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
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 .
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 .
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.