# Complex Numbers

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

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Defining Complex Numbers

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, 11\begin{align*}\sqrt{11}\end{align*} , and 13\begin{align*}\frac{1}{3}\end{align*} are all examples of real numbers. With what we have previously learned, we cannot find 25\begin{align*} \sqrt{-25}\end{align*} 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 25\begin{align*} \sqrt{-25}\end{align*}.

25=251=51\begin{align*} \sqrt{-25}= \sqrt{25 \cdot -1}=5 \sqrt{-1}\end{align*}

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

25=251=51=5i\begin{align*} \sqrt{-25}= \sqrt{25 \cdot -1}=5 \sqrt{-1}=5i\end{align*}

All complex numbers have the form a+bi\begin{align*}a + bi\end{align*}, where a\begin{align*}a\end{align*} and b\begin{align*}b\end{align*} are real numbers. a\begin{align*}a\end{align*} is the real part of the complex number and b\begin{align*}b\end{align*} is the imaginary part. If b=0\begin{align*}b = 0\end{align*}, then a\begin{align*}a\end{align*} is left and the number is a real number. If a=0\begin{align*}a = 0\end{align*}, then the number is only bi\begin{align*}bi\end{align*} and called a pure imaginary number. If b0\begin{align*}b \ne 0\end{align*} and a0\begin{align*}a \ne 0\end{align*}, the number will be an imaginary number.

Let's find 162\begin{align*}\sqrt{-162}\end{align*} .

First pull out the i\begin{align*}i\end{align*}. Then, simplify 162\begin{align*}\sqrt{162}\end{align*} .

162=1162=i162=i812=9i2\begin{align*}\sqrt{-162}=\sqrt{-1} \cdot \sqrt{162}=i\sqrt{162}=i\sqrt{81 \cdot 2}=9i\sqrt{2}\end{align*}

#### Powers of i

In addition to now being able to take the square root of a negative number, i\begin{align*}i\end{align*} also has some interesting properties. Try to find i2,i3,\begin{align*}i^2,i^3,\end{align*} and i4\begin{align*}i^4\end{align*}.

Step 1: Write out i2\begin{align*}i^2\end{align*} and simplify. i2=ii=11=12=1\begin{align*}i^2=i \cdot i = \sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1}^2=-1\end{align*}

Step 2: Write out i3\begin{align*}i^3\end{align*} and simplify. i3=i2i=1i=i\begin{align*}i^3=i^2 \cdot i = -1 \cdot i = -i\end{align*}

Step 3: Write out i4\begin{align*}i^4\end{align*} and simplify. i4=i2i2=11=1\begin{align*}i^4=i^2 \cdot i^2 = -1 \cdot -1 = 1\end{align*}

Step 4: Write out i5\begin{align*}i^5\end{align*} and simplify. i5=i4i=1i=i\begin{align*}i^5=i^4 \cdot i = 1 \cdot i = i\end{align*}

Step 5: Write out i6\begin{align*}i^6\end{align*} and simplify. i6=i4i2=11=1\begin{align*}i^6=i^4 \cdot i^2 = 1 \cdot -1 = -1\end{align*}

Step 6: Do you see a pattern? Describe it and try to find i19\begin{align*}i^{19}\end{align*}.

You should see that the powers of i\begin{align*}i\end{align*} repeat every 4 powers. So, all the powers that are divisible by 4 will be equal to 1. To find i19\begin{align*}i^{19}\end{align*}, divide 19 by 4 and determine the remainder. That will tell you what power it is the same as.

i19=i16i3=1i3=i\begin{align*}i^{19}=i^{16} \cdot i^3 = 1 \cdot i^3=-i \end{align*}

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

1. i32\begin{align*}i^{32}\end{align*}

32 is divisible by 4, so i32=1\begin{align*}i^{32}=1\end{align*}.

1. i50\begin{align*}i^{50}\end{align*}

50÷4=12\begin{align*}50 \div 4=12\end{align*}, with a remainder of 2. Therefore, i50=i2=1\begin{align*}i^{50}=i^2=-1\end{align*}.

1. i7\begin{align*}i^7\end{align*}

7÷4=1\begin{align*}7 \div 4=1\end{align*}, with a remainder of 3. Therefore, i7=i3=i\begin{align*}i^7=i^3=-i\end{align*}

Finally, let's simplify the following complex expressions.

1. \begin{align*}(6-4i)+(5+8i)\end{align*}

\begin{align*}(6-4i)+(5+8i)={\color{red}6}{\color{blue}-4i}+{\color{red}5}+{\color{blue}8i}={\color{red}11}+{\color{blue}4i}\end{align*}

1. \begin{align*}9-(4+i)+(2-7i)\end{align*}

\begin{align*}9-(4+i)+(2-7i)={\color{red}9-4}{\color{blue}-i}+{\color{red}2}{\color{blue}-7i}={\color{red}7}{\color{blue}-8i}\end{align*}

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 \begin{align*}a + bi\end{align*}.

### Examples

#### Example 1

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

We're looking for \begin{align*}\sqrt{-460}\end{align*} .

First we need to pull out the \begin{align*}i\end{align*}. Then, we need to simplify \begin{align*}\sqrt{460}\end{align*} .

\begin{align*}\sqrt{-460}=\sqrt{-1} \cdot \sqrt{460}=i\sqrt{460}=i\sqrt{4 \cdot 115}=2i\sqrt{115}\end{align*}

#### Example 2

Simplify \begin{align*}\sqrt{-49}\end{align*}.

Rewrite \begin{align*}\sqrt{-49}\end{align*} in terms of \begin{align*}i\end{align*} and simplify the radical.

\begin{align*}\sqrt{-49}=i\sqrt{49}=7i\end{align*}

#### Example 3

Simplify \begin{align*}\sqrt{-125}\end{align*}.

Rewrite \begin{align*}\sqrt{-125}\end{align*} in terms of \begin{align*}i\end{align*} and simplify the radical.

\begin{align*}\sqrt{-125}=i\sqrt{125}=i\sqrt{25 \cdot 5}=5i\sqrt{5}\end{align*}

#### Example 4

Simplify \begin{align*}i^{210}\end{align*}.

\begin{align*}210 \div 4=52\end{align*}, with a remainder of 2. Therefore, \begin{align*}i^{210}=i^2=-1\end{align*}.

#### Example 5

Simplify \begin{align*}(8-3i)-(12-i)\end{align*}.

Distribute the negative and combine like terms.

\begin{align*}(8-3i)-(12-i)=8-3i-12+i=-4-2i\end{align*}

### Review

Simplify each expression and write in standard form.

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

### Answers for Review Problems

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

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### Vocabulary Language: English

TermDefinition
$i$ $i$ 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 $bi$.
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.