The number i and complex numbers

Warm-up

Three quick questions from earlier lessons.

Discover

No real number squares to $-1$ . Mathematicians define $i$ with $i^2 = -1$ . What is $(3i)^2$ ?

$-9$
$9$

Key idea

A new number that makes negative square roots possible.

i^2 = -1

. For

a > 0

\sqrt{-a} = i\sqrt{a}

. A complex number is

a + bi

with real part

a

and imaginary part

b

. Powers of

i

cycle:

i, -1, -i, 1, \dots

(period 4).

Worked example

Simplify $\sqrt{-49}$ and $i^{10}$ .

\sqrt{-49} = 7i

√49 · √−1.

i^{10} = i^2

10 = 4·2 + 2.

i^{10} = -1

i² = −1.

Your turn, guided

Your turn: $i^7$ .

-i

7 = 4 + 3: i³ = −i.

Practice ★

Write $\sqrt{-121}$ in terms of $i$ .
Write $\sqrt{-4}$ in terms of $i$ .

Which number is real?
1. $(9i)^2$
2. $\sqrt{-9}$
3. $9 + i$

Watch out!

Practice ★★

Simplify $i^{ 10 }$ .
Simplify $i^{ 13 }$ .
Simplify $i^{ 5 }$ .

Exit ticket

Three questions to check you have got it.

Write $\sqrt{-121}$ in terms of $i$ .
Write $\sqrt{-4}$ in terms of $i$ .
Write $\sqrt{-25}$ in terms of $i$ .

Summary card

The number i

$i^2 = -1$ ; $\sqrt{-a} = i\sqrt{a}$ ; powers of $i$ repeat every 4.

Watch out: $\sqrt{-16}$ is $4i$ , not $-4$ .

End of the lesson

Next lesson: Adding, subtracting and multiplying complex numbers Back to the chapter

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