The real numbers do not contain a solution to

$$x^2=-1.$$

Introducing the imaginary unit

$$i^2=-1$$

extends the number system to the complex numbers.

A complex number has the form

$$z=a+bi,$$

where $a$ is the real part and $b$ the imaginary part.

The complex plane

The number $a+bi$ can be represented by the point $(a,b)$ in the complex plane. This turns a complex number into both an algebraic quantity and a two-dimensional geometric object.

Arithmetic

Addition is componentwise:

$$(a+bi)+(c+di)=(a+c)+(b+d)i.$$

Multiplication uses $i^2=-1$:

$$(a+bi)(c+di)=(ac-bd)+(ad+bc)i.$$

Conjugate and magnitude

The complex conjugate of $z=a+bi$ is

$$\bar z=a-bi.$$

Their product is real:

$$z\bar z=a^2+b^2.$$

The magnitude is

$$|z|=\sqrt{a^2+b^2}.$$

Complex numbers make equations such as $x^2+1=0$ solvable and later provide a compact language for rotations, oscillations and AC circuits.