Complex numbers: arithmetic with rotation

The symbol $i$ is defined by $i^2=-1$. A complex number has the form $a+ib$, where $a$ and $b$ are real. Its real part is $a$, and its imaginary part is the real coefficient $b$, not the whole term $ib$. “Imaginary” is historical terminology; it does not imply the arithmetic is less legitimate than arithmetic with negative numbers.

Add real parts to real parts and imaginary coefficients to imaginary coefficients. For multiplication, distribute just as in algebra and then replace each $i^2$ with $-1$. A complex number can be drawn as the point $(a,b)$ in a plane. Addition therefore has the same geometry as adding two-dimensional vectors, although complex multiplication adds extra structure that ordinary vector addition alone does not provide.

The complex conjugate reverses the sign of the imaginary part. In the complex plane this reflects the point across the real axis. Multiplying a number by its conjugate produces the sum of the squared coordinates, which is real and nonnegative. The modulus is the nonnegative square root of that quantity. It measures distance from the origin, not distance along either individual coordinate axis.

Conjugation also makes division manageable. Multiply numerator and denominator by the denominator's conjugate to replace a complex denominator with its squared modulus. Division remains undefined when that denominator is zero. Notice that the modulus and the squared modulus are different quantities. In quantum mechanics, it is a normalized amplitude's squared modulus that gives a probability for a specified measurement outcome, not its modulus alone.

A nonzero complex number can be described by its modulus $r$ and angle $\theta$: $z=r(\cos\theta+i\sin\theta)$. Its angle is called its argument or phase, and adding a complete turn leaves the same number. Multiplying two complex numbers multiplies their moduli and adds their phases. In particular, multiplication by $i$ rotates a point counterclockwise by a quarter turn without changing its modulus.

The shorthand $e^{i\theta}=\cos\theta+i\sin\theta$ is Euler's formula. We use it here as a useful connection; its exponential interpretation can be justified through calculus or convergent series later. A phase can change the sum of two amplitudes even when neither individual squared modulus changes. That is the essential distinction between adding complex amplitudes first and adding classical probabilities directly.