From Taylor Series to Euler Equation

When I was in middle school, I was always puzzled why those mathematicians would invent complex, irrational number, negative number on top of positive number. Why bother and make the route to grasp math more difficult?

I took some research on Euler equation, the famously most elegant one

I am amazed by and greatly interested in the interconnection, necessity and importance of these inventions.

We know Taylor’s expansion is a powerful tool to approximate a function, it is said the Euler, a genius, just by staring at these Taylor expansion

e raised to i*theta has the geometrical meaning as is the circumference of a circle, cited from wiki of below illustration

when theta = pie,

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