For as long as I can remember, I had issues remembering new information. But at about 8, I discovered that if the information could be derived by some logical steps, I could remember those very well.
In high school, mathematics was done in what I now recognize as a very non-mathematical way. We were told to dogmatically memorize formulas and laws, so I struggled with many tests and got bad grades. That was until I realized what math truly was. It wasn’t about memorizing stuff, it was about understanding it.
Suddenly everything became simple. Apart from the few definitions, I derived pretty much everything and optimized my brain’s storage space. And later, at the university, this way of studying has turned out to be the right one.
This post is about clearing up some easy proofs that most teachers skip. They’re not difficult but can open a door to studying mathematics the way it was meant to be studied.
Arithmetic sums
Let’s consider the following sum
s(n) = 1 + 2 + 3 + 4 + 5 + 6 + … + (n-1) + n.
Its simplification is based on one easy step, discovered by an 8-year old (ok, he was a genius). The step is
We have to notice that adding the same colors always results in (n+1). Finally, dividing by 2 gives us
At this point, deriving the generalized arithmetic sum is trivial:
Geometric sums
For a given number q, the geometric sum is of the form
I remember my teacher telling me that deriving the formula was complicated. It’s weird thinking about how easily I accepted this ‘fact’. When I eventually did find out the derivation, it was incredibly easy and in a way beautiful:
The trick is to notice that the same colors cancel out. To finish, we divide by (1 - q) on both sides:
I always forget the above formula, while the idea of multiplying the geometric sum into a nicer form is engraved in my mind forever. It’s so natural!
Chances are, you’ve also encountered the below formula for the infinite geometric sum
As this is the limit of the previous sum (n goes to infinity), you can see exactly how it’s derived and therefore what condition q must satisfy (|q| < 1) for it to hold.
Chain rule
Most people will recognize the chain rule:
Unfortunately, few have seen how it’s derived. The derivation reveals that math is not just about reducing expressions. Expansions can often be extremely useful. In the below proof, we add two terms that cancel each other, in order to obtain the rule:
We’ve used that the limit of a product is a product of two limits (if they both exist). This proof works in most cases but overall has a (removable) flaw, which is discussed in the comment section.
Product rule
As with the chain rule, this one should also look familiar:
Its proof is similar to that of the chain rule, except now we expand the expression by adding two terms that add up to zero:
We used the fact that + and × can be taken outside the limits.
Power rule
We all know the derivative of the following function.
Let’s learn how to prove it. By definition of the derivative, we have
It will be helpful to generalize the previous result from the geometric sums section:
Again, the same colors cancel out. Now switch x for a and x0 for b to get
You may have noticed that our derivation also holds for all real exponents k and complex numbers x. So we’ve proven quite a lot:
Conclusion
Memorizing formulas is tedious and many wrongly associate it with math. Math is about understanding things. There are formulas in mathematics that no living soul could possibly remember, while their derivation is natural and can be done with some effort. All you have to do is think for yourself and not let people tell you to just accept mathematical statements. Ask them why.
