The Square Root of a Negative Number

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<\overbrace<\ \ \ \sqrt\ \ \ >^>> $ equals, by definition, the unique nonnegative real number which, when squared, equals $\,k\,.$

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

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

• it is nonnegative (not negative; i.e., greater than or equal to zero)
• when squared, it gives $\,k\,$

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

• $\,3\,$ is nonnegative
• $\,3\,,$ when squared, gives $\,9\,$: $\,3^2 = 9$

So, what is (say) $\,\sqrt\ $? 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 $\,\sqrt\,.$ Remember the key fact: $\,i:=\sqrt\,,$ so that $\,i^2=-1\,.$

Let $\,p\,$ be a positive real number, so that $\,-p\,$ is a negative real number. Then:

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

Observe that $\,i\sqrtp>\,,$ when squared, does indeed give $\,-p\,$:

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 $\,i\,$!

Examples

Here are some examples:

• $\sqrt = i\sqrt = i2 = 2i\,$ It is conventional to write $\,2i\,,$ not $\,i2\,.$ When there's no possible misinterpretation (see below), write a real number multiplier before the $\,i\,.$
• $\sqrt = i\sqrt$ In this situation, it's actually better to leave the $\,\sqrt\,$ after the $\,i\,.$ Why? If you pull the $\,\sqrt\,$ to the front, it can lead to a misinterpretation, as follows: The numbers $\,\sqrti\,$ and $\,\sqrt\,$ are different numbers—look carefully! In $\ \sqrti\ $, the $\,i\,$ is outside the square root. In $\ \sqrt\ $, the $\,i\,$ is inside the square root. Unless you look carefully, however, you might mistake one of these for the other. By writing $\,i\sqrt\,,$ you eliminate any possible confusion.

Two Different Questions; Two Different Answers

Recall these two different questions, with two different answers:

• What is $\,\sqrt\,$? Answer: By definition, $\,\sqrt = 2\,.$ The number $\ \sqrt\ $ is the nonnegative number which, when squared, gives $\,4\,.$
• What are the solutions to the equation $\,x^2 = 4\,$? Answer: $\,x= \pm\ 2\,.$ There are two different real numbers which, when squared, give $\,4\,.$

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

• What is $\,\sqrt\,$? Answer: $\sqrt = 2i\,$
• What are the (complex) solutions to the equation $\,x^2 = -4\,$? Answer: $\,x=\pm \sqrt = \pm\ 2i\,.$ There are two different complex numbers which, when squared, give $\,-4\,.$ Verifying the second solution: $$ \cssId\cssId\cssId\cssId$$

Square Roots of Products

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 $\,-4\,$:

Here is the correct approach:

Similarly, $\,\sqrt\sqrt\,$ is not equal to $\,\sqrt\,.$ Instead, here is the correct simplication: