Skill 6 of 12

The imaginary unit and square roots of negative numbers

Use $i=\sqrt{-1}$ and $i^2=-1$ to write square roots of negative numbers as $bi$ , simplify powers of $i$ , and identify the real and imaginary parts of $a+bi$ .

Common Core standards: N-CN.1
Builds on: Nothing earlier in this unit. This is where it starts.

Common mistakes

What students most often get wrong with this skill:

Computing $\sqrt{-4}\cdot\sqrt{-9}$ as $\sqrt{36}=6$ instead of $-6$
Treating $i^2$ as 1
Saying the imaginary part of $3+4i$ is $4i$ instead of 4

A worked example

From the lesson, with every step explained.

Example 1: A perfect square

Problem. Write $\sqrt{-25}$ in the form $bi$ . Then give the real and imaginary parts of $3+\sqrt{-25}$ .

Plan. Split off $-1$ , then take the root of 25.

1) Rule: $\sqrt{-b}=\sqrt{b}\,i$ . Why: it pulls the negative sign into $i$ .

\sqrt{-25}=\sqrt{25}\cdot i

2) Since $\sqrt{25}=5$ :

\sqrt{-25}=5i

3) Now rewrite the expression:

3+\sqrt{-25}=3+5i

4) The real part is 3. The imaginary part is 5 (not $5i$ ).

Answer. $\sqrt{-25}=5i$ . The real part is 3 and the imaginary part is 5.

Check. Square it:

(5i)^2=25i^2=25(-1)=-25

This matches, so the answer is correct.

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