# How to visualize math?

by Late Bloomer
Tags: math, visualize
HW Helper
Thanks
P: 9,091
1. 4=2piR A=pi(R^2) 2. R= 4/2pi = 2/pi 3. A=pi(4/pi^2) A= 4/pi

 So at 3. i processed the 4pi/pi^2 like this. I visualized 4 atoms that i labeled pi so 4 pi atoms to represent the numerator. And they are moving towards a small colony of pi atoms to represent the denominator. They collide and a column of the colony is annihilated. Then we get 4/pi which is the numerical value of the missing column in proportion to the colony before the collision.
Sounds like Calvin math :)
http://calvinandhobbes.wikia.com/wik...ystem_Planet_6
$$C=2\pi r \Rightarrow r=\frac{C}{2\pi}\\ A=\pi r^2 = \pi \left ( \frac{C}{2\pi} \right )^2 = \frac{1}{4}\frac{\pi}{\pi\pi}C^2 = \frac{1}{4\pi}C^2$$ ... so the idea that the numerator somehow annihilates one of the pis in the denominator is not that bad. Some people would have the pi in the numerator elope with one of the pis in the denominator leaving an unpaired pi all alone.
After a while, you don't need these narratives to do math - you just do the math.

 Anyways the point here is this. Were you able to visualize math or physics equations far more complex then the one i used as a example? If so was it difficult? and are there any tricks i should know in advance? Are there other ways besides visualization that have helped you perform with physics problems?
Yes.
Yes.
Yes: Learn to draw pictures.
Yes. You should treat math as a language for describing physical situations rather than as an end in itself. Fluency in the language will allow you to do manipulations like the example above without having to resort to elaborate scenarios.

When you understand the physics involved, you can write down the math without resorting to many memorized equations requiring abstract processing.
P: 8
 Quote by Simon Bridge $$C=2\pi r \Rightarrow r=\frac{C}{2\pi}\\ A=\pi r^2 = \pi \left ( \frac{C}{2\pi} \right )^2 = \frac{1}{4}\frac{\pi}{\pi\pi}C^2 = \frac{1}{4\pi}C^2$$
I like to think about the cancellation in a few more steps, however I never write it down.

$$A=\pi r^2 = \pi \left ( \frac{C}{2\pi} \right )^2 = \frac{1}{4}\frac{\pi}{\pi\pi}C^2 = \frac{1}{4}\frac{1}{\pi}\frac{\pi}{\pi}C^2=\frac{1}{4}\frac{1}{\pi}(1)C ^2= \frac{1}{4\pi}C^2$$

HW Helper
For completeness: $$y=a\frac{b}{a^2} = \frac{a}{1}\frac{b}{a^2} = \frac{ab}{aa} = \frac{a}{a}\frac{b}{a} = (1)\frac{b}{a} = \frac{b}{a}$$ .. which is basically a lot of tautology :)
what you get taught is usually:$$y=a\!\!\!/ \frac{b}{a^{2\!\!/}}=\frac{b}{a}$$ ... the middle bit shows your reasoning while the last bit tidys up the notation.