Abstract We introduce the operations of ‘norm’ and ‘conjugation’ of a complex number. This enables us to define division by any nonzero complex number.
Outline Is it possible to divide by a complex number? – To appreciate this question, let us recall that division is inversion of multiplication, and we know that inverting such an operation in a given number system can not be taken for granted. For example, addition is not always invertible in \(\NNr\), the system of natural numbers. We need to expand it and form \(\ZNr\), the number system of integers, for subtraction to be always possible. Similarly, in \(\ZNr\) multiplication is not always invertible. We need to expand it and form \(\QNr\), the system of rational numbers, so that division by a nonzero number is always possible. So, in the light of these facts, how does \(\CNr\) fare with respect to division?
The answer is: in \(\CNr\) division by a nonzero number is always possible. To explain how, it helps to first introduce conjugation of a complex number: if \(z=x+yi\), then its conjugate is
\[\CCnjgt{z}\DefEq x - yi\]Geometrically, this means we reflect \(z\) over the \(x\)-axis. Algebraically, conjugation interacts nicely with addition and multiplication, meaning: conjugation commutes with addition as well as with multiplication. Here is an essential observation about conjugating \(z=x+yi\):
\[z\CCnjgt{z} = (x+yi)(x-yi) = x^2 + y^2\]is always a nonnegative real number, namely the square of the distance of \(z\) from \(0\). In particular, this means that \(z\CCnjgt{z}>0\) whenever \(z\neq 0\). In this situation we find
\[1 = \dfrac{z \CCnjgt{z} }{z \CCnjgt{z} } = z\cdot \dfrac{\CCnjgt{z} }{z \CCnjgt{z} }\]But this means that \(z^{-1} = \tfrac{\CCnjgt{z} }{z \CCnjgt{z} }\). Consequently, if we want to divide \(w\in \CNr\) by \(z\neq 0\), we multiply by \(z^{-1}\): \(\tfrac{w}{z} = w\cdot z^{-1}\). – Geometrically, the passage from \(z\) to \(z^{-1}\) is: reflect \(z\) over the \(x\)-axis to obtain \(\CCnjgt{z}\), then reflect \(\CCnjgt{z}\) over the unit circle centered at \(0\).
Complex norm operation – It is also possible to extend the absolute value operation on \(\RNr\) over all of \(\CNr\): the norm, modulus, or absolute value of \(z=x+yi\) is \(\CAbs{z}\DefEq \sqrt{x^2+y^2} = \sqrt{ z \CCnjgt{z} }\). Algebraically, the most important property of the norm operation is that it commutes with multiplication of complex numbers \(\CAbs{wz} = \CAbs{w}\CAbs{z}\), while geometrically useful is the triangle inequality \(\CAbs{w+z} \leq \CAbs{w}+\CAbs{z}\).
We show next that conjugation commutes with addition and multiplication.
For complex numbers \(w\) and \(z\) the following hold:
| \(\CCnjgt{w+z}\) | \(=\) | \(\CCnjgt{w} + \CCnjgt{z}\) |
| \(\CCnjgt{w\cdot z}\) | \(=\) | \(\CCnjgt{w} \cdot \CCnjgt{z}\) |
Moreover, the real and imaginary parts of \(z\) are given by
\[\RePrt{z} = \tfrac{1}{2}(z + \CCnjgt{z})\quad \text{and}\quad \ImPrt{z} = -\tfrac{i}{2}(z-\CCnjgt{z})\]The key to further geometric properties of addition, multiplication, and conjugation of complex numbers is the norm operation:
The norm operation on complex numbers has the following properties
The triangle inequality: \(\CAbs{ w + z } \leq \CAbs{w} + \CAbs{z}\).
Non degeneracy: \(\CAbs{z} = 0\) if and only if \(z=0\).
\(\CAbs{ \CCnjgt{z} } = \CAbs{ z }\).
\(\CAbs{ z }^2 = z\CCnjgt{z}\)
Commutes with complex multiplication: \(\CAbs{wz} = \CAbs{w} \CAbs{z}\).
From the identity \(\CAbs{z}^2 = z\CCnjgt{z}\) we see that \(z\CCnjgt{z}\in \RNr\) is positive, so long as \(z\neq 0\). Therefore
For a nonzero complex number \(z\),
\[1 = \dfrac{z\CCnjgt{z}}{z\CCnjgt{z}} = z\cdot \dfrac{\CCnjgt{z}}{\CAbs{z}^2},\quad \text{so}\quad z^{-1} = \dfrac{\CCnjgt{z}}{ z \CCnjgt{z} }\]