you zero sin, -1 cos. So: e^iπ = -1. You quickly realize that inputting any multiple of pi will give you a real number.

Trig angle-addition formulas can be derived: substitute (a + b) for θ. Convert to cis(a + b) form. Perform the complex multiplication. The terms with i will correspond to the angle addition formula for sin, since the sine axis in the unit circle and the imaginary axis are both vertical. The terms without i then, of course, correspond to the angle addition formula for cos.

De Moivre's formula follows from Euler's formula. (cosθ + isinθ)ⁿ is (e^iθ)ⁿ, which by simple exponent rules is e^iθn, which by definition is cos(nθ) + isin(nθ).

De Moivre's formula🔗

(cosθ + isinθ)ⁿ is equal to cos(nθ) + isin(nθ). You can state the cisθ in terms of Euler's formula to prove it.

Angle addition

To get the formula for tan, merely do the angle addition for sin, divided by angle addition for cos. Tan is sin/cos.

Multiply both numerator and denominator by node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 and you will get a simplified form.

What links here

Math

Created 2016-Nov-17 (8 years ago)

Combinatorics

Types of selection

	With rearrangement	Without rearrangement
With repeat	n^k	node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1
Without repeat	node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1	node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

See 9. Urval med återläggning

Multiplication principle

The basis for all combinatorics.

Sequence

An array of numbers.

arithmetic

geometric

Series

The "sum" of an array of numbers.

arithmetic

An arithmetic sequence changes by the same amount per term.

Let's describe some arbitrary series. Each term is made up of the "initial amount" plus the indice times the "step size".

To teach clearly, we won't use the variable i for the indice, but instead say n − 1, n − 2 and so on.

Writing it out – if there are n terms where the first term is a, the sum can be expressed:

[a]  +  [a + 1]  +  [a + 2]  + ... +  [a + (n-2)]  +  [a + (n-1)]

Note how this series does not reach a + n, and falls just short, because the very first term is at indice zero. You can of course write this differently, counting from indice one to indice n, rather than from zero to (n-1). You must change the initial amount (or "starting value").

It is common for the "step size", or increment, to be just one. If you have a different increment, you have to multiply the indice. Say the increment is d and we get the general expression (the one above was specifically for d=1 so that d disappeared):

[a]  +  [a + d]  +  [a + 2d]  + ... +  [a + (n-2)d]  +  [a + (n-1)d]

On our quest to sum this thing, we're going to place an indentical line below it, with the order reversed. Then think about what happens if you add the lines together.

 S =            a   +    a + 1d       +    a + 2d       + ... +    a + (n-1)d
 S =   a + (n-1)d   +    a + (n-2)d   +    a + (n-3)d   + ... +    a
2S =  2a + (n-1)d   +   2a + (n-1)d   +   2a + (n-1)d   + ... +   2a + (n-1)d

What just happened? The indice in the first row increases, but the indice of the second row decreases, so the summed indice stays.

You know how to factor. I hand the reins to you. Do this every time you need the summation formula, derive it.

n(a) + n (n-1) d 1/2
n(a) + n(n-1)d
(n + n)(a + nd/2-d/2)

geometric

A geometric sequence increases by the same factor per term. (?)

Binomial theorem

You can expand (x + y)ⁿ as follows.

This is not the same thing as (xⁿ + yⁿ)! That is a sum of e.g. cubes like x³ + 1, where the exponents are "inside the parentheses", for which there is a different formula (Sum/difference of cubes).

Explanation

Binomials are expressions like (x + y)². You memorize quadratics in high school, but how do you expand (x + y)¹⁴?

You can use Pascal's Triangle, but suppose you have (x + y)¹²⁷³³. Find facts about a specific term in that massive polynomial.

Uses the notion of Combination.

Pascal's Triangle

    1       zeroth row -- a constant: (x + y)^0
   1 1      first row --- a binomial
  1 2 1     second row -- a binomial to the second degree
 1 3 3 1    third row
1 4 6 4 1   fourth row

Explanation

You can write Pascal's Triangle from scratch – for each entry, add the two entries above it.

Take an example binomial of fourth degree, (x + y)⁴. When expanded, it produces a polynomial with coefficients from the fourth row.

We have the row, 1 4 6 4 1. Place your x'es and y'es. The fourth-degree x-term goes to the left, the fourth-degree y goes go the right.

Once both are placed out, we end up with the completed polynomial:

Which can be viewed with redundancy, for clarity:

Combination🔗

Finding combinations of k items from n, node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 "n choose k", is node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

For example, node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

A startling symmetry is that node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 which is the same as the previous identity. This symmetry, that node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 works for any numbers node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 and node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 you want.

Permutation

A combination lock should be called a permutation lock. The order matters, 2-7-1 isn't 2-1-7.

There are more permutations than combinations.

Factorials

Note that 10! / 8! is the same as 10 * 9. If it's not obvious why, think.

What links here

Math

Created 2016-Nov-17 (8 years ago)

Algebra

Least common multiple🔗

The LCM is called a MGN (minsta gemensamma nämnare) in Swedish because it's for use in a denominator. You have several fractions with different denominators and want to expand them to use the same. In principle it doesn't have to be about denominators, so call it LCM.

To find the LCM of two numbers, construct a number that contains all the prime factors of both.

Bigger than the Greatest common factor (GCF), in the sense that you use every prime you have access to (though not in their full multiplicity). Note the naming of LCM versus GCF. "Multiple" refers to a product of possibly many numbers, "factor" refers to something you can factor out of a single number.

Greatest common factor🔗

To find the GCF of two numbers, find what primes they share.

Use Divisibility tests or the Euclidean algorithm.

Smaller than the Least common multiple (LCM) in a sense, that you don't use all the primes you have access to. In reality it's apples and oranges. The LCM is to expand two fractions to have the same denominator, so that you can then add them. The GCF is to deexpand a single ratio (fraction or equation) to its most simplified form.

In short, you shrink a denominator to the GCF, or you expand a denominator to the LCM.

Triangle inequality🔗

See Anki cards. Basically |x| + |y| ≥ |xy|, because x and y, being vectors or complex numbers, take a long way around while xy takes the shortest way.

Triangle inequality for integrals (PB 294)

Modular arithmetic

The rules are braindead. This wand just works everywhere.

(A)^B mod C ≡ (A mod C)^B mod C

(A + B) mod C ≡ A mod C + B mod C

(A * B) mod C ≡ A mod C * B mod C

Division algorithm🔗

You've used long division. You're familiar with getting a quotient and a remainder? This is just a realization of that, which we can then use for other things. Warning: the polynomial q(x) has an unfortunate name, it is not the quotient. Without further ado, realize that:

for two polynomials p(x) and q(x), there are unique polynomials r and k such that

which can be written

which is the same as

See the Polynomial Remainder Theorem on how to use this last form.

Polynomial Remainder Theorem🔗

Divide the polynomial p(x) with (x - a). The remainder will be identical to the value of plugging a into the original: p(a). This lets us know all sorts of things. If the remainder is zero, then (x - a) did divide the polynomial cleanly, and a is a root.

Restate the original Division algorithm, node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

in these terms. New division algorithm: node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

Which leaves only k(x) as the unknown.

Example

Suppose a polynomial p(x). Someone divides it with (x-1)(x-2)(x-6). What's the remainder, if p(1)=2, p(2)=1, p(6)=-4?

Wow, I have no idea….

The first step is to see how you can fit this into the division algorithm. The divisor is q(x), so:

Now, the remainder r(x) must have a degree less than 3, because by definition, writing p(x)/q(x) as k(x) + r(x)/q(x), means r(x) has a lower degree than q(x). It doesn't matter what degree p(x) and k(x) are, the "leftover" gets shunted into k. I think.

So r(x) can be written as a second-degree expression, so our equation is: node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1

Moving on. Let's gather some facts. The exercise supplies us the remainder for three different values of x. We will insert those values into the expression for r(x).

You recognize what to do with this: it's a system of equations.

1	1	1	2
4	2	1	1
36	6	1	-4

1	1	1	2
36	6	1	-4
0	-2	-3	-7

1	1	1	2
0	-30	-35	-76
0	-2	-3	-7

1	1	1	2
0	-2	-3	-7
0	0	10	29

1	1	1	2
0	-2	-3	-7
0	0	1	29/10

Simplified third row

1	1	1	2
0	-2	0	17/10
0	0	1	29/10

1	0	1	57/20
0	-2	0	17/10
0	0	1	29/10

1	0	0	-1/20
0	-2	0	17/10
0	0	1	29/10

1	0	0	-1/20
0	1	0	-17/20
0	0	1	29/10

Simplified second row

So we found values for a, b, and c. We're done: the remainder r(x) is node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 .

ANSWER NOT VERIFIED

Example 2

What's the remainder when you divide x⁸⁶ - 3x⁷ + 6 by (x² - 1)?

Try the Polynomial Remainder Theorem. Test the cases where x is 1 or negative 1, because this makes the divisor zero.

Plugging the data into the division algorithm, We have x⁸⁶ - 3x⁷ + 6 = (x² - 1)k(x) + r(x).

p(1) = 1 - 3 + 6 = 4
p(-1) = 1 - -3 + 6 = 10

At p(1) and p(-1), the RHS just becomes r(x). So we know that at x=1, not only that the remainder r(x) is 4, but that the whole polynomial evaluates to 4. Remember that we are dividing by a specific divisor in this exercise, so we better not pick some willy-nilly x when we analyze the remainder.

The remainder r(x) will be at most degree one, see other example. So we'll write it as ax + b.

For x=1, we have a(1) + b = 4 For x=-1, we have a(-1) + b = 10

We can find a and b, and thereafter, we'll be able to write the remainder.

Key trick: Pick an x that makes the divisor zero.

Euclidean algorithm🔗

Let's find GCF(209,77). Realize that 209 can be restated in units of 77, plus something to make up the remainder.

209 = (2 * 77) + 55

This rewrite is the basic step. Repeat: restate 77 in terms of the new thing.

77 = (1 * 55) + 22
55 = (2 * 22) + 11
22 = (2 * 11) + 0

The algorithm is: Re-apply division until you get zero remainder. Then the divisor you just tried is the GCF.

Diophantine equation

Diophantine equations come in many forms. Fermat's Last Theorem is a famous example. What we mean with this word, 'Diophantine', is that we're interested in integer solutions only. This is why it has applications for e.g. proving the Fundamental Theorem of Arithmetic, which is about prime numbers, which of course are integers. We're going to solve a simple kind, the linear Diophantine equation.

Note: it may seem pointless to run the Euclidean algorithm when you can detect that two numbers share no common factor. But you do it for the steps in the process, not the end result (that GCF=1).

Take an equation of the form ax + by = c, where a, b, and c are known integers, and we're interested in integer solutions. For example 43x + 19y = 4.

Normally equations of this form describe a continuous line, crossing an infinite number of points in the real plane, and some such equations even manage to skip any integer pair (x, y).

Crux: For there to be an integer solution to ax + by = c, then all GCF(a,b) must be a factor of c. If a and b have a common factor not shared by c, then stop. There are no integer solutions. This means you can't e.g. have even numbers for both a and b and an odd number for c, but one of a or b may be even, and c may be too.

Aside: a statement (ka + lb) is a linear combination of (a + b). Therefore, ax + by is a linear combination, too. You can put in any x and y. Anything that divides (a + b) will retain its ability to divide (ax + by).

EXAMPLE: My seminarieuppgift

Bestäm den lösning till den diofantiska ekvationen node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , som har minst positiva x-värde.

Jag verifierar att node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 är 1.

Vi skriver en hjälpekvation och får partikulärlösning node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , vilket multiplicerat med node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 kan stoppas in i ursprungsekvationen.

Då återstår att bestämma den allmänna lösningen till en lite annorlunda ekvation (en generell): node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 där node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , vilket säger att node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 och node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 inte delar varandra. Ekvationen kan skrivas node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 . Om a inte delar node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 måste det dela node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , eller hur? Förklaring: pga likhetstecknet (vi struntar i minus) måste precis samma faktorer ingå i node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 som i node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , eftersom de är samma tal. Så givet att node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 inte har faktorerna i node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , så måste node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 ha faktorer i node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 (inte nödvändigtvis alla, men node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 innehåller alla faktorer till node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , vilket kan uttryckas med bokstaven node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 för ''resten av faktorerna/de som inte ingår i node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 '' så att node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 ).

Vi har att node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 kan uttryckas node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , så vi byter node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 mot node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 i ekvationen och med algebra får att node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 .

Vi tittar på den specifika ekvationen (ytterligare en ny ekvation för oss) node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 . Vi har nu lösningar till den eftersom node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 och node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 är definierade. Lösningarna var som sagt node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 och node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , så till den specifika ekvationen är det node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 och node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 .

Dessa är ingen speciell lösning utan en allmän, där node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 är något heltal. Nu applicerar vi detta till ursprungsekvationen, vilken hade partikulärlösningarna node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 och node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 , vilket kan ses som vårt ''offset'' från origo. Ta dessa, plus eller minus node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 något antal (node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 ) gånger, så rör vi oss längs linjen som utgör ekvationen. Varje steg är en till heltalslösning: de ligger node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 ifrån varandra.

Vilken är den heltalslösning med minst positiva x-värde? Vi skriver en formel för x: node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 . Vi sätter node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 till node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 vilket ger ett x-värde på node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 . För samma node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 blir y-värdet node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 .

\begin{svar*} Den lösning med minst positiva x-värde är $(33, 27)$. \end{svar*}

Fundamental Theorem of Arithmetic🔗

All numbers are either primes, or a combination of primes.

Fundamental Theorem of Algebra

Minimally stated: Every positive-degree polynomial has a complex zero.

Alternatively: An nth-degree polynomial has n zeros, if you include complex zeros and count multiplicity (duplicate zeros).

That follows from the minimal statement, because once you know you have a zero, you can factor it out and are left with a new (lower-degree) polynomial, which of course also has a zero.

Divisibility tests🔗

Divisible by 2 if it's even.
Divisible by 3 if the sum of the digits are.
Divisible by 5 if it ends in zero or five.
- An odd number containing the factor five will end in five.
- An even number containing the factor five will end in zero.

Divisibility by three

Intuition for divisibility by three can be helped by that 1000 can be rewritten (1 + 999), 100 as (1 + 99), etc. The nine-rich terms leave no remainder after division and can hence be deleted.

400 can be written 4(1+99), meaning 4 + 4*99, and the latter term can be deleted as any multiple of 99 will keep on being divisible by 3. What remains is the value of the hundredth-place digit, in this case… four.

Convert the number 462 yourself: 4 + (4*99) + 6 + (6*9) + 2. After deleting the parenthesed terms, you are left with… the digits in the original number!

99 is also divisible by 9, so the same test can be used for divisibility by 9.

Set theory

domain and range

take a function f: A → B
- the domain is A, can refer to it by D_f
- the målmängd is B
  - the range (värdemängd) V_f is a subset of B

find the range of a function

see if it has a min or max

The range of a function that only takes on the values 1 and 2 is written {1,2}, not [1,2], which would be the full interval.

Interval

A closed interval [a,b] includes the ends. node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 .

[a,b]  --------[a----------b]--------

An open interval ]a,b[ excludes the ends. Unfortunate notation. node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 .

]a,b[  ---------a[--------]b---------

Sum/difference of cubes🔗

Memory key: the sum will have a binomial factor that contains a plus sign, just like the sum itself.

a³ + b³ = (a + b)( … )

a³ - b³ = (a - b)( … )

Complete formula

a³ + b³ = (a + b)(a² − ab + b²)

a³ - b³ = (a − b)(a² + ab + b²)

Inequality

Well-ordering principle

Complex binomial

Solve x³ − 125i = 0 for node:internal/modules/cjs/loader:1228 throw err; ^ Error: Cannot find module 'katex' Require stack: - /home/kept/private-dotfiles/.config/emacs/texToMathML.js at Module._resolveFilename (node:internal/modules/cjs/loader:1225:15) at Module._load (node:internal/modules/cjs/loader:1051:27) at Module.require (node:internal/modules/cjs/loader:1311:19) at require (node:internal/modules/helpers:179:18) at Object. (/home/kept/private-dotfiles/.config/emacs/texToMathML.js:1:15) at Module._compile (node:internal/modules/cjs/loader:1469:14) at Module._extensions..js (node:internal/modules/cjs/loader:1548:10) at Module.load (node:internal/modules/cjs/loader:1288:32) at Module._load (node:internal/modules/cjs/loader:1104:12) at Function.executeUserEntryPoint [as runMain] (node:internal/modules/run_main:173:12) { code: 'MODULE_NOT_FOUND', requireStack: [ '/home/kept/private-dotfiles/.config/emacs/texToMathML.js' ] } Node.js v20.18.1 .

Complex quadratic

Transcendental numbers

First, the notion of algebraic numbers: a number that can be a solution to some equation with rational coefficients. This would be most numbers that you actually use.

One example is √2 because it is the solution to x² - 2 = 0. It is algebraic.

Though there are many more transcendental numbers than algebraic, it is hard to prove that a number isn't algebraic. Examples of proven transcendentals include π and e.

What links here

Math

Created 2016-Nov-17 (8 years ago)

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