
# 18.3: Conjugation and absolute value


Let $$z=x+i\cdot y$$; that is, $$z$$ is a complex number with real part $$x$$ and imaginary part $$y$$. If $$y=0$$, we say that the complex number $$z$$ is real and if $$x=0$$ we say that $$z$$ is imaginary. The set of points with real (imaginary) complex coordinates is a line in the plane, which is called real (respectively imaginary) line. The real line will be denoted as $$\mathbb{R}$$.

The complex number

$$\bar z := x-i\cdot y$$

is called the complex conjugate of $$z=x+i\cdot y$$. Let $$Z$$ and $$\bar Z$$ be the points in the plane with the complex coordinates $$z$$ and $$\bar z$$ respectively. Note that the point $$\bar Z$$ is the reflection of $$Z$$ across the real line.

It is straightforward to check that

\begin{aligned} x&=\text{Re} z=\frac{z+\bar z}2, & y&=\text{Im} z=\frac{z-\bar z}{i\cdot2}, & x^2+y^2&=z\cdot\bar z. \end{aligned}

The last formula in 18.3.1 makes it possible to express the quotient $$\tfrac{w}{z}$$ of two complex numbers $$w$$ and $$z=x+i\cdot y$$:

$$\frac{w}{z}=\tfrac{1}{z\cdot\bar z}\cdot w\cdot\bar z=\tfrac{1}{x^2+y^2}\cdot w\cdot\bar z.$$

Note that

\begin{aligned} \overline {z+ w}&=\bar z+\bar w, & \overline {z- w}&=\bar z-\bar w, & \overline {z\cdot w}&=\bar z\cdot\bar w, & \overline {z/w}&=\bar z/\bar w.\end{aligned}

That is, the complex conjugation respects all the arithmetic operations.

The value

\begin{aligned} |z|&:=\sqrt{x^2+y^2}=\sqrt{(x+i\cdot y)\cdot(x-i\cdot y)} = \sqrt{z\cdot\bar z}\end{aligned}

is called the absolute value of $$z$$. If $$|z|=1$$, then $$z$$ is called a unit complex number.

Exercise $$\PageIndex{1}$$

Show that $$|v\cdot w|=|v|\cdot |w|$$ for any $$v,w\in\mathbb{C}$$.

Hint

Use that $$|z|^2 = z \cdot \bar{z}$$ for $$z = v, w$$, and $$v \cdot w$$.

Suppose that $$Z$$ and $$W$$ are points with complex coordinates $$z$$ and $$w$$. Note that

$ZW=|z-w|.$

The triangle inequality for the points with complex coordinates $$0$$, $$v$$, and $$v+w$$ implies that

$$|v+w|\le |v|+|w|$$

for any $$v,w\in\mathbb{C}$$; this inequality is also called triangle inequality.

Exercise $$\PageIndex{2}$$

Use the identity

$$u\cdot (v-w)+v\cdot (w-u)+w\cdot(u-v)=0$$

for $$u,v,w\in\mathbb{C}$$ and the triangle inequality to prove Ptolemy’s inequality (Theorem 6.4.1).

Hint

Given a quadrangle $$ABCD$$, we can choose the complex coordinates so that $$A$$ has complex coordinate 0. Rewrite the terms in the Ptolemy’s inequality in terms of the complex coordinates $$u, v$$, and $$w$$ of $$B, C$$, and $$D$$; apply the identity and the triangle inequality.