The number i

What you will be able to do

• explain why no real number solves $x^2 + 1 = 0$;
• use the number $i$, whose square is $-1$;
• write the square root of a negative number as a multiple of $i$;
• name the real and the imaginary parts of $a + ib$, and use the equality of two complex numbers.

An equation with no real root

In Class 10 you met quadratic equations with no real roots. The simplest is $x^2 + 1 = 0$. It asks for a number whose square is $-1$.

The square of a real number is never negative: $3^2 = 9$, and $(-3)^2 = 9$ as well. So no real number solves this equation. On a graph, the curve $y = x^2 + 1$ stays above the $x$-axis and never meets it.

Mathematicians in Italy in the 1500s met roots like these while they solved cubic equations. At first they called them impossible. Yet when they calculated with them boldly, the impossible parts cancelled and correct real answers came out. So the new numbers were kept, and given rules.

The number i

We bring in one new number, written $i$, with one rule:

So $i$ is a square root of $-1$. The other square root is $-i$, because $(-i)^2 = i^2 = -1$.

Now every negative number has square roots. For a positive real number $a$:

For example, $\sqrt{-36} = 6i$ and $\sqrt{-5} = \sqrt{5}\, i$. So the equation $x^2 + 36 = 0$ has the two roots $6i$ and $-6i$.

Python writes the number $i$ as 1j, the letter that electrical engineers use. The number $6i$ is 6j in Python.

Complex numbers

A complex number is a number of the form

where $a$ and $b$ are real numbers. The number $a$ is the real part of $z$, written $\operatorname{Re} z$. The number $b$ is the imaginary part, written $\operatorname{Im} z$.

• For $z = 4 - 7i$, the real part is $4$ and the imaginary part is $-7$. The imaginary part is the real number $-7$, not $-7i$.
• A real number such as $5$ is the complex number $5 + 0i$. So the real numbers sit inside the complex numbers.
• A number such as $2i = 0 + 2i$, with real part $0$, is called purely imaginary.

When are two complex numbers equal?

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

So one equation between complex numbers gives two equations between real numbers.

Example. Find the real numbers $x$ and $y$ for which $(3x - 1) + (y + 2)i = 8 - 5i$.

• Real parts: $3x - 1 = 8$, so $x = 3$.
• Imaginary parts: $y + 2 = -5$, so $y = -7$.

There is one thing that complex numbers cannot do. They cannot be put in order. A statement such as $2 + i > 1$ has no meaning, and Python refuses to compare them.

A worked example

Solve $4x^2 + 25 = 0$.

1. Make $x^2$ stand alone. $x^2 = -\dfrac{25}{4}$.
2. Take square roots. $x = \pm\sqrt{-\dfrac{25}{4}} = \pm\dfrac{5}{2}\, i$.
3. Check one root. $\left(\dfrac{5}{2}i\right)^2 = \dfrac{25}{4}\, i^2 = -\dfrac{25}{4}$, and $4 \times \left(-\dfrac{25}{4}\right) + 25 = 0$.

The roots are $\dfrac{5}{2}i$ and $-\dfrac{5}{2}i$. Both are purely imaginary.

Practise with new numbers

Try it in Python

The program solves $x^2 + c = 0$ for several values of $c$. It uses the module cmath, the complex cousin of math. A small function show writes each answer in the form $a + bi$.

Common mistakes

• Saying that $x^2 + 4 = 0$ has no solution. It has no real solution. In the complex numbers it has two: $2i$ and $-2i$.
• Giving $i$ as part of the imaginary part. In $6 - 2i$ the imaginary part is $-2$, a real number, not $-2i$.
• Using math for a negative square root. math.sqrt(-1) stops with ValueError: expected a nonnegative input, got -1.0. Use cmath.sqrt(-1), which gives 1j.
• Equating only one part. $a + ib = 5$ means $a = 5$ and also $b = 0$.
• Comparing complex numbers. 3 + 4j < 5 stops with TypeError: '<' not supported between instances of 'complex' and 'int'. Complex numbers have no order.

Remember these

• $i$ is a number with $i^2 = -1$. The two square roots of $-1$ are $i$ and $-i$.
• For $a > 0$, $\sqrt{-a} = \sqrt{a}\, i$.
• A complex number is $a + ib$: real part $a$, imaginary part $b$, both real.
• $a + ib = c + id$ means $a = c$ and $b = d$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Write $\sqrt{-81}$ and $\sqrt{-12}$ as multiples of $i$.

2. What are the real and imaginary parts of $-\sqrt{2} + 0.5i$?

3. Find the real numbers $p$ and $q$ if $(2p + 1) + 3i = 9 + (q - 4)i$.

4. Solve $9x^2 + 4 = 0$.

5. Is $7$ a complex number? Is $3i$ a real number?