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Complex Numbers

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Defining Complex Numbers
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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, \sqrt{11} , and \frac{1}{3} are all examples of real numbers. Look at #1 from the Review Queue. 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 \ne 0 and a \ne 0 , the number will be an imaginary number.

Example A

Find \sqrt{-162} .

Solution: 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}

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 i^2,i^3, and i^4 .

1. Write out i^2 and simplify. i^2=i \cdot i = \sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1}^2=-1

2. Write out i^3 and simplify. i^3=i^2 \cdot i = -1 \cdot i = -i

3. Write out i^4 and simplify. i^4=i^2 \cdot i^2 = -1 \cdot -1 = 1

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

5. Write out i^6 and simplify. i^6=i^4 \cdot i^2 = 1 \cdot -1 = -1

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

Example B

Find:

a) i^{32}

b) i^{50}

c) i^7

Solution:

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

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

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

Example C

Simplify the complex expressions.

a) (6-4i)+(5+8i)

b) 9-(4+i)+(2-7i)

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 a + bi .

a) (6-4i)+(5+8i)={\color{red}6}{\color{blue}-4i}+{\color{red}5}+{\color{blue}8i}={\color{red}11}+{\color{blue}4i}

b) 9-(4+i)+(2-7i)={\color{red}9-4}{\color{blue}-i}+{\color{red}2}{\color{blue}-7i}={\color{red}7}{\color{blue}-8i}

Intro Problem Revisit 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}

Guided Practice

Simplify.

1. \sqrt{-49}

2. \sqrt{-125}

3. i^{210}

4. (8-3i)-(12-i)

Answers

1. Rewrite \sqrt{-49} in terms of i and simplify the radical.

\sqrt{-49}=i\sqrt{49}=7i

2. Rewrite \sqrt{-125} in terms of i and simplify the radical.

\sqrt{-125}=i\sqrt{125}=i\sqrt{25 \cdot 5}=5i\sqrt{5}

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

4. Distribute the negative and combine like terms.

(8-3i)-(12-i)=8-3i-12+i=-4-2i

Vocabulary

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

Practice

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. -9i^{65}
  10. i^{365}
  11. 2i^{91}
  12. \sqrt{-\frac{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

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