THE SQUARE ROOT OF A NEGATIVE NUMBER

LESSON READ-THROUGH
by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!
Thanks for your support!
 
Square roots of negative numbers were introduced in a prior lesson:
Arithmetic with Complex Numbers in the Algebra II materials.
For your convenience, that material is repeated (and extended) here.

Firstly, recall some information from beginning algebra:

The Square Root of a Nonnegative Real Number

For a nonnegative real number $\,k\,$, $$ \cssId{s7}{\overbrace{\ \ \ \sqrt{k}\ \ \ }^{\text{the square root of $\,k\,$}}} \cssId{s8}{:=} \ \ \cssId{s9}{\text{the unique nonnegative real number which, when squared, equals $\,k\,$}} $$

(Note: when the distinction becomes important in higher-level mathematics, then $\,\sqrt{k}\,$ is called the principal square root of $\,k\,$.)

Thus, the number that gets to be called the square root of $\,k\,$ satisfies two properties:

For example, $\,\sqrt{9} = 3\,$ since the number $\,3\,$ satisfies these two properties:

So, what is (say) $\,\sqrt{-4}\ $?
There does not exist a nonnegative real number which, when squared, equals $\,-4\ $.
Why not?
Every real number, when squared, is nonnegative:   for all real numbers $\,x\,$,   $\,x^2 \ge 0\,$.
Complex numbers to the rescue!

The Square Root of a Negative Number

Complex numbers allow us to compute the square root of negative numbers, like $\,\sqrt{-4}\ $.
Remember the key fact:   $\,i:=\sqrt{-1}\,$,   so that   $\,i^2=-1\,$

THE SQUARE ROOT OF A NEGATIVE NUMBER
Let $\,p\,$ be a positive real number, so that $\,-p\,$ is a negative real number. Then: $$ \cssId{s31}{\overbrace{\ \ \ \sqrt{-p}\ \ \ }^{\text{the square root of negative $\,p\,$}}} \ \cssId{s32}{:=}\ \cssId{s33}{i\,\sqrt{\vphantom{h}p}} $$
(Note: when the distinction becomes important in higher-level mathematics, then $\,\sqrt{-p}\,$ is called the principal square root of $\,-p\,$.)

Observe that $\,i\sqrt{\vphantom{h}p}\,$, when squared, does indeed give $\,-p\ $: $$ \cssId{s36}{(i\sqrt{\vphantom{h}p})^2} \ \cssId{s37}{=\ i^2(\sqrt{\vphantom{h}p})^2} \ \cssId{s38}{=\ (-1)(p)} \ \cssId{s39}{=\ -p} $$

Some of my students like to think of it this way:
You can slide a minus sign out of a square root, and in the process, it turns into the imaginary number $\,i\,$!

Here are some examples:

TWO DIFFERENT QUESTIONS; TWO DIFFERENT ANSWERS

Recall these two different questions, with two different answers:

Similarly, there are two different questions involving complex numbers, with two different answers:

Square Roots of Products

The following statement is true:   for all nonnegative real numbers $\,a\,$ and $\,b\,$, $\,\sqrt{ab} = \sqrt{a}\sqrt{b}\ $.
For nonnegative numbers, the square root of a product is the product of the square roots.

Does this property work for negative numbers, too?
The answer is NO, as shown next.

Certainly, anything called ‘the square root of $\,-4\,$’ must have the property that, when squared, it equals $\,-4\ $.
Unfortunately, the following incorrect reasoning gives the square as $\,4\,$, not $\,-4\,$: $$ \cssId{s87}{\text{? ? ? ? }}\ \ \ \ \cssId{s88}{(\sqrt{-4})^2} \ \ \cssId{s89}{=\ \ \sqrt{-4}\sqrt{-4}} \cssId{s90}{\overbrace{\ \ =\ \ }^{\text{this is the mistake}}}\ \ \cssId{s91}{\sqrt{(-4)(-4)}}\ \ \cssId{s92}{=\ \ \sqrt{16}}\ \ \cssId{s93}{=\ \ 4} \ \ \ \ \cssId{s94}{\text{? ? ? ? }} $$ Here is the correct approach: $$ \cssId{s96}{(\sqrt{-4})^2} \ \ \cssId{s97}{=\ \ \sqrt{-4}\sqrt{-4}}\ \ \cssId{s98}{=\ \ i\sqrt{4}\ i\sqrt{4}} \ \ \cssId{s99}{=\ \ i^2(\sqrt{4})^2} \ \ \cssId{s100}{=\ \ (-1)(4)} \ \ \cssId{s101}{=\ \ -4} $$

Similarly, $\,\sqrt{-5}\sqrt{-3}\,$ is NOT equal to $\,\sqrt{(-5)(-3)}\,$.
Instead, here is the correct simplication: $$ \cssId{s104}{\sqrt{-5}\sqrt{-3}} \ \ \cssId{s105}{=\ \ i\sqrt{5}\ i\sqrt{3}} \ \ \cssId{s106}{=\ \ i^2\sqrt{5}\sqrt{3}} \ \ \cssId{s107}{=\ \ (-1)\sqrt{(5)(3)}} \ \ \cssId{s108}{=\ \ -\sqrt{15}} $$ Be careful about this!

Master the ideas from this section
by practicing the exercise at the bottom of this page.


When you're done practicing, move on to:
the complex conjugate

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9 10 11 12 13
AVAILABLE MASTERED IN PROGRESS

(MAX is 13; there are 13 different problem types.)