Learning Outcomes
- Express roots of negative numbers in terms of [latex]i[/latex]
- Express imaginary numbers as [latex]bi[/latex] and complex numbers as [latex]a+bi[/latex]
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—they seem pretty real to us! When something is not real, we often say it is imaginary. We call this new number [latex]i[/latex] and it is used 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\,\cdot \,i=-1[/latex]. We also know that [latex] i\,\cdot \,i={{i}^{2}}[/latex], so we can conclude that [latex] {{i}^{2}}=-1[/latex].
[latex] {{i}^{2}}=-1[/latex]
The number [latex]−1[/latex] allows us to work with roots of all negative numbers, not just [latex] \sqrt{-1}[/latex]. There are two important rules to remember: [latex] \sqrt{-1}=i[/latex], and [latex] \sqrt{ab}=\sqrt{a}\sqrt{b}[/latex]. You will use these rules to rewrite the square root of a negative number as the square root of a positive number times [latex] \sqrt{-1}[/latex]. Next you will simplify the square root and rewrite [latex] \sqrt{-1}[/latex] as [latex]i[/latex]. Let us try an example.
Example
Simplify. [latex] \sqrt{-4}[/latex]
Example
Simplify. [latex] \sqrt{-18}[/latex]
Example
Simplify. [latex] -\sqrt{-72}[/latex]
You may have wanted to simplify [latex] -\sqrt{-72}[/latex] using different factors. Some may have thought of rewriting this radical as [latex] -\sqrt{-9}\sqrt{8}[/latex], or [latex] -\sqrt{-4}\sqrt{18}[/latex], or [latex] -\sqrt{-6}\sqrt{12}[/latex] for instance. Each of these radicals would have eventually yielded the same answer of [latex] -6i\sqrt{2}[/latex].
In the following video, we show more examples of how to use imaginary numbers to simplify a square root with a negative radicand.
Rewriting the Square Root of a Negative Number
- Find perfect squares within the radical.
- Rewrite the radical using the rule [latex] \sqrt{ab}=\sqrt{a}\cdot \sqrt{b}[/latex].
- Rewrite [latex] \sqrt{-1}[/latex] as [latex]i[/latex].
Example: [latex] \sqrt{-18}=\sqrt{9}\sqrt{-2}=\sqrt{9}\sqrt{2}\sqrt{-1}=3i\sqrt{2}[/latex]
Try It
Complex Numbers
A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written [latex]a+bi[/latex] where [latex]a[/latex] is the real part and [latex]bi[/latex] is the imaginary part. For example, [latex]5+2i[/latex] is a complex number. So, too, is [latex]3+4i\sqrt{3}[/latex].
Imaginary numbers are distinguished from real numbers because a squared imaginary number produces a negative real number. Recall, when a positive real number is squared, the result is a positive real number and when a negative real number is squared, again, the result is a positive real number. Complex numbers are a combination of real and imaginary numbers. You can use the usual operations (addition, subtraction, multiplication, and so on) with imaginary numbers. You will see more of that later.
| Complex Number | Real Part | Imaginary Part |
|---|---|---|
| [latex]3+7i[/latex] | [latex]3[/latex] | [latex]7i[/latex] |
| [latex]18–32i[/latex] | [latex]18[/latex] | [latex]−32i[/latex] |
| [latex] -\frac{3}{5}+i\sqrt{2}[/latex] | [latex] -\frac{3}{5}[/latex] | [latex] i\sqrt{2}[/latex] |
| [latex] \frac{\sqrt{2}}{2}-\frac{1}{2}i[/latex] | [latex] \frac{\sqrt{2}}{2}[/latex] | [latex]-\frac{1}{2}i[/latex] |
In a number with a radical as part of b, such as [latex]-\frac{3}{5}+i\sqrt{2}[/latex] above, the imaginary [latex]i[/latex] should be written in front of the radical. Though writing this number as [latex] -\frac{3}{5}+\sqrt{2}i[/latex] is technically correct, it makes it much 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. Look at these last two examples.
| Number | Complex Form: [latex]a+bi[/latex] |
Real Part | Imaginary Part |
|---|---|---|---|
| [latex]17[/latex] | [latex]17+0i[/latex] | [latex]17[/latex] | [latex]0i[/latex] |
| [latex]−3i[/latex] | [latex]0–3i[/latex] | [latex]0[/latex] | [latex]−3i[/latex] |
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 imaginary number can be expressed as a complex number. By making [latex]a=0[/latex], any imaginary number [latex]bi[/latex] can be written as [latex]0+bi[/latex] in complex form.
Example
Write [latex]83.6[/latex] as a complex number.
Try It
Example
Write [latex]−3i[/latex] as a complex number.
In the next video, we show more examples of how to write numbers as complex numbers.
Summary
Square roots of negative numbers can be simplified using [latex] \sqrt{-1}=i[/latex] and [latex] \sqrt{ab}=\sqrt{a}\sqrt{b}[/latex]. 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]. Imaginary numbers have the form [latex]bi[/latex] and can also be written as complex numbers by setting [latex]a=0[/latex].