Mathematics

This is such an undescriptive title because there is so much covered under this branch

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.

ax2+bx+c⏞Generic Quadratic=0x2+bax+ca=0(x+b2a)2−(b2a)2+ca=0(x+b2a)2=(b2a)2−ca=b2−4ac4a2x+b2a=±b2−4ac2a⟶x=−b±b2−4ac2a⏞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.

What is the result of 1−1+1−1+1…?\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 X\text{X} and notice that it contains itself.

X=1−1+1−1+1−1…⏞−XX=1−X2X=1→X=1/2\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. 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.

DivisorRule

1

...all numbers are divisible by 1

2

Last digit is even

3

Sum of the digits is divisible by 3

4

Last two digits are divisible by 4

5

Last digit ends in 0 or 5

6

Satisfies both 2 and 3

8

Last three digits divisible by 8

9

Sum of the digits are divisible by 9

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