Complex Numbers

A complex number is a number of the form a + i*b, where a and b are real numbers, and i is defined to be the square root of negative one. Addition of two complex numbers c₁ and c₂ is defined to be:

c₁ = a₁ + i*b₁

c₂ = a₂ + i*b₂

c₁ + c₂ = a₁ + a₂ + i*(b₁ + b₂)

The complex conjugate of a complex number c, denoted c^* is defined to be:

c = a + i*b

c^* = a - i*b

Multiplication of two complex numbers c₁ and c₂ is defined to be:

c₁ = a₁ + i*b₁

c₂ = a₂ + i*b₂

c₁*c₂ = a₁*a₂ - b₁*b₂ + i*(a₁*b₂ + a₂*b₁)

Euler's Formula for complex numbers states that e^ix = cos x + i*sin x, this relationship is used in the discrete Fourier transform of Shor's algorithm.