Famous Five

Euler’s Identity Gautam Sethi

The numbers 0, 1, π, i, and e are fundamental to mathematics. It can be shown that these five famous numbers can be expressed in a single equation. Let the variable z be defined as z ≡ cos(x) + i sin(x)

(1)

Differentiating with respect to x, dz = − sin(x) + i cos(x) dx = i2 sin(x) + i cos(x) = i (cos(x) + i sin(x)) = iz Rearranging terms, we have dz = i dx z

(2)

Integrating both sides, Z

1 dz = z

Z i dx

implies log z = ix + c Solving for c, we have c = log z − ix. Letting x = 0 in equation (1) and noting that z(0) = 1, we get c = log 1 − 0 = 0. Thus, the complete solution of the differential equation (2) is log z = ix.

Exponentiating both sides, z = eix Combining equation (1) with equation (3), eix ≡ cos(x) + i sin(x) Letting x = π, we have eiπ = cos(π) + i sin(π) = −1 + 0 which leads us to eiπ + 1 = 0 which is amazing and beautiful!

For more information, see http://en.wikipedia.org/wiki/Euler’s_Identity.

2

(3)