Exponential form

The trigonometric form is correct but heavy. A more compact notation is about to simplify everything — and turn multiplication into addition.

The starting observation

Multiply two complex numbers of modulus 1 written in trigonometric form:

(cos α + i sin α)(cos β + i sin β)
   = cos α cos β - sin α sin β  +  i(sin α cos β + cos α sin β)
   = cos(α + β) + i sin(α + β)

These are the addition formulas of trigonometry. The result is striking: multiplying two complex numbers of modulus 1 amounts to adding their angles.

And there is a well-known function that turns sums into products: the exponential, with e^x × e^y = e^(x+y). The notation suggests itself.

Euler's notation

e^(iθ) = cos θ + i sin θ

Every non-zero complex number can then be written in exponential form:

z = r e^(iθ)          with r = |z| > 0  and  θ = arg(z)

This is not a mere notational convention: expanding e^x, cos x and sin x as power series proves the identity. The exponential notation is legitimate, not analogical.

The rules of computation become trivial

r e^(iα) × r' e^(iβ) = r r' e^(i(α+β))        moduli ×, arguments +

     r e^(iα)      r
   ------------ = ---- e^(i(α-β))             moduli ÷, arguments -
     r' e^(iβ)     r'

   (r e^(iθ))^n = r^n e^(inθ)                 modulus ^n, argument ×n

   conjugate of r e^(iθ) = r e^(-iθ)          the angle changes sign
   inverse   of r e^(iθ) = (1/r) e^(-iθ)

Compare with algebraic form: raising (1 + i) to the power 10 would need ten successive expansions. In exponential form:

1 + i = √2 e^(iπ/4)      ->      (1+i)^10 = (√2)^10 e^(i·10π/4)
                                          = 32 e^(i·5π/2)
                                          = 32 e^(iπ/2)  =  32 i

Three lines, and no risk of a sign slip.

The picture that sums it up

             multiplication by r e^(iθ)
        =    rotation by θ  +  scaling by a factor r

     ^                              ^
     |   • z                        |      • z' = 2 e^(iπ/3) × z
     |  /                           |     /
     | /                            |    /   (rotated by 60°,
     |/________>                    |___/_____  twice as far out)

This is the key to the whole chapter: a complex number is a transformation of the plane, both rotation and dilation. Multiplying by i = e^(iπ/2) is a quarter-turn.

Euler's identity

For θ = π the formula gives:

e^(iπ) = cos π + i sin π = -1        hence        e^(iπ) + 1 = 0

In five symbols this ties together the fundamental constants of analysis (e), geometry (π), algebra (i), and the two neutral elements 0 and 1. It is regularly cited as the most beautiful formula in mathematics.

Summary

• e^(iθ) = cos θ + i sin θ; every non-zero complex is z = r e^(iθ).
• Product: moduli multiplied, arguments added.
• Power: (r e^(iθ))^n = r^n e^(inθ) — the computation becomes immediate.
• Conjugate: r e^(-iθ); inverse: (1/r) e^(-iθ).
• Multiplying by r e^(iθ) means rotating by θ and scaling by r.
• e^(iπ) + 1 = 0: Euler's identity.