Up Main

What is a complex number?

The number \(i\) is defined to be such that \(i^2 = -1\). It is called an imaginary number because, unlike the natural numbers and the real numbers, it has no immediate physical interpretation. (Note that some books use \(j\) instead of \(i\). So beware!)

Even though \(i\) is imaginary, it is useful in many engineering applications.

We can mix \(i\) with the real numbers using addition and multiplication to form the set of complex numbers, denoted by \(\mathbb{C}\), with the special properties \(i \cdot i = -1\) and \(0\cdot i = 0\).

Just as irrational numbers were invented to solve equations such as \(x^2-2=0\), the number \(i\) was invented to solve equations that have no real solutions, such as \(x^2+1=0\). For a more detailed motivation for complex numbers, click here.

A complex number is written in the form \(a+bi\) where \(a,b\in \mathbb{R}\). Here, \(a\) is the real part of the complex number and \(b\) is the imaginary part. (Note that \(i\) is not included when we refer to the imaginary part.) We also use the following notation to extract the real and imaginary parts:

Where \(z = a+bi\), \(\textbf{Re}(z) = a\) and \(\textbf{Im}(z) = b\).

For convenience, if \(b=1\), we simply write \(a + i\) instead of \(a + 1 i\). If \(a = 0\), we simply write \(bi\) instead of \(0 + bi\). Of course, when \(a = 0\) and \(b = 1\), we get just the imaginary number \(i\).


  1. If \(z = 3 - 4i\), then \(\textbf{Re}(z) = 3\) and \(\textbf{Im}(z) = -4\).

  2. If \(w = 2i\), then \(\textbf{Re}(w) = 0\) and \(\textbf{Im}(w) = 2\).

  3. If \(u = -5\), then \(\textbf{Re}(u) = -5\) and \(\textbf{Im}(u) = 0\).

Quick quiz


For each of the following complex numbers, give its real part and imaginary part.

  1. \(1+3i\)  

  2. \(\sqrt{2}i\)  

  3. \(-\pi - 4i\)  

  4. \(2\)