While at University many years ago, I noticed that many subjects would provide you with “givens”, but not explain how to derive and understand these yourself. Some examples I recall: the Pythagorean Theorem, Shannon’s formula for entropy, and the origin of Euler’s Number, e.
I personally set out to create my own answers for these and others, not with the eye to publishing them but just to understand things for myself and seeing if I could originate such ideas without any other help than knowing that such answer did exist. Now, that is a ton of help, for it is far more difficult to originate such ideas from complete scratch, than to confidently know they exist and just try to reverse-engineer how the originator might have come up with them. I will note in passing that this is a huge gaping hole in our education system: they pump us full of ideas, but they don’t give us deep understanding. They teach us to be able to perform operations as if we were a computer, not become creative human beings who know how ideas are originated.
I was able to invent my own proof of the Pythagorean theorem from scratch, without having ever observed a proof or knowing that the way to proceed was geometric rather than algebraic (being trained to use formulas, my first instinct was to try using formulas). How I finally went about it was to notice that the formula was comprised of squares. I reasoned that there must be a geometric relation of squares that would prove it, and so played around with drawing squares and right triangles until I found a shape that could be viewed in two ways: one large square and an equal area composed of a middle square and four equal right triangles. I called a friend and told him about it, and he attempted and completed the same. One of us had created proof #3, the other proof #4, as shown here, but we don’t recall who came up with which.
Euler’s number, e, is readily drawn from the idea of a system whereby its rate of change is proportional to its value. For example, the rate of growth of a colony of bacteria is proportional to the number of bacteria present (assuming no limiting factors). Newton’s Law of Cooling is another such example and fairly intuitive. Mathematically, we posit a function whereby the rate of change at a given point is exactly equal to the value of the function at that point multiplied by a constant, i.e.:
d/dx f(x) = a * f(x). The simplest form of this is when
d/dx f(x) = f(x).
This formula is called a “differential equation”, and the constant e arises directly and naturally from this idea and from fundamental ideas about calculus. The general rule for differential equations is that if you find a solution to this formula, then it is the solution. Ergo, you’re allowed to “guess” an answer, and if the guess is correct then that is the answer.
Someone who had never even heard of “e” could readily guess that the following infinite series is the solution to this equation:
f(x) = 1 + x + x^2/2! + x^3/3! ... (take its derivative and see for yourself). It turns out that if you compute f(1), you get e. In other words, the value of e itself can directly originate from a trivial application of calculus and a simple guess.
This simple demonstration doesn’t give you the full significance of e – exploring that would constitute a lifelong devotion to the subject – but it does very simply demonstrate how the number results from a very basic application of calculus.
Furthermore, knowing that
f(1) = e is one thing, but knowing the correct general solution,
f(x) = e^x, is quite another. I know of three ways to prove that the foregoing infinite series is identical to
d/dx e^x = e^x, then we know also that that constitutes a solution to the original differential equation, and since the infinite series is too, then the infinite series must also be equal to e^x.
d/dx e^x = e^x, we find that the Taylor Series expansion of
e^xexactly matches the infinite series, and therefore that
e^xmay be substituted for our original general solution.
e^xis identical to the foregoing infinite series (which, incidentally and by virtue of the foregoing, also proves that
d/dx e^x = e^x).