A Geometric Proof of Euler's Identity
- Tyler Haslam
- Complex Numbers
A geometric proof of the identity, e^(pi*i)=-1
As n increases the complex number (1+pi*i/n)^n approaches -1+0i. Incidentally, if you change the pi to a 2 and look at (1+Ai/n)^n, you see it approaching cos(A)+i sin(A). This is because as n gets larger, the argument of 1+Ai/n gets smaller, making it an approximate sector line of the unit circle with an angle approaching A/n radians. When we raise it to the nth power (by adding the angles n times) then we get an angle of A radians.