# [Learn] Mathematics : Index

Clean mathematical web pages

It was first here to link to other mathematical webpage but as there wasn't much of stuff, I want it in one webpage, because I tend to better make sense of informations when it's presented in the same page, like when I'm programming though. Later if things grow to much I will make a new file, like when programming though.

## How to self study pure math - a step-by-step guide

[YT] How to self study pure math - a step-by-step guide

[Learn] Mathematics Trigonometrie: Sine Cosine [Learn] Mathematics : todo list

Take the units you have side to side, multiply them or divide them as needed to cancel units. If you need to flip units, just do : `1/unit`

## Algebra

### Scale as analogy

I found it a really great way to understand some concepts, like in high school we tend to put a value in the other side of the equation and inverse its sign without thinking. But when you realize that if you do the same thing with a weight, it feels obvious that you need to cancel the same weight on the other side in order to keep it balanced.

### Multiply the terms in order to delete one unknown

It's a good way to get ride of one unknown. don't forget that value without unknown can be nulled, it permit to have two expressions equal, it makes it really easy to delete unknown. When a pair of equations both have the same expression on one side, we can set their opposite sides equal

When you have fractions in an expression with different denominator, most of the time you can know the number to multiply them in order to get rid of the fraction by multiplying the denominator (when two different. denominators). Note that you can multiply by the multiple of the number find previously. It's useful when mutliplying the denominators gives you a big value. So you get the number from both denominators and you take a multiple of it (smaller). This value is called the least common multiple

Rules from brilliant.org :
"When equations involves fractions, we can scale to remove the fractions.
- if an equation only has one fraction, we can scale by its denominator.
- if an equation has multiple fractions, we can scale by the least common multiple of the denominators."

### Replace an unknown by another (substitution)

We are talking about two unknown here.

Id est if you have `X` and `Y` values only in two equations, you can try to get something like `X = Y + 2` then replace all `X` terms by it and ends up with only one unknown.
There is two general paths:

1. if we want `X`, we can isolate `Y` and substitute `Y`
2. If we want `X`, and `Y` is not easily isolated, we can first isolate `X` and then solve for `Y` instead. Then solve for `X`

### Variables change consequence in a formula

"When one variable is a product of other variables, we say it varies jointly with them. That means changing just one of the factors makes it vary directly with the product."
`x.y = z` means if we double `x` or `y` we double `z`
"A product varies jointly with its factors. When we make one of the factors constant, the product varies directly with the other factor."

## Calculus

### Is the most important concept in Calculus is that the more you move the `X'` from `ΔX (X' - X'')` the more you approach `2X` in `ΔA / ΔX`

In the context of a square.

while learning Calculus in Brilliant.org, they told me "As we push the two points infinitely closer together `ΔA / ΔX` gets infinitely closer to 2. This leads to perhaps the most important concept in Calculus. Is it true ?"
The reason is because Δx2 shrinks to zero

@To-Learn: Is it true?

### As you decrease ΔA, Δx2 shrinks to zero

In order to retrieve the formula `ΔA = 2 X ΔX + ΔX²` just draw the square and find the formula.

### Is calculus wrong?

As I am learning it, I found the statement that infinitely small is not to be bother wrong. Even if a very small value is almost nothing why not counting it? I know Newton create this field and of course I don't doubt the truth in it but I am wondering if in atomic quantities it's still true? Is calculus only useful in everyday physics?

### Calculus is like "get ride of this"

@To-Learn: Is it a physicist thing? On wikipedia and related science does not write mathematics in it but phycics

What does the derivative is that it delete terms in order to simplify the expression of the parabole (function). It's funny as it makes me think of the recurrent joke between physicists and mathematiciens, indeed physicist is like "ho it's true enought !" when they do mathematics

### A trick: if you need X count derivative, you need X power in the first

[YT] Handmade Hero 043 - 'Can you derive the 1/2at^2 part of the position equation again?' explains how a non-math person would be always right when writing derivative.

You can be always right on the power you use in the first derivative by understanding that in order to keep a variable in the last derivative you will have to keep by having `X` power of it. Id est you have `f`, `f'`, `f''` you will have to have `X²` in `f` because you will still have it in `f''` as everytime add one, you take one power to it.

## Basic mathematics

### Equations of lines

#### Y intercept

Think of it as a linear function in wich you move up or down each points from the "root" linear equation.

### Distributive property

It's way more useful to understand visually what it means than learning by heart a formula. [Brilliant.org] Distributive property (not free) Only works if you have a squared operation. Indeed it won't work with: `(a² b³)(a⁵b²)` [Wikipedia] Distributive property