The Imaginary Unit

A new number

No real number squares to −1. So a new number is defined to do exactly that. It is called i, the imaginary unit, and its one defining property is $i^{2} = -1$.

Apart from that, i obeys the ordinary rules of algebra: it can be added, multiplied, put in brackets and collected like a letter. The one extra rule is that $i^{2}$ is replaced by −1 wherever it appears.

Half of a half turn

Multiplying by −1 turns the number line through half a turn about 0. It sends 3 to −3 and 1 to −1, each to the point the same distance from 0 on the other side. Doing it twice brings every number back where it started, because (−1) × (−1) = 1.

Multiplying by i twice is multiplying by $i^{2} = -1$, which is that half turn. So multiplying by i once must be half of a half turn: a quarter turn. A quarter turn about 0 takes 1 off the number line altogether, which is why i is not a real number. Where it lands needs a plane rather than a line, which is the subject of a later lesson, The Argand Diagram.

Multiplying by −1 carries 3 over 0 to −3. Multiplying by −1 again brings it back to 3. Two multiplications by i must make this same move.

Square roots of negative numbers

With i defined, every negative number has a square root. $\sqrt{-9} = \sqrt{9} \times \sqrt{-1} = 3 \times i = 3i$. Check by squaring: $\left(3i\right)^{2} = 3^{2} \times i^{2} = 9 \times \left(-1\right) = -9$.

−3i is also a square root of −9, because $\left(-3i\right)^{2} = 9 \times i^{2} = -9$. So −9 has two square roots, 3i and −3i, and $\sqrt{-9}$ means 3i, just as $\sqrt{9}$ means 3 rather than −3.

In general $\sqrt{-a} = i\sqrt{a}$ for any positive number a. So $\sqrt{-16} = 4i$, $\sqrt{-25} = 5i$, and $\sqrt{-7} = i\sqrt{7}$, which is about 2.646i.

Equations that had no real solution now have two. $x^{2} = -1$ gives x = i or x = −i. $x^{2} + 9 = 0$ gives $x^{2} = -9$, so x = 3i or x = −3i.

One rule that fails

For numbers that are not negative, $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$: $\sqrt{4} \times \sqrt{9} = 2 \times 3 = 6 = \sqrt{36}$. The rule fails when both numbers are negative.

$\sqrt{-4} \times \sqrt{-9}$ is not $\sqrt{36}$. Write each root with i first: $\sqrt{-4} = 2i$ and $\sqrt{-9} = 3i$, so $\sqrt{-4} \times \sqrt{-9} = 2i \times 3i = 6i^{2} = -6$. Multiplying under the root gives $\sqrt{\left(-4\right) \times \left(-9\right)} = \sqrt{36} = 6$, which is wrong by a sign.

The safe order is to turn every root of a negative into i times a real root, $\sqrt{-a} = i\sqrt{a}$, and only then multiply. Splitting $\sqrt{-9}$ as $\sqrt{9} \times \sqrt{-1}$, with only one negative under a root, is that same step.

Complex numbers

A real number added to a multiple of i makes a complex number, such as 3 + 2i. In a + bi, a is the real part and b is the imaginary part. Both a and b are real numbers: the imaginary part of 3 + 2i is 2, not 2i.

3 + 2i does not simplify, any more than 3 + 2x does. The real part and the imaginary part stay separate, and arithmetic with complex numbers keeps track of each.

Every real number is a complex number whose imaginary part is 0: 7 = 7 + 0i. A number such as 5i, whose real part is 0, is called imaginary.

Five complex numbers split into their two parts. The imaginary part is the number that multiplies i, so for 3 + 2i it is 2.

The usual mistakes

Dropping the i. $\sqrt{-49}$ is not 7, because $7^{2} = 49$. It is 7i, since $\left(7i\right)^{2} = 49 \times \left(-1\right) = -49$.

Giving the negative real root. $\left(-7\right)^{2} = 49$ as well, so −7 is a square root of 49, not of −49.

Multiplying two roots of negatives under one root sign. $\sqrt{-4} \times \sqrt{-9} = 2i \times 3i = -6$, not $\sqrt{36} = 6$.

Writing the imaginary part with its i. The imaginary part of −1 + 4i is 4.