Complex numbers: what i is, and the parts of a complex number

Try to solve $x^2 = -1$. You need a number which, multiplied by itself, gives $-1$. But a positive number squared is positive, and a negative number squared is also positive ($(-3)^2 = 9$). Zero squared is zero. So no real number works.

For a long time mathematicians simply said "no solution". Then they tried something bold: invent a new number that does the job, and see whether the maths still hangs together. They called it $i$:

$i^2 = -1$

It turned out to hang together beautifully, and these numbers are now used every day in electrical engineering, signal processing and physics. The word "imaginary" is just an old name; it doesn't mean the numbers are fake.

Once you have $i$, square roots of any negative number make sense. For example, $\sqrt{-9} = 3i$, because $(3i)^2 = 9 \times i^2 = 9 \times (-1) = -9$ ✓.

Powers of $i$ go round in a cycle of four. $i^1 = i$, $i^2 = -1$, then $i^3 = i^2 \times i = -i$, and $i^4 = i^2 \times i^2 = (-1)(-1) = 1$. After that the pattern repeats: $i^5 = i$, $i^6 = -1$, and so on.

A complex number is a real number plus a real multiple of $i$:

$z = x + iy$

For example, $3 - 4i$ has real part $3$ and imaginary part $-4$. We write $\operatorname{Re} z = 3$ and $\operatorname{Im} z = -4$.

Two things that trip people up:

• The imaginary part is the number in front of $i$, so it is $-4$, not $-4i$. It is a real number.
• Ordinary real numbers are complex numbers too, just with imaginary part $0$. For example, $5 = 5 + 0i$.

The set of all complex numbers is written $\mathbb{C}$.

Two complex numbers are equal exactly when their real parts are equal and their imaginary parts are equal:

$a + bi = c + di \quad\text{means}\quad a = c \ \text{ and } \ b = d$

Why? The real part and the imaginary part are independent of each other. No amount of real number can ever make up for a difference in the $i$ part, because a real number has no $i$ in it.

This is surprisingly powerful: one equation between complex numbers gives you two ordinary equations. That is how you will later find square roots of complex numbers and solve for unknowns.