> For the complete documentation index, see [llms.txt](https://nkintc.gitbook.io/brainless/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://nkintc.gitbook.io/brainless/steam/mathematics.md).

# Mathematics

Mathematics is such a broad term for what I want to cover. There are many that are broken out that should be covered under this but I digress. These are concepts based in mathematics which may or may not have practical applications in the real world. But in any case, I think that my favorite is stuff that everyone can at least be somewhat familiar with. 

## Screwing Around

A fun example is that two concepts that are taught in high school are intimately related to each other: the quadratic formula can be derived by completing the square of a generic quadratic equation.

$$
\begin{align\*}
\overbrace{ax^2 + bx + c }^\texttt{Generic Quadratic}&= 0
\\
x^2 + \frac{b}{a}x + \frac{c}{a} &= 0 \\
\left( x + \frac{b}{2a} \right )^2 - \left( \frac{b}{2a}  \right )^2 + \frac{c}{a} &= 0 \\
\left( x + \frac{b}{2a} \right )^2 &=  \left( \frac{b}{2a}  \right )^2 - \frac{c}{a}\\
& =  \frac{b^2-4ac}{4a^2}\\
x + \frac{b}{2a} & = \frac{\pm\sqrt{b^2-4ac}}{2a}
\quad\longrightarrow\quad
\overbrace{
\boxed{x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}}
}^\texttt{Quadratic Equation}
\end{align\*}
$$

Even though these symbols are interchangeable, by manipulating them in the same context, the 

### Alternating Summation of Ones

Here is a pretty common high school problem that shows the weirdness that happen with infinite series. 

$$
\text{What is the result of } 1-1+1-1+1\ldots\text{?}
$$

We want to know what the value is so set equal to some unknown, like $$\text{X}$$ and notice that it contains itself. 

$$
\begin{align\*}
& \text{X} = 1 \overbrace{ - 1 + 1 - 1 + 1-1\ldots}^{-\text{X}}\\
& \text{X} = 1 -\text{X}\\
& 2\text{X} = 1 \rightarrow \boxed{\text{X} = 1/2}
\end{align\*}
$$

### Divisibility Tricks

Ran into a situation where you needed to know if a number had a specfic factor? Here are some fast ways check the divisibility of large numbers, courtesy of reddit user [u/BlueEmu](<https://www.reddit.com/r/lifehacks/comments/msj1sq/if_you_want_to_know_if_a_number_can_be_divided_by/guum6x1&#xA;>). Remember that you can also check divisibility by larger numbers using factors. A number divisible by 15 would be divisible by both 3 and 5.&#x20;

<table><thead><tr><th width="150">Divisor</th><th>Rule</th></tr></thead><tbody><tr><td>1</td><td>...all numbers are divisible by 1</td></tr><tr><td>2</td><td>Last digit is even</td></tr><tr><td>3</td><td>Sum of the digits is divisible by 3</td></tr><tr><td>4</td><td>Last two digits are divisible by 4</td></tr><tr><td>5</td><td>Last digit ends in 0 or 5 </td></tr><tr><td>6</td><td>Satisfies both 2 and 3</td></tr><tr><td>8</td><td>Last three digits divisible by 8</td></tr><tr><td>9</td><td>Sum of the digits are divisible by 9</td></tr></tbody></table>

## Links

* [Complex Analysis Intro](https://complex-analysis.com/)