Question

Best way to learn the theory and why formulas are applied rather than just memorize them?

Answer 1

Math has an uncanny property. The "fundamental" questions can be the most difficult to understand. It is a good thing to be curious and to explore and ask.

Certain things are simply defined that way, i = sqrt(-1) for example. Why or what complex numbers are, for example, can be thought of as a solution space for all algebraic polynomials. This might not be a very satisfactory answer since it is mostly an expression "why it is useful". Or we could just say that complex numbers are a type of number that can be expressed as a+bi without attempting to answer "what are they". To go down a rathole, most students won't even be exposed to a rigorous approach to "what is a real number" until they're probably into upper college level maths.  

Formulas are a way to summarize some mathematical property, relationship or behavior. Rather than focusing on the memorizing them it is probably more useful to make sure you know how to use them. This ultimately involves memorizing the formula, but math is about doing stuff rather than regurgitating facts, for the most part.

Not sure this helps...

Answer 2

i^2 =-1
That is really about all there is. We wanted a number with that property and we invented one. From there we have a few interesting things. I was confused for a while how we know i from -i and we don’t really. Can we use i with our usual rules? Yes pretty much, except rules about exponents. It is also surprising that complex numbers are algebraically closed. Can we use complex numbers with transcendental functions? Sure no problem.

Answer 3

Don't overthink things. Precal is not deep.

The fact that i^(2)=-1 IS what imaginary numbers are. You write them down, and FOIL things like normal. Understanding IS memorizing their properties and connections. You get intuition by doing problems well. That is a kind of knowledge that you have to read. It's not something divine or philosophical.

Similarly, usually the "why" is "that's the definition" or "because that's what it's for." The reason you use the quadratic formula to solve quadratic equations is because the quadratic formula solves quadratic equations. It's really more about reading instructions and knowing what you want to accomplish, not some deep why.

If you want to understand more of that, make sure you know your theorems. For example, why do we factor polynomials in equations like polynomial=0? It's because there's a theorem that says if you have AB=0, then you know A=0 or B=0. How do you know to apply it? Because the theorem starts with "AB=0", and you have a thing=0.

If you really want something to build an intuition around, I say focus that energy on visual stuff. For example, it's good to practice a lot of problems involving graph shifting/stretching. Because visualization is actually skill (although incidentally it involves just memorizing some graphs), unlike calculations which is just following instructions.