Euler's identity : simple and beautiful

This formula is one of the most famous ones out there, along with E = mc^2 and the roots of a second degree polynomial.

Extremely simple in itself it, manages to unite 3 very important and apparently unrelated mathematical constants.

The exponential function, the symbol e, is defined as the only function that equals its derivative. It's used in different models to simulate an exponential growth and appears in differential equations as a potential solution. It can be defined for any complex number as the following infinite sum :

More on this equation here

This is used to prove Euler's identity by plugging in iπ and computing the sum which magically gives -1.

It also features the world renowned π, somewhat of a mythical symbol that dates back to ancient Greece where mathematicians discovered its link with a circle, the circumference of a circle is π times its diameter. Roughly, π = 3.141592...

Finally, comes the symbol i which is the imaginary unit, defined as i^2 =-1 . It was introduced in the world of mathematics to find a solution to second degree equations that had no real number solution like the equation x^2 +1 = 0 which doesn't have any solution on the real number line but has both i and -i as complex solutions.

These 3 constants come together as one in Euler's identity to create what we now commonly refer to as the exponential notation of complex numbers, this result is used all the time alongside complex numbers to simplify the way they're written and to help in various proofs.

Before discovering this equation, these constants were totally unrelated, this is the beauty behind this formula, the link to an entirely new world, the world of complex numbers.

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