Euler’s Fabulous Formula | Maths and Musings - Medium

Photo: Leonhard Euler

Leonhard Euler (1707–83), who was one of the most prolific mathematicians ever, continuing to publish large numbers of papers even after going blind in 1766.

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Here we investigate and understand the brilliant formula which unlocks the power of complex numbers by Maths and Musings in Cantor’s Paradise.

Photo: https://www2.clarku.edu/faculty/djoyce/complex/mult.html

In some sense, it is obvious why it is famous. Like, what! e raised to the square root of -1 multiplied by pi gives -1. How!??!! I remember being astounded that this was the case.

Yet this turns out to almost be a definition of complex numbers, the idea that we can write:

It extends the definition of e to make sense for complex numbers while still making sense with the definition of e for real numbers. This will all make a lot of sense later, but is mind-boggling at first!

First, let’s really understand the right hand side of the equation, which ties in beautifully with elementary geometry...

Epilogue
Who would have thought it! The functions created by the Greeks to describe coordinates on circles (cos and sin) have a mystical link with the function which differentiates to itself, e, once we expand numbers to include the square root of negative 1.

It’s a wonderful world.
Read more... 

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 Source: Medium