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3.1 Complex Numbers

Square roots of negative numbers rewritten as multiples of i, plotting a + bi as a point on the complex plane, and the four arithmetic operations on such numbers.

Precalculus 2e, 3.1 Complex Numbers

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40 exercises

Expressing an Imaginary Number in Standard Form

Express −9 \sqrt{- 9} in standard form.

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3.1 Exercise 2

Express −24 \sqrt{- 24} in standard form.

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Adding Complex Numbers

Add 3−4i 3 - 4 i and 2+5i. 2 + 5 i .

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3.1 Exercise 6

Subtract 2+5i 2 + 5 i from 3–4i. 3 – 4 i .

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Multiplying a Complex Number by a Real Number

Find the product 4(2+5i). 4 \left(2 + 5 i\right) .

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3.1 Exercise 8

Find the product −4(2+6i). - 4 \left(2 + 6 i\right) .

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Multiplying a Complex Number by a Complex Number

Multiply ( 4+3i )(2−5i). \left(4 + 3 i\right) \left(2 - 5 i\right) .

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3.1 Exercise 10

Multiply (3−4i)(2+3i). \left(3 - 4 i\right) \left(2 + 3 i\right) .

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Finding Complex Conjugates

Find the complex conjugate of each number.

  1. ⓐ 2+i 5 2 + i \sqrt{5}
  2. ⓑ − 1 2 i - \frac{1}{2} i
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Dividing Complex Numbers

Divide ( 2+5i ) \left(2 + 5 i\right) by ( 4−i ). \left(4 - i\right) .

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3.1 Exercise 14

Let f(x)=2 x 2 −3x. f \left(x\right) = 2 x^{2} - 3 x . Evaluate f( 8−i ). f \left(8 - i\right) .

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Substituting an Imaginary Number in a Rational Function

Let f( x )= 2+x x+3 . f \left(x\right) = \frac{2 + x}{x + 3} . Evaluate f( 10i ). f \left(10 i\right) .

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3.1 Exercise 16

Let f(x)= x+1 x−4 . f \left(x\right) = \frac{x + 1}{x - 4} . Evaluate f( −i ). f \left(- i\right) .

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Simplifying Powers of i

Evaluate i 35 . i^{35} .

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3.1 Exercise 18

Explain how to add complex numbers.

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3.1 Exercise 20

Give an example to show the product of two imaginary numbers is not always imaginary.

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3.1 Exercise 22

Evaluate the algebraic expressions.

Iff(x)= x 2 +x−4, \text{If} f \left(x\right) = x^{2} + x - 4 , evaluate f(2i). f \left(2 i\right) .

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3.1 Exercise 24

Evaluate the algebraic expressions.

Iff(x)= x 2 +3x+5, \text{If} f \left(x\right) = x^{2} + 3 x + 5 , evaluate f(2+i). f \left(2 + i\right) .

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3.1 Exercise 26

Evaluate the algebraic expressions.

Iff(x)= x+1 2−x , \text{If} f \left(x\right) = \frac{x + 1}{2 - x} , evaluate f(5i). f \left(5 i\right) .

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3.1 Exercise 34

Perform the indicated operation and express the result as a simplified complex number.

( 3+2i )+(5−3i) \left(3 + 2 i\right) + \left(5 - 3 i\right)

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3.1 Exercise 36

Perform the indicated operation and express the result as a simplified complex number.

( −5+3i )−(6−i) \left(- 5 + 3 i\right) - \left(6 - i\right)

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3.1 Exercise 38

Perform the indicated operation and express the result as a simplified complex number.

(−4+4i)−(−6+9i) \left(- 4 + 4 i\right) - \left(- 6 + 9 i\right)

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3.1 Exercise 40

Perform the indicated operation and express the result as a simplified complex number.

( 5−2i )(3i) \left(5 - 2 i\right) \left(3 i\right)

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3.1 Exercise 42

Perform the indicated operation and express the result as a simplified complex number.

( −2+4i )( 8 ) \left(- 2 + 4 i\right) \left(8\right)

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3.1 Exercise 44

Perform the indicated operation and express the result as a simplified complex number.

( −1+2i )(−2+3i) \left(- 1 + 2 i\right) \left(- 2 + 3 i\right)

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3.1 Exercise 46

Perform the indicated operation and express the result as a simplified complex number.

( 3+4i )( 3−4i ) \left(3 + 4 i\right) \left(3 - 4 i\right)

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3.1 Exercise 48

Perform the indicated operation and express the result as a simplified complex number.

6−2i 3 \frac{6 - 2 i}{3}

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3.1 Exercise 50

Perform the indicated operation and express the result as a simplified complex number.

6+4i i \frac{6 + 4 i}{i}

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3.1 Exercise 52

Perform the indicated operation and express the result as a simplified complex number.

3+4i 2−i \frac{3 + 4 i}{2 - i}

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3.1 Exercise 54

Perform the indicated operation and express the result as a simplified complex number.

−9 +3 −16 \sqrt{- 9} + 3 \sqrt{- 16}

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3.1 Exercise 56

Perform the indicated operation and express the result as a simplified complex number.

2+ −12 2 \frac{2 + \sqrt{- 12}}{2}

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3.1 Exercise 58

Perform the indicated operation and express the result as a simplified complex number.

i 8 i^{8}

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3.1 Exercise 60

Perform the indicated operation and express the result as a simplified complex number.

i 22 i^{22}

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3.1 Exercise 62

Use a calculator to help answer the questions.

Evaluate (1−i) k \left(1 - i\right)^{k} for k=2, 6, and 10. k = \text{2},\textrm{ }\text{6},\textrm{ }\text{and}\textrm{ }\text{10} . Predict the value if k=14. k = 14.

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3.1 Exercise 64

Use a calculator to help answer the questions.

Show that a solution of x 6 +1=0 x^{6} + 1 = 0 is 3 2 + 1 2 i. \frac{\sqrt{3}}{2} + \frac{1}{2} i .

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3.1 Exercise 66

Evaluate the expressions, writing the result as a simplified complex number.

1 i + 4 i 3 \frac{1}{i} + \frac{4}{i^{3}}

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3.1 Exercise 68

Evaluate the expressions, writing the result as a simplified complex number.

i 7 ( 1+ i 2 ) i^{7} \left(1 + i^{2}\right)

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3.1 Exercise 70

Evaluate the expressions, writing the result as a simplified complex number.

( 2+i )( 4−2i ) (1+i) \frac{\left(2 + i\right) \left(4 - 2 i\right)}{\left(1 + i\right)}

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3.1 Exercise 72

Evaluate the expressions, writing the result as a simplified complex number.

( 3+i ) 2 ( 1+2i ) 2 \frac{\left(3 + i\right)^{2}}{\left(1 + 2 i\right)^{2}}

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3.1 Exercise 74

Evaluate the expressions, writing the result as a simplified complex number.

4+i i + 3−4i 1−i \frac{4 + i}{i} + \frac{3 - 4 i}{1 - i}

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Adapted from College Algebra © 2021 Rice University. Licensed under CC BY 4.0. Sources: College Algebra 2e, Algebra and Trigonometry 2e, Precalculus 2e, College Algebra with Corequisite Support 2e. Imported at revision 4922e46.