Chapter 3 · Complex Numbers and Equations · Concept 1
3-4 Square Roots of Negative Numbers
Once i exists, √(−9) is no longer impossible — it is 3i. But something else happens at the same moment, and almost nobody notices: a rule you have used since Grade VIII quietly stops being true.
Guiding question
What can go wrong once roots are allowed to be negative inside?
Learning outcome. Calculate square roots of negative numbers and of complex numbers.
Roots of negatives
for any a greater than 0
√(−a) = i√a
And the instruction that matters more than the rule:
express everything in terms of i before calculating
Not afterwards, not halfway through. First. The next section shows what happens to people who leave it until later.
“√(−9)” and “the square roots of −9” are different questions. The first is 3i. The second is ±3i, because a number has two square roots. Read which one is being asked.
Both columns below look like reasonable school algebra. They give opposite answers, so one of them must be wrong — and it is the one that feels most natural.
Warm Up 2(a)
turn them into i FIRST
√(-12) × √(-6)
= i√12 × i√6
= i2 × √72
multiply under one root
√(-12) × √(-6)
= √(-12 × -6)
= √72
Two misconceptions the book names.
- · “√(−n) is impossible.” It was, before i. Complex numbers were built precisely so that it is not.
- · “√(−n) = −√n.” No. −√9 is −3, an ordinary negative number; √(−9) is 3i, which is not on the number line at all. The minus is inside the root, and that changes everything.
2(a) Calculate √(−12) × √(−6).
- 1
Before anything else, write both in terms of i.
√(−12) = i√12 and √(−6) = i√6This is the whole instruction of the lesson. Do it first and nothing can go wrong.
Work with a partner. Before either of you multiplies anything, both write the i form of every root on the page. If one is missed, the sign at the end will be wrong.
1. Express in terms of i.
2. Calculate.
1. Express in terms of i.
2. Calculate.
Part (h) is lesson 3-3 in disguise.
(2 + √(−3))(2 − √(−3)) is (2 + i√3)(2 − i√3) — a pair of conjugates. So the answer has to be real without any working at all: 4 + 3 = 7.
Solve (2x − 8)2 = −10
Complex numbers let engineers model tissue impedance and audio signals — real, measurable things. So does calling one part 'imaginary' and the other 'real' actually mean anything?
Ask who chose those words, and when.
Multiplying under one root.
√(−12) × √(−6) is NOT √72. Convert both to i first, or you lose the minus sign that the two i's create.
Writing √(−9) = −3.
It is 3i. The minus is inside the root, which puts the answer off the number line entirely.
Dropping the ± on "the square roots".
"√(−6)" is i√6, but "the square roots of −6" are ±i√6. Read which one was asked.
Leaving the root untidied.
√(−45) is 3i√5, not i√45. Pull out the square factor as you always would.
Leaving an i on the bottom.
2/(3i) is not finished. Multiply top and bottom by i to get −(2/3)i, exactly as in lesson 3-3.
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