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5.8: Defining Complex Numbers

Difficulty Level: At Grade Created by: CK-12
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The coldest possible temperature, known as absolute zero is almost –460 degrees Fahrenheit. What is the square root of this number?

Watch This

First, watch this video.

Khan Academy: Introduction to i and Imaginary Numbers

Then, watch this video.

Khan Academy: Complex Numbers

Guidance

Before this concept, all numbers have been real numbers. 2, -5, 11 , and 13 are all examples of real numbers. Look at #1 from the Review Queue. With what we have previously learned, we cannot find 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 25.

25=251=51

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

25=251=51=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 b0 and a0, the number will be an imaginary number.

Example A

Find 162 .

Solution: First pull out the i. Then, simplify 162 .

162=1162=i162=i812=9i2

Investigation: 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 i2,i3, and i4.

1. Write out i2 and simplify. i2=ii=11=12=1

2. Write out i3 and simplify. i3=i2i=1i=i

3. Write out i4 and simplify. i4=i2i2=11=1

4. Write out i5 and simplify. i5=i4i=1i=i

5. Write out i6 and simplify. i6=i4i2=11=1

6. Do you see a pattern? Describe it and try to find i19.

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 i19, divide 19 by 4 and determine the remainder. That will tell you what power it is the same as.

i19=i16i3=1i3=i

Example B

Find:

a) i32

b) i50

c) i7

Solution:

a) 32 is divisible by 4, so i32=1.

b) 50÷4=12, with a remainder of 2. Therefore, i50=i2=1.

c) 7÷4=1, with a remainder of 3. Therefore, i7=i3=i

Example C

Simplify the complex expressions.

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

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

Solution: 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*}.

a) \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*}

b) \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*}

Intro Problem Revisit 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*}

Guided Practice

Simplify.

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

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

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

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

Answers

1. 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*}

2. 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*}

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

4. Distribute the negative and combine like terms.

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

Vocabulary

Imaginary Numbers
Any number with an \begin{align*}i\end{align*} associated with it. Imaginary numbers have the form \begin{align*}a + bi\end{align*} or \begin{align*}bi\end{align*}.
Complex Numbers
All real and imaginary numbers. Complex numbers have the standard form \begin{align*}a + bi\end{align*}, where \begin{align*}a\end{align*} or \begin{align*}b\end{align*} can be zero. \begin{align*}a\end{align*} is the real part and \begin{align*}bi\end{align*} is the imaginary part.
Pure Imaginary Numbers
An imaginary number without a real part, only \begin{align*}bi\end{align*}.

Practice

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*}

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Vocabulary

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.

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Difficulty Level:
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Date Created:
Mar 12, 2013
Last Modified:
Sep 07, 2016
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