Imaginary and Complex Numbers

Learning Outcomes

Express roots of negative numbers in terms of i
Express imaginary numbers as bi and complex numbers as $a+bi$

You really need only one new number to start working with the square roots of negative numbers. That number is the square root of $−1,\sqrt{-1}$ . 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. So let us call this new number i and use it to represent the square root of $−1$ .

$i=\sqrt{-1}$

Because $\sqrt{x}\,\cdot \,\sqrt{x}=x$ , we can also see that $\sqrt{-1}\,\cdot \,\sqrt{-1}=-1$ or $i\,\cdot \,i=-1$ . We also know that $i\,\cdot \,i={{i}^{2}}$ , so we can conclude that ${{i}^{2}}=-1$ .

${{i}^{2}}=-1$

The number i allows us to work with roots of all negative numbers, not just $\sqrt{-1}$ . There are two important rules to remember: $\sqrt{-1}=i$ , and $\sqrt{ab}=\sqrt{a}\sqrt{b}$ . You will use these rules to rewrite the square root of a negative number as the square root of a positive number times $\sqrt{-1}$ . Next you will simplify the square root and rewrite $\sqrt{-1}$ as i. Let us try an example.

Example

Simplify. $\sqrt{-4}$

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Use the rule $\sqrt{ab}=\sqrt{a}\sqrt{b}$ to rewrite this as a product using $\sqrt{-1}$ .

$\sqrt{-4}=\sqrt{4\cdot -1}=\sqrt{4}\sqrt{-1}$

Since $4$ is a perfect square $(4=2^{2})$ , you can simplify the square root of $4$ .

$\sqrt{4}\sqrt{-1}=2\sqrt{-1}$

Use the definition of i to rewrite $\sqrt{-1}$ as i.

$2\sqrt{-1}=2i$

The answer is $2i$ .

Example

Simplify. $\sqrt{-18}$

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Use the rule $\sqrt{ab}=\sqrt{a}\sqrt{b}$ to rewrite this as a product using $\sqrt{-1}$ .

$\sqrt{-18}=\sqrt{18\cdot -1}=\sqrt{18}\sqrt{-1}$

Since $18$ is not a perfect square, use the same rule to rewrite it using factors that are perfect squares. In this case, $9$ is the only perfect square factor, and the square root of $9$ is $3$ .

$\sqrt{18}\sqrt{-1}=\sqrt{9}\sqrt{2}\sqrt{-1}=3\sqrt{2}\sqrt{-1}$

Use the definition of i to rewrite $\sqrt{-1}$ as i.

$3\sqrt{2}\sqrt{-1}=3\sqrt{2}i=3i\sqrt{2}$

Remember to write i in front of the radical.

The answer is $3i\sqrt[{}]{2}$ .

Example

Simplify. $-\sqrt{-72}$

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Use the rule $\sqrt{ab}=\sqrt{a}\sqrt{b}$ to rewrite this as a product using $\sqrt{-1}$ .

$-\sqrt{-72}=-\sqrt{72\cdot -1}=-\sqrt{72}\sqrt{-1}$

Since $72$ is not a perfect square, use the same rule to rewrite it using factors that are perfect squares. Notice that $72$ has three perfect squares as factors: $4, 9$ , and $36$ . It is easiest to use the largest factor that is a perfect square.

$-\sqrt{72}\sqrt{-1}=-\sqrt{36}\sqrt{2}\sqrt{-1}=-6\sqrt{2}\sqrt{-1}$

Use the definition of i to rewrite $\sqrt{-1}$ as i.

$-6\sqrt{2}\sqrt{-1}=-6\sqrt{2}i=-6i\sqrt{2}$

Remember to write i in front of the radical.

The answer is $-6i\sqrt[{}]{2}$

You may have wanted to simplify $-\sqrt{-72}$ using different factors. Some may have thought of rewriting this radical as $-\sqrt{-9}\sqrt{8}$ , or $-\sqrt{-4}\sqrt{18}$ , or $-\sqrt{-6}\sqrt{12}$ for instance. Each of these radicals would have eventually yielded the same answer of $-6i\sqrt{2}$ .

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 $\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}$ .
Rewrite $\sqrt{-1}$ as i.

Example: $\sqrt{-18}=\sqrt{9}\sqrt{-2}=\sqrt{9}\sqrt{2}\sqrt{-1}=3i\sqrt{2}$

Complex Numbers

Showing the real and imaginary parts of 5 + 2i. In this complex number, 5 is the real part and 2i is the complex part.

A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written a + bi where a is the real part and bi is the imaginary part. For example, $5+2i$ is a complex number. So, too, is $3+4i\sqrt{3}$ .

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
$3+7i$	$3$	$7i$
$18–32i$	$18$	$−32i$
$-\frac{3}{5}+i\sqrt{2}$	$-\frac{3}{5}$	$i\sqrt{2}$
$\frac{\sqrt{2}}{2}-\frac{1}{2}i$	$\frac{\sqrt{2}}{2}$	$-\frac{1}{2}i$

In a number with a radical as part of b, such as $-\frac{3}{5}+i\sqrt{2}$ above, the imaginary i should be written in front of the radical. Though writing this number as $-\frac{3}{5}+\sqrt{2}i$ is technically correct, it makes it much more difficult to tell whether i is inside or outside of the radical. Putting it before the radical, as in $-\frac{3}{5}+i\sqrt{2}$ , clears up any confusion. Look at these last two examples.

Number	Complex Form: $a+bi$	Real Part	Imaginary Part
$17$	$17+0i$	$17$	$0i$
$−3i$	$0–3i$	$0$	$−3i$

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

Example

Write $83.6$ as a complex number.

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Remember that a complex number has the form $a+bi$ . You need to figure out what a and b need to be.

$a+bi$

Since $83.6$ is a real number, it is the real part (a) of the complex number $a+bi$ .

$83.6+bi$

A real number does not contain any imaginary parts, so the value of $b$ is $0$ .

The answer is $83.6+0i$ .

Example

Write $−3i$ as a complex number.

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Remember that a complex number has the form $a+bi$ . You need to figure out what a and b need to be.

$a+bi$

Since $−3i$ is an imaginary number, it is the imaginary part $bi$ of the complex number $a+bi$ .

$a–3i$

This imaginary number has no real parts, so the value of $a$ is $0$ .

The answer is $0–3i$ .

In the next video, we show more examples of how to write numbers as complex numbers.

Summary

Complex numbers have the form $a+bi$ , where a and b are real numbers and i is the square root of $−1$ . All real numbers can be written as complex numbers by setting $b=0$ . Imaginary numbers have the form bi and can also be written as complex numbers by setting $a=0$ . Square roots of negative numbers can be simplified using $\sqrt{-1}=i$ and $\sqrt{ab}=\sqrt{a}\sqrt{b}$ .

Module 3: Quadratic Equations and Functions