# 5.3.2: The Complex Numbers

- Page ID
- 4160

\(a + bi\), 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

**of the complex number and b is the**

**real part****. If b=0, then a is left and the number is a**

**imaginary part****If a=0, then the number is only bi and called a**

**real number.****. If b≠0 and a≠0, the number will be an imaginary number.**

**pure 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**

**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.

- \(i^{32}\)

32 is divisible by 4, so \(i^{32}=1\).

- \(i^{50}\)

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

- \(i^7\)

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

Finally, let's simplify the following complex expressions.

- \((6−4i)+(5+8i)\)

\((6−4i)+(5+8i)=6−4i+5+8i=11+4i\)

- \(9−(4+i)+(2−7i)\)

\(9−(4+i)+(2−7i)=9−4−i+2−7i=7−8i\)

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.

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

**Answers for Review Problems**

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. |

## Additional Resources

Interactive Element

Video: Writing Complex Numbers in Standard Form - Example 1

Practice: The Complex Numbers

Real World: Getting Lost in a Pineapple