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Negative Square Root: Definition & Overview

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  • 0:00 What Is a Square Root?
  • 1:00 What Is the Negative…
  • 2:46 The Imaginary Unit
  • 4:17 Operations With…
  • 5:57 Powers of the Imaginary Unit
  • 7:26 Lesson Summary
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Lesson Transcript
Instructor: David Liano

David has a Master of Business Administration, a BS in Marketing, and a BA in History.

After completing this lesson, you will know how to write the square root of a negative number. You will also be able to complete mathematical operations that involve square roots of negative numbers.

What Is A Square Root?

Before we learn what a negative square root is, let's first define what a square root is. The number a is the square root of b in the expression a^2 = b. This means that if you multiply a by itself, or a by a, you will get b. Let's plug real numbers into that equation, where a is 4:

4^2=16

This means that 4 times 4 is 16, and therefore 4 is the square root of 16.

A positive number has two square roots: one is positive and one is negative. If we have a positive number b, then its square roots are written as shown in Figure 1. The negative square root of b has the negative sign.


Figure 1
square root property


Let's again look at an actual number. The two square roots of 16 are 4 and -4 because 4^2 = 16 and (-4)^2 = 16 as seen in the following Figure 2.


Figure 2
example


What Is The Negative Square Root?

As shown earlier, a negative square root is one of two square roots of a positive number. For the number 25, its negative square root is -5 because (-5)^2 = 25.

We can solve certain equations by finding the square root of a number. Let's consider the equation of x^2 = 121. We want to solve for x, so we need to take the square root of both sides of the equation as shown in Figure 3.


Figure 3
example


We use the symbol ± because we need to consider both square roots of 121. This symbol is read as 'plus or minus the square root of 121.' The solution to the problem is +11 or -11. We can check this by plugging the answers into the original equation:

11^2 = 121

(-11)^2 = 121

What about the square root of a negative number? For instance, what is the square root of -9? We can try 3, but 3 x 3 = 9. We can try -3, but (-3) x (-3) = 9. This dilemma is due to the fact that the square root of any real number x cannot be negative. Therefore, the square root of a negative number does not exist, at least not within the system of real numbers.

We should recall that real numbers include all the rational numbers (e.g., the whole numbers 0 and 7, the integer -5, and the fraction 2/3) as well as the irrational numbers (like pi and square root of 3).

However, mathematicians overcame this problem of square roots of negative numbers by creating the imaginary unit.

The Imaginary Unit

The imaginary unit i is defined as the square root of -1.

A primary reason for creating the imaginary unit was for solving quadratic equations that have no real number solutions. Let's consider a simple quadratic equation such as the following:

x^2 + 4 = 0

If we solve for x, we will get x = ± square root of -4. What then are possible values of x?

2 x 2 does not equal -4, and (-2) x (-2) does not equal -4.

This tells us that the graph of this quadratic equation does not have any solutions that are real numbers. In other words, it does not cross the x axis.

However, we can give it imaginary solutions. We can use the product property of square roots and re-write the square root of -4 as shown in Figure 4. We isolated the imaginary number, which gave us a positive number 4 under the other square root symbol. We replace the square root of -1 with i and finish simplifying as normal.


Figure 4
imaginary number


There, we have found the solutions for the quadratic equation x^2 + 4 = 0, albeit they are imaginary solutions.

Operations With Imaginary Numbers

Now let's look at operations with imaginary numbers and start off with a simple example:

  • Simplify the square root of -18 plus the square root of -50.

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