2.1: Basic Notions
The set of complex numbers is obtained algebraically by adjoining the number \(i\) to the set \(\mathbb{R}\) of real numbers, where \(i\) is defined by the property that \(i^2=−1\). We will take a geometric approach and define a complex number to be an ordered pair \((x,y)\) of real numbers. We let \(\mathbb{C}\) denote the set of all complex numbers,
\[\mathbb{C} =\{(x,y) | x,y∈ \mathbb{R}\}.\]
Given the complex number \(z=(x,y)\), \(x\) is called the real part of \(z\), denoted Re\((z)\); and \(y\) is called the imaginary part of \(z\), denoted Im\((z)\). The set of real numbers is a subset of \(\mathbb{C}\) under the identification \(x↔(x,0)\), for any real number \(x\).
Addition in \(\mathbb{C}\) is componentwise,
\[(x,y)+(s,t)=(x+s,y+t),\]
and if \(k\) is a real number, we define scalar multiplication by
\[k⋅(x,y)=(kx,ky).\]
Within this framework, \(i=(0,1)\), meaning that any complex number \((x,y)\) can be expressed as \(x+yi\) as suggested here:
\[ \begin{array} (x,y) &= (x,0)+(0,y) \\ &= x(1,0)+y(0,1) \\ &= x + yi \end{array} \]
The expression \(x+yi\) is called the Cartesian form of the complex number. This form can be helpful when doing arithmetic of complex numbers, but it can also be a bit gangly. We often let a single letter such as \(z\) or \(w\) represent a complex number. So, \(z=x+yi\) means that the complex number we're calling \(z\) corresponds to the point \((x,y)\) in the plane.
It is sometimes helpful to view a complex number as a vector, and complex addition corresponds to vector addition in the plane. The same holds for scalar multiplication. For instance, in Figure \(\PageIndex{1}\) we have represented \(z=2+i\), \(w=−1+1.5i\), as well as \(z+w=1+2.5i\), as vectors from the origin to these points in \(\mathbb{C}\). The complex number \(z−w\) can be represented by the vector from \(w\) to \(z\) in the plane.
We define complex multiplication using the fact that \(i^2=−1\).
\[ \begin{array} (x+yi)⋅(s+ti) &= xs+ysi+xti+yti^2 \\ &= (xs−yt)+(ys+xt)i \end{array} \]
The modulus of \(z=x+yi\), denoted \(|z|\), is given by
\[|z|= \sqrt{x^2+y^2}.\]
Note that \(|z|\) gives the Euclidean distance of \(z\) to the point \((0,0)\).
The conjugate of \(z=x+yi\), denoted \(\overline{z}\), is
\[\overline{z} =x−yi\]
In the exercises, the reader is asked to prove various useful properties of the modulus and conjugate.
Suppose \(z=3−4i\) and \(w=2+7i\).
Then \(z+w=5+3i\), and
\[ \begin{equation} \begin{array} \cdot z \cdot w &= (3−4i)(2+7i) \\ &= 6+28−8i+21i \\ &= 34+13i \end{array} \end{equation} \]
A few other computations:
\[ \begin{array} 4z &= 12−16i \\ |z| &= \sqrt{3^2 + (-4)^2} = 5 \\ \overline{zw} &= 34-13i \end{array} \]
Exercises
In each case, determine \(z+w\), \(sz\), \(|z|\), and \(z⋅w\).
- \(z=5+2i\), \(s=−4\), \(w=−1+2i\)
- \(z=3i\), \(s= \dfrac{1}{2}\), \(w=−3+2i\)
- \(z=1+i\), \(s=0.6\), \(w=1−i\)
- Answer
-
a. \(z+w=4+4i\), \(sz=−20−8i\), \(|z|=\sqrt{29}\), and \(zw=−9+8i\).
Show that \(z⋅ \overline{z} =|z|^2\), where \(\overline{z}\) is the conjugate of \(z\).
- Hint
-
Let \(z=a+bi\), and show that the two sides of the equation agree.
Suppose \(z=x+yi\) and \(w=s+ti\) are two complex numbers. Prove the following properties of the conjugate and the modulus.
- \(|w⋅z|=|w|⋅|z|\).
- \(\overline{zw}= \overline{z} \cdot \overline{w}\)
- \(\overline{z+w}= \overline{z} + \overline{w}\)
- \(z+\overline{z}= 2 \text{ Re} (z).\) (Hence, \(z+ \overline{z}\) is a real number.)
- \(z- \overline{z}= 2 \text{ Im} (z)i.\)
- \(|z|=|\overline{z}|.\)
A Pythagorean triple consists of three integers \((a,b,c)\) such that \(a^2+b^2=c^2\). We can use complex numbers to generate Pythagorean triples. Suppose \(z=x+yi\) where \(x\) and \(y\) are positive integers. Let
\(a= \text{ Re}(z^2) \;\;\;\;\; b= \text{ Im}(z^2) \;\;\;\;\; c=z \overline{z}.\)
- Prove that \(a^2+b^2=c^2\).
- Find the complex number \(z=x+yi\) that generates the famous triple \((3,4,5)\).
- Find the complex number that generates the triple \((5,12,13)\).
- Find five other Pythagorean triples, generated using complex numbers of the form \(z=x+yi\), where \(x\) and \(y\) are positive integers with no common divisors.