The Beauty of Math

e^(iπ) = -1

That’s a weird equation. Two seemingly unrelated constants and an imaginary number can be combined to equal a real number? And something as pretty as -1? This formula is known as Eueler’s Identity, or one of the most beautiful equations in math. You might be wondering how on earth these separate mathematical variables make a true statement.

Let’s begin to deconstruct this equation. [e^(iπ)] is equivalent to cos(x) + i sin(x) via de Moivre's formula. For any value of x, this equation is equal to -1. This is one step closer, but this still seems to relate seemingly seperate relationships in math We can further break this down by evaluating what e to any power is. This is determined to be an infinite sum show below.

We can now fill in (ix) for x. When we do this, the equation simplifies to

Now, this is kind of ugly. Luckily, another calculus theory provides us with enough information to simplify this. Each trigonometric relationship has its own Maclaurin series or an expansion of the Taylor series for some function. This sum contained the first parentheses happens to be the cos x summation and the second one happens to be the sin x function. This means that this summation again simplifies to cos (x) + i * sin (x). Even though there are a lot of steps to this equation that require calculus, this equation combines imaginary numbers, trigonometric relationships, summations, and constants making one true statement.