We have noted many times in this chapter that square roots of negative numbers do not exist as real numbers. However, these “non-real numbers” do still have applications in many areas of math and also in the real world in fields such as electrical engineering.

You really need only one new number to start working with the square roots of negative numbers. That number is the square root of [latex]−1,\sqrt{-1}[/latex]. The real numbers are those that can be shown on a number line. When something is not real, we often say it is imaginary. So let us call this new number [latex]i[/latex] and use it to represent the square root of [latex]−1[/latex].

[latex]i=\sqrt{-1}[/latex]

Because [latex]\sqrt{x}\,\cdot \,\sqrt{x}=x[/latex], we can also see that [latex]\sqrt{-1}\,\cdot \,\sqrt{-1}=-1,[/latex] or [latex]i^2=-1[/latex]. Another way to say this is that [latex]i[/latex] is a solution to the equation [latex]x^2=-1[/latex] which previously did not have solutions.

The number [latex]i[/latex] allows us to work with roots of all negative numbers, not just [latex]\sqrt{-1}[/latex]. We will be using our previous Product Rule for Radicals, [latex]\sqrt{ab}=\sqrt{a}\sqrt{b}[/latex]. Be very careful using this rule with imaginary numbers though! Remember when we introduced the rule, we required that both [latex]\sqrt{a}[/latex] and [latex]\sqrt{b}[/latex] exist, which is not the case in this section! For this section we redefine the product rule to work for imaginary numbers.

Let us try an example of using this rule. If a radicand of a square root is negative, always apply this rule first before any other steps.

In the following video, we show more examples of how to use imaginary numbers to simplify a square root with a negative radicand.

Complex Numbers

For example, [latex]5+2i[/latex] is a complex number. So, too, is [latex]-3+4i\sqrt{3}[/latex]. Here are some examples of complex numbers, with real part and imaginary part labeled. Note that [latex]i[/latex] is not included in the imaginary part.

In a number with a radical as part of [latex]b[/latex], such as [latex]-\frac{3}{5}+i\sqrt{2}[/latex] above, it is acceptable to write the imaginary unit [latex]i[/latex] in front of the radical. Though writing this number as [latex]-\frac{3}{5}+\sqrt{2} \; i[/latex] is technically correct, it makes it more difficult to tell whether [latex]i[/latex] is inside or outside of the radical. Putting it before the radical, as in [latex]-\frac{3}{5}+i\sqrt{2}[/latex], clears up any confusion. While usually we will require answers to be in the form of a complex number [latex]a+bi[/latex], we make an exception if [latex]b[/latex] is a radical. Look at these last two examples.

By making [latex]b=0[/latex], any real number can be expressed as a complex number. The real number [latex]a[/latex] is written as [latex]a+0i[/latex] in complex form. Similarly, any pure imaginary number can be expressed as a complex number. By making [latex]a=0[/latex], any pure imaginary number [latex]bi[/latex] can be written as [latex]0+bi[/latex] in complex form.

In practice if the real or imaginary part of a complex number is $$0$$ we will not write that part at all (so we would leave it as $$-3i$$ rather than writing the real part of $$0$$).

Adding and Subtracting Complex Numbers

Our goal for the rest of this section is to prove the claim that the complex numbers form a number system just like the real numbers – they can be added, subtracted, multiplied, and divided. We will need some of these skills in the next chapter. In order for this claim to be true, we must be able to write each final answer as a complex number, in the form [latex]a+bi.[/latex]

First, consider the following expression.

[latex](7+6i)+(2+4i)[/latex]

If we think of [latex]i[/latex] as a variable for a moment, we just combine the like terms,

[latex](7+6i)+(2+4i)=9+10i[/latex]

The same rules apply to adding and subtracting complex numbers. You combine the corresponding parts and write the final answer again in the form [latex]a+bi.[/latex]

In the following video, we show more examples of how to add and subtract complex numbers (Note in this video that it incorrectly calls [latex]bi[/latex] the imaginary part instead of just [latex]b[/latex]).

Multiplying Radicals with Negative Radicands

We have to be careful when multiplying radicals with negative radicands. Remember to use the product rule for imaginary numbers first before any other steps.

Multiplying Complex Numbers

Summary

Complex numbers have the form [latex]a+bi[/latex], where [latex]a[/latex] and [latex]b[/latex] are real numbers and [latex]i[/latex] is the square root of [latex]−1[/latex]. All real numbers can be written as complex numbers by setting [latex]b=0[/latex]. Square roots of negative numbers can be simplified using [latex]\sqrt{-a}=i\sqrt{a}[/latex] where [latex]a>0.[/latex] We can perform arithmetic operations on complex numbers in much the same way as working with polynomials and combining like terms, making sure to simplify [latex]i^2=-1[/latex] when appropriate. Powers of [latex]i[/latex] form a cycle that repeats every four powers. Division of complex numbers is done by first making the denominator a real number using the complex conjugate multiplied on the top and bottom of the fraction.