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 $c_{1}$ and $c_{2}$ is defined to be:

$$c_{1} = a_{1} + i * b_{1}$$

$$c_{2} = a_{2} + i * b_{2}$$

$$c_{1} + c_{2} = a_{1} + a_{2} + i * (b_{1} + b_{2})$$

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 $c_{1}$ and $c_{2}$ is defined to be:

$$c_{1} = a_{1} + i * b_{1}$$

$$c_{2} = a_{2} + i * b_{2}$$

$$c_{1}*c_{2}=a_{1}*a_{2}-b_{1}*b_{2}+i*(a_{1}*b_{2}+a_{2}*b_{1})$$

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.