Visualizing Math: Tips & Tricks for Newbies

[Late Bloomer]

Greetings Newbie here,

I'd like to ask the engineers and physicists here about their experience with visualizing math.

What i mean by that is to compartmentalize mathematical equations and then translate it into the images of objects. And the equations would then become templates to explain the activity around those objects. It sounds a bit abstract i know but first let me give my personal account. And also know that for me at least this is more inquiry then delegation.

I am a very active spatial thinker and my grades in math have been decent to excellent depending on my effort. I am talking about high school math though. But i at least know that my arithmetic is above average.

However when it comes down to physics problems i have great difficulty trying to connect the two together.

It's as if i become dyslexic and have no idea what the formulas mean anymore. I second guess myself frantically and create scenarios in my head to help to interpret the question. But it rarely gets me any closer to solving the problem. To make this short i'll say this, i will holistically juggle everything at once and fail miserably.

Now in two months i'll go back to school to start fresh in a engineering science program. I have a disdain for academia but being at the end of my adolescence i know i can't mooch off my parents forever.

So i decided to best solve my dilemma i'll have to rethink how math works. As a start i decided to revisit math i had already learned and figure out how to visually process them in the way that was described in the first paragraph. One instance was this:

If the circumference of a circle is 4cm find the Area. I processed the mathematics mentally:
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.

This should be basic intuition i know. But unfortunately i have had second guessed even basic intuition when confronted with a physics problem.

Also if anyone wants to know i discerned the circumference and area of a circle formulas using images. For C i imagined a ruler the length of the circle's diameter spinning inside it. For area i imaged the the radius line become an actual square with a portion extended outside the circle and also span.

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?

 

[Simon Bridge]

Per your example:
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/wiki/Mysterio_System_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.

 

[James Wine]

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

I like to think about the cancellation in a few more steps, however I never write it down.

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

 

[Simon Bridge]

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.

 

[fluidmech]

I use stories to memorizing kanji characters myself - if it works for you, then do it. After awhile and practice, it'll be second nature.

As you progress, a lot of equations will become confusing. When can I use that one vs this one, etc. It would be useful to make flash cards with the restrictions on each equations - especially for thermodynamics later one.

As an aside - what are you reasons for engineering science? I would double check that your major is ABET accredited :-) Good luck!