# 5.3.2: The Complex Numbers

$$a + bi$$, the sum of a real and an imaginary number.

The coldest possible temperature, known as absolute zero is almost –460 degrees Fahrenheit. What is the square root of this number?

### Complex Numbers

Before this concept, all numbers have been real numbers. 2, -5, $$\sqrt{11}$$, and $$\dfrac{1}{3}$$ are all examples of real numbers. With what we have previously learned, we cannot find $$\sqrt{−25}$$ because you cannot take the square root of a negative number. There is no real number that, when multiplied by itself, equals -25. Let’s simplify $$\sqrt{−25}$$.

$$\sqrt{−25}=\sqrt{25\cdot −1}=5\sqrt{−1}$$

In order to take the square root of a negative number we are going to assign $$\sqrt{−1}$$ a variable, $$i$$. $$i$$ represents an imaginary number. Now, we can use i to take the square root of a negative number.

$$\sqrt{−25}=\sqrt{25\cdot −1}=5\sqrt{−1}=5i$$

All complex numbers have the form a+bi, where a and b are real numbers. a is the real part of the complex number and b is the imaginary part. If b=0, then a is left and the number is a real number. If a=0, then the number is only bi and called a pure imaginary number. If b≠0 and a≠0, the number will be an imaginary number.

Let's find $$\sqrt{−162}$$.

First pull out the i. Then, simplify $$\sqrt{−162}$$.

$$\sqrt{−162}=\sqrt{−1}\cdot \sqrt{162}=i\sqrt{162}=i \sqrt{81\cdot 2}=9i\sqrt{2}$$

#### Powers of i

In addition to now being able to take the square root of a negative number, i also has some interesting properties. Try to find $$i^2$$, $$i^3$$, and $$i^4$$.

Step 1: Write out $$i^2$$ and simplify. $$i^2=i\cdot i=\sqrt{−1} \cdot \sqrt{−1}=\sqrt{−1}^2=−1$$

Step 2: Write out $$i^3$$ and simplify. $$i^3=i^2\cdot i=−1\cdot i=−i$$

Step 3: Write out $$i^4$$ and simplify. $$i^4=i^2\cdot i^2=−1\cdot −1=1$$

Step 4: Write out $$i^5$$ and simplify. $$i^5=i^4\cdot i=1\cdot i=i$$

Step 5: Write out $$i^6$$ and simplify. $$i^6=i^4\cdot i^2=1\cdot −1=−1$$

Step 6: Do you see a pattern? Describe it and try to find $$i^{19}$$.

You should see that the powers of i repeat every 4 powers. So, all the powers that are divisible by 4 will be equal to 1. To find i^{19}, divide 19 by 4 and determine the remainder. That will tell you what power it is the same as.

$$i^{19}=i^{16}\cdot i^3=1\cdot i^3=−i$$

Now, let's find the following powers of i.

1. $$i^{32}$$

32 is divisible by 4, so $$i^{32}=1$$.

1. $$i^{50}$$

$$50\div 4=12$$, with a remainder of 2. Therefore, $$i^{50}=i^2=−1$$.

1. $$i^7$$

$$7\div 4=1$$, with a remainder of 3. Therefore, $$i^7=i^3=−i$$

Finally, let's simplify the following complex expressions.

1. $$(6−4i)+(5+8i)$$

$$(6−4i)+(5+8i)=6−4i+5+8i=11+4i$$

1. $$9−(4+i)+(2−7i)$$

$$9−(4+i)+(2−7i)=9−4−i+2−7i=7−8i$$

To add or subtract complex numbers, you need to combine like terms. Be careful with negatives and properly distributing them. Your answer should always be in standard form, which is a+bi.

Example $$\PageIndex{1}$$

Earlier, you were asked to find the square root of -460 degrees.

Solution

We're looking for $$\sqrt{−460}$$ .

First we need to pull out the i. Then, we need to simplify $$\sqrt{−460}$$.

$$\sqrt{−460}=\sqrt{−1}\cdot \sqrt{460}=i\sqrt{460}=i\sqrt{4\cdot 115}= 2i \sqrt{115}$$

Example $$\PageIndex{2}$$

Simplify $$\sqrt{−49}$$.

Solution

Rewrite $$\sqrt{−49}$$ in terms of i and simplify the radical.

$$\sqrt{−49}=i\sqrt{49}=7i$$

Example $$\PageIndex{3}$$

Simplify $$\sqrt{−125}$$.

Solution

Rewrite $$\sqrt{−125}$$ in terms of i and simplify the radical.

$$\sqrt{−125}=i\sqrt{125}=i\sqrt{25\cdot 5}=5i\sqrt{5}$$

Example $$\PageIndex{4}$$

Simplify $$i^{210}$$.

Solution

$$210\div 4=52$$, with a remainder of 2. Therefore, $$i^{210}=i^2=−1$$.

Example $$\PageIndex{5}$$

Simplify $$(8−3i)−(12−i)$$.

Solution

Distribute the negative and combine like terms.

$$(8−3i)−(12−i)=8−3i−12+i=−4−2i$$

### Review

Simplify each expression and write in standard form.

1. $$\sqrt{−9}$$
2. $$\sqrt{−242}$$
3. $$6\sqrt{−45}$$
4. $$−12i \sqrt{98}$$
5. $$\sqrt{−32}\cdot \sqrt{−27}$$
6. $$7i\sqrt{-126}$$
7. $$i^8$$
8. $$16i^{22}$$
9. $$−9 i^{65}$$
10. $$i^{365}$$
11. $$2i^{91}$$
12. $$-\sqrt{16}{80}$$
13. $$(11−5i)+(6−7i)$$
14. $$(14+2i)−(20+9i)$$
15. $$(8−i)−(3+4i)+15i$$
16. $$−10i−(1−4i)$$
17. $$(0.2+1.5i)−(−0.6+i)$$
18. $$6+(18−i)−(2+12i)$$
19. $$−i+(19+22i)−(8−14i)$$
20. $$18−(4+6i)+(17−9i)+24i$$

To see the Review answers, open this PDF file and look for section 5.8.

## Vocabulary

Term Definition
i i is an imaginary number. $$i=\sqrt{−1}$$.
Absolute Value The absolute value of a number is the distance the number is from zero. The absolute value of a complex number is the distance from the complex number on the complex plane to the origin.
Complex Conjugate Complex conjugates are pairs of complex binomials. The complex conjugate of $$a+bi$$ is $$a−bi$$. When complex conjugates are multiplied, the result is a single real number.
i i is an imaginary number. $$i=\sqrt{−1}$$.
Imaginary Number An imaginary number is a number that can be written as the product of a real number and i.
imaginary part The imaginary part of a complex number $$a+bi$$ is $$b$$i.
Pure Imaginary Numbers The pure imaginary numbers are the subset of complex numbers without real parts, only bi.
Real Number A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.
real part The real part of a complex number $$a+bi$$ is $$a$$.
rectangular coordinates A point is written using rectangular coordinates if it is written in terms of $$x$$ and $$y$$ and can be graphed on the Cartesian plane.
rectangular form The rectangular form of a point or a curve is given in terms of $$x$$ and $$y$$ and is graphed on the Cartesian plane.
standard form The standard form of a complex number is $$a+bi$$ where $$a$$ and $$b$$ are real numbers.