## 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. $\sqrt{-4}$

### Example

Simplify. $\sqrt{-18}$

### Example

Simplify. $-\sqrt{-72}$

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

A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written + 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.

### Example

Write $−3i$ as a complex number.

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 $\sqrt{ab}=\sqrt{a}\sqrt{b}$.