Complex Numbers

Problem: 1

Write using i:

Problem: 3

Write using i:

Problem: 5

Write using i:

Problem: 7

Simplify.

6i²

Problem: 9

Simplify.

–6i²

Problem: 11

Simplify.

Problem: 13

Simplify.

Problem: 15

Simplify.

(–2i) (–3i)(7i)

Problem: 17

Simplify.

Problem: 19

Simplify.

Problem: 21

Perform the addition and give the result as real numbers or as multiples of i.

5i + 6i

Problem: 23

Multiply and give the result as real numbers or as multiples of i.

(6 i)(7i)

Problem: 25

Perform the division and give the result as real numbers or as multiples of i.

Problem: 27

Evaluate the given expression and give your answer in a + bi form.

5i(4 + 7i)

Problem: 29

Evaluate the given expression and give your answer in a + bi form.

(–6 + i)(7 – 9i)

Problem: 31a

The value of i² by definition is –1. From this, we get, i³ = i² ·i = –1 ·i = –i, i⁴ = i³ ·i = –i ·i = –i² = –(–1) = 1. Similarly, i⁵ = i,i⁶ = –1, i⁷ = –i and i⁸ = 1. By continuing this pattern, evaluate the values of i⁹, i¹⁰,i¹¹, and i¹².

Problem: 31b

Generalize the result obtained in part (a) of this question and predict the values of i²⁰⁰⁴, i⁴⁰⁰¹ and i⁸⁰⁰⁰³.

Problem: 33

Simplify.

(2 – i) + (7 + 4i)

Problem: 35

Simplify.

(9 – 5i) – (3 + 5i)

Problem: 37

Simplify.

Problem: 39

Simplify.

(5 – i)(6 + 3i)

Problem: 41

Simplify.

Problem: 43

Simplify.

Problem: 45

Simplify.

Problem: 47

Simplify.

Problem: 49

Simplify.

Problem: 51

Graph the number in the complex plane.

5 – i

Problem: 53

Graph the number in the complex plane.

–4 + 3i

Problem: 55

Graph the number in the complex plane.

–4i

Problem: 57

Determine the absolute value of the complex number.

6 + 8i

Problem: 59

Determine the absolute value of the complex number.

–5 + 2i

Problem: 61

Determine the absolute value of the complex number.

7 – 9i

Problem: 63

Determine the absolute value of the complex number.

Problem: 65

Find if the complex number c belongs to the Mandelbrot set. Use absolute value to justify your answer.

c = –1

Problem: 67

Find if the complex number c belongs to the Mandelbrot set. Use absolute value to justify your answer.

c = –2 – i

Problem: 69

Simplify, and graph the number on the Mandelbrot set. Does the number appear to be in the set?

(8.2 – 1.7i) – (9.5 – 1.7i)

Problem: 71

Simplify, and graph the number on the Mandelbrot set. Does the number appear to be in the set?

(1 + 0.07i)(0.251 + 0.907i)

Problem: 73

Find x and y if 5x + 4yi = –15 + 20i.

Problem: 75

Find x and y if –18 – 7i = 3x + yi

Problem: 1

Write using i:

Problem: 3

Write using i:

Problem: 5

Write using i:

Problem: 7

Simplify.

6i²

Problem: 9

Simplify.

–6i²

Problem: 11

Simplify.

Problem: 13

Simplify.

Problem: 15

Simplify.

(–2i) (–3i)(7i)

Problem: 17

Simplify.

Problem: 19

Simplify.

Problem: 21

Perform the addition and give the result as real numbers or as multiples of i.

5i + 6i

Problem: 23

Multiply and give the result as real numbers or as multiples of i.

(6i)(7i)

Problem: 25

Perform the division and give the result as real numbers or as multiples of i.

Problem: 27

Evaluate the given expression and give your answer in a + bi form.

5i(4 + 7i)

Problem: 29

Evaluate the given expression and give your answer in a + bi form.

(–6 + i)(7 – 9i)

Problem: 31a

The value of i² by definition is –1. From this, we get, i³ = i² ·i = –1 ·i = –i, i⁴ = i³ ·i = –i ·i = –i² = –(–1) = 1. Similarly, i⁵ = i, i⁶ = –1, i⁷ = –i and i⁸ = 1. By continuing this pattern, evaluate the values of i⁹, i¹⁰,i¹¹, and i¹².