# Cool Things

Math is cool. Here are some cool things in math that I don’t think are really extensive enough for their own post but I still want to share.

1. $\inline&space;0^0=1$

The following is a great reason why.

Let’s say we have a set $\inline&space;M$ of distinct elements. Using some function, we can form another set $\inline&space;N$ which can contain repetitions of elements, no occurrence of some elements, and any other possible combination of the ones in $\inline&space;M$.

$M={a,b,c,d}\rightarrow&space;N=\{\_,\_,\_,\_,\_,\_\}$

The sets do not necessarily have the same size so there are $\inline&space;|M|^{|N|}$ functions that do this because $\inline&space;N$ has $\inline&space;|N|$ elements each of which can have $\inline&space;|M|$ possible values. In fact, we can use this idea to define exponentiation. $\inline&space;m^n$ is the number of functions one can define that creates a set of $\inline&space;n$ elements using $\inline&space;m$ distinct elements. If $\inline&space;m=0$, there are no functions that create a set $\inline&space;N$ so $\inline&space;0^n=0$. If $\inline&space;n=0$, the only function is the empty function that takes a set to the empty set so $\inline&space;m^0=1$. If $\inline&space;n=m=0$, then there is still one function which is the identity function taking an empty set to an empty set so $\inline&space;0^0=1$.

2. Spinny Infinity Proof

This one is a kind of a joke but a note-worthy one. Why does $\inline&space;\frac{1}{\infty}=0$?

Well, we know the following intuitively.

$\frac{1}{0}=\infty$

Because doing the same thing to both sides of the equals sign holds the statement true, we can rotate both sides counter-clockwise 90 degrees (physically rotate each character).

$-10=8$

We then subtract.

$-18=0$

Rotating back, we come to our desired equality.

$\frac{1}{\infty}=0$

3. Cross Products

Sometimes it becomes tedious to compute a big determinant and where that determinant comes from seems really arbitrary. It seems there is not intuitive and nice way to compute cross products. But wait, there is! Let’s list out the three basic properties of the cross product.

a. $\inline&space;v\times&space;v=0$
b. $\inline&space;v\times&space;w=-w\times&space;v$
c. $\inline&space;v\times(w+y)=v\times&space;w+v\times&space;y$

It turns out, this is all you need! Let me show you an example of a cross product. Let’s say we want to cross $\inline&space;(1,0,-2)$ and $\inline&space;(1,3,1)$. Well, we can write it in terms of its basis directions first (the $\inline&space;\times$ is omitted for now).

$(\hat{i}-2\hat{k})(\hat{i}+2\hat{j}+\hat{k})$

Using linearity, one can distribute.

$\hat{i}\hat{i}-2\hat{k}\hat{i}+3\hat{i}\hat{j}-6\hat{k}\hat{j}+\hat{i}\hat{k}-2\hat{k}\hat{k}$

Using reflexivity, we can cancel some terms.

$-2\hat{k}\hat{i}+3\hat{i}\hat{j}-6\hat{k}\hat{j}+\hat{i}\hat{k}$

Antisymmetry allows for more simplification.

$-3\hat{k}\hat{i}+3\hat{i}\hat{j}-6\hat{k}\hat{j}$

Knowing that $\inline&space;\hat{i}\times\hat{j}=\hat{k},\hat{k}\times\hat{k}=\hat{i},\hat{k}\times\hat{i}=\hat{j}$, we finish our computation.

$-3\hat{j}+3\hat{k}-6\hat{i}$

Although this may seem like a lot of steps, it is really fast and feels very natural like actual multiplication instead of obscure determinants. By simply having knowledge of the properties of the operator and some basic identities, we can easily calculate the result.

In fact, determinants can be calculated in the same way but are less convenient. As a mental exercise, see if you can figure out how (Note: a determinant is the area of the parallelogram/parallelepiped formed by the column vectors of a matrix).

4. Tic-Tac-Toe Integral Trick

This integration trick works when you are integrating a polynomial multiplied by some easily integrated function (usually trigonometric or exponential functions). Note: this does not help you integrate functions that are difficult to integrate but merely provides a systematic method for speeding up multiple integration by parts steps.

This is best shown through example. Consider the integral $\inline&space;\int&space;x^3\sin(x)$.

We create the following table where we take derivatives along the first column, integrals through the second, and alternate signs on the third. We color in the jagged patterns shown.

Multiplying the ones of similar color and adding them, we get our final integral.

$-x^3\cos(x)+3x^2\sin(x)+6x\cos(x)-6\sin(x)+C$

If you want to learn more, then just have awesome teachers like I do. I am sure you can find someone 🙂