Break out of your cage!
Forget about the lines that have been placed around the equation in your mind! They aren’t real! You may have been taught a certain formula or equation that you hold sacred in your belief in the immutable structure of math. Please, do away with these structures that hold you tight. The only things that are not fluid in math are the basic rules; the properties inherently found. These are things like the commutative property, where two plus five equals five plus two and so on. The majority of mathematical discovery has been the result of people willing to play around with a known equation.
For example, the law of cosines is able to be turned into the dot product of vectors. These two formulas are separated in instruction by different classes; trigonometry and calculus 3, respectively. However they are really separated only by a little algebraic manipulation.
“The definition of the dot product incorporates the law of cosines, so that the length of the vector from X to Y is given by
|X-Y|^2 = (X-Y)·(X-Y)
= X·X-2X·Y+Y·Y
= |X|^2+|Y|^2-2|X||Y|cos(θ),
where θ is the angle between X and Y.”
~https://mathworld.wolfram.com/LawofCosines.html
We are taught to stay on the path that has been built by centuries of achievement prior to our instruction, but all of that achievement was accomplished exactly by people who wandered off of the beaten path. If you want to learn all that there has already been discovered, then by all means stay in your predetermined path. It is much safer there. If you want to truly discover the full power of math, though, then let your mind become fluid, only allowing the basic laws of mathematical nature to be your bounds. This is the difference between a painter and an artist. Even if you do not become the next Rembrandt and your work looks more like Picasso’s, you will have done one seemingly small but immensely important thing: You allowed yourself to question, to explore, to have a brain that sees things in a scientific light.
If you never discover a formula for anything new under the sun, you might, through the mere fact of having developed an inquisitive mind, have allowed yourself to see EVERYTHING in that same fluid manner. Art, business, sports, tactics, law, medicine….the sky is the limit.
See, math isn’t just for counting; Math is for THINKING and in a manner that literally restructures one’s mind to see all of the universe in a new light. So, cast off those chains of the rigid equations! Play with your numbers! Move things to the “wrong” side of that equal sign! Remake your mind.