A complex number is an expression of the form \[ z = x + iy = x + yi \] where \(x, y\) are real numbers and \(i\) is a symbol satisfying \[ i^2 = ii = i \cdot i = -1 \] \(x\) is the real part of \(z\) and \(y\) is the imaginary part of \(z\), and are denoted as

\begin{align} x = \texttt{Re}z,&& y = \texttt{Im}z \end{align}

Operations

Equality

We identify two complex numbers \(z\) and \(w\) if an only if \(Rez = Rew\) and \(Imz = Imw\). Additionally we write

\begin{align} x + 0i = x && 0+yi = yi \end{align}

Thus real numbers are exactly those complex numbers whose imaginary part is zero.

Absolute value

The modulus (absolute value) of \(z\) is defined by

\[|z| = \sqrt{x^2 + y^2}\; \text{ if } \;z = x + iy\]

The complex conjugate of \(z = x + iy\) is defined by \(\bar{z} = x - iy\).

Complex conjugate

The complex conjugate is the number with an equal real part and an imaginary part equal in magnitude, but opposite in sign. The complex conjugate of \(z = x + iy\) is defined by \[ \bar{z} = x - iy \] and,

\begin{align} \text{Re}z = \frac{1}{2}(z+\bar{z}) = \text{Re}\bar{z}, && \text{Im}z = \frac{1}{2i}(z - \bar{z}) = -\text{Im}\bar{z} \end{align}

and for \(z = x + iy\),

\begin{align} |x| \leq |z|, && |y|\leq|z|; && |\bar{z}| = |z| \end{align}

Addition, subtraction, multiplication, division

For \(z = x+iy\) and \(w = s+it\)

• \(z+w = (x+s) + i(y+t)\)
• \(z - w = (x-s)+i(y-t)\)
• \(z \cdot w = (xs - yt) + i(xt+yx)\)
• if \(s^2 + t^2 \neq 0\) then \[ \frac{z}{w} = \frac{\bar{w}z}{\bar{w}w} = \frac{(xs + yt) + i(ys - xt)}{s^2 + t^2} \]

Under these operations the set of all complex numbers becomes a field, with \(0 = 0 +0i\) and \(1 = 1 + 0i\).

Note that for all complex numbers \(z, w\) $$

\begin{align} z\bar{z} = |z|^2; && \overline{z + w} = \bar{z} + \bar{w}; && \overline{zw} = \bar{z}\bar{w}; && |zw| = |z||w| \end{align}

$$

As vectors

A complex number \(z = x+iy\) can be identifies as a point \(P(x, y)\) in the xy-plane. All the rules for geometry can be recast in terms of complex numbers. E.g. let \(w = s+it\) be another complex number. The point of \(z + w\) becomes the vector sum of \(P(x,y)\) and \(Q(s, t)\), and \(|z -w|\) is exactly the distance between \(P(x, y)\) and \(Q(s, t)\).

Geometry

Locus of points

The locus of points is the set of points that satisfy a general equation \(F(z) = 0\). It is sometimes easier to use the xy-coordinates by setting \(z = x+iy\) and to study the definitions defined by \(F(x+iy) = 0\).

Straight lines and circles

Lines

The equation of a straight line can be written as

\[|z - p| = |z-q|\]

where \(p\) and \(q\) are two distinct complex numbers. This line is the bisecting line of the line segments joining \(p\) and \(q\). This is the geometric way for the line equation, the algebraic way is

\[\text{Re}(az + b) = 0\]

where \(a\) and \(b\) are two complex numbers and \(a \neq 0\).

In the xy-coordinates, the line has an equation of the form \[ Ax + By = C \] where \(A, B, C\) are real constants and \(A^2 + B^2 \neq 0\).

Circle

A circle is the set (locus) of points equidistant from a given point. The distance is the radius. The equation for a circle of radius \(r\) and center \(z_0\) is

\[|z - z_0| = r\]

A useful characterization

A circle is also a locus of points satisfying the equation

\[ |z - p| = \rho|z-q|\]

where \(p, q\) are distinct complex numbers and \(\rho \neq 1\) is a positive real number.