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E to the pi i equals negative 1 is one of the most famous equations in math, but it's also one of the most confusing. Those watching this video likely fall into one of three categories 1) you know what each term means, but the statement as a whole seems nonsensical, 2) you were lucky enough to see what this means and some long formulae explaining why it works in a calculus class, but it still feels like black magic, or 3) it's not entirely clear with the terms themselves are. Those in this last category might be in the best position to understand the explanation I'm about to give since it doesn't require any calculus or advanced math, but will instead require an open-minded to reframing how we think about numbers. Once we do this, it will become clear what the question means, why it's true and most importantly why it makes intuitive sense. First let's get one thing straight, what we write as e to the x is not repeated multiplication that would only make sense when x is a number that we can count 1, 2, 3 and so on, and even then you'd have to define the number e first. To understand what this function actually does, we first need to learn how to think about numbers as actions. We are first taught to think about numbers as counting things, and addition and multiplication are thought of with respect counting. However, is more thinking becomes tricky when we talk about fractional amounts, very tricky when we talk about irrational amounts, and downright nonsensical when we introduce things like the square root of -1. Instead, we should think of each number as simultaneously being three things a point on an infinitely extending line, an action which slides that line along itself, in which case we call it an “adder”, and an action which stretches the line in which case we call it a “multiplier”. When you think about numbers as adders, you could imagine adding it with all numbers as points on the line all at once. But instead, forget that you already know anything about addition so that we can reframe how you think about it. Thing of adders purely as sliding the line with the following rule: You slide until the point corresponding to zero ends up where the point corresponding with the adder itself started When you successively applied two adders, the effect will be the same as just applying some other adder. This is how we define their sum. Likewise, forget the already know anything about multiplication, and think of a multiplier purely as a way to stretch the line. Now the rule is to fix zero in place, and bring the point corresponding with one, to where the point corresponding with the multiplier itself started off, keeping everything evenly spaced as you do so. Just as with adders we can now redefine multiplication as the successive application of two different actions. The life's ambition of e to the x is to transform adders into multipliers, and to do so is naturally as possible. For instance, if you take two adders, successfully apply them, then...