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
\(i\).

Examples

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

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

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