Mathspresso
Math II · N-CN.2

Adding, Subtracting, and Multiplying Complex Numbers

Complex arithmetic shows that numbers of the form `a + bi` are not just symbols; they form a working number system with consistent operations.

Concept Number and Quantity
Domain The Complex Number System
Read time 1 minutes

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Problem Library

Problems in the App From This Objective

144 problems across 12 archetypes in the app.

combine real parts and imaginary parts.
12 problems Warmup Practice Mixed Review Assessment
Problem 1

Add complex numbers 3+4i and 5+2i.

Problem 2

Add complex numbers -2+7i and 6-3i.

Problem 3

Add complex numbers 9-i and -4+5i.

Problem 4

Add complex numbers -3-2i and 1-6i.

Problem 5

Add complex numbers 7+3i and 2-5i.

Problem 6

Add complex numbers -1+8i and 4+2i.

Problem 7

Add complex numbers -5-i and -3+7i.

Problem 8

Add complex numbers 10 and 6i.

Problem 9

Add complex numbers 2-9i and -7.

Problem 10

Add complex numbers -4+3i and -2i.

Open in simulator
Problem 11

Add complex numbers 15-12i and -8+10i.

Problem 12

Add complex numbers -1-i and -2-3i.

distribute negative and combine parts.
12 problems Warmup Practice Mixed Review Assessment
Problem 13

Subtract complex numbers 7+5i - (2+3i).

Problem 14

Subtract complex numbers 4-6i - (-1+2i).

Problem 15

Subtract complex numbers -3+i - (5-4i).

Problem 16

Subtract complex numbers 2+7i - (2+10i).

Problem 17

Subtract complex numbers 10+3i - (4-2i).

Open in simulator
Problem 18

Subtract complex numbers -5+8i - (-2+3i).

Problem 19

Subtract complex numbers 6-9i - (3-5i).

Problem 20

Subtract complex numbers 8i - (3+2i).

Problem 21

Subtract complex numbers 7 - (-1-4i).

Problem 22

Subtract complex numbers -2-3i - (5-i).

Problem 23

Subtract complex numbers 9+4i - (2+4i).

Problem 24

Subtract complex numbers 5-2i - (5+6i).

distribute scalar to both parts.
12 problems Warmup Practice Mixed Review Assessment
Problem 25

Multiply complex number 3+4i by real scalar 2.

Problem 26

Multiply complex number -2+5i by real scalar -3.

Open in simulator
Problem 27

Multiply complex number 6-9i by real scalar 1/3.

Problem 28

Multiply complex number 4+i by real scalar -1/2.

Problem 29

Multiply complex number 1+i by real scalar 5.

Problem 30

Multiply complex number 7-2i by real scalar -1.

Problem 31

Multiply complex number -3-6i by real scalar 4.

Problem 32

Multiply complex number 5i by real scalar 0.5.

Problem 33

Multiply complex number -10 by real scalar -0.2.

Problem 34

Multiply complex number 0 by real scalar 7.

Problem 35

Multiply complex number 2/3 + 1/2i by real scalar 6.

Problem 36

Multiply complex number -4+3i by real scalar -1/4.

use `i^2=-1`.
12 problems Warmup Practice Mixed Review Assessment
Problem 37

Multiply complex number 3+4i by pure imaginary number 2i.

Problem 38

Multiply complex number 5-i by pure imaginary number 3i.

Problem 39

Multiply complex number -2+6i by pure imaginary number -i.

Problem 40

Multiply complex number 1+7i by pure imaginary number 4i.

Problem 41

Multiply complex number 2+5i by pure imaginary number 3i.

Problem 42

Multiply complex number -4-i by pure imaginary number 2i.

Problem 43

Multiply complex number 7+2i by pure imaginary number -5i.

Problem 44

Multiply complex number -3+i by pure imaginary number -4i.

Problem 45

Multiply complex number 1-6i by pure imaginary number i.

Problem 46

Multiply complex number -5-3i by pure imaginary number -2i.

Open in simulator
Problem 47

Multiply complex number 8 by pure imaginary number 0.5i.

Problem 48

Multiply complex number -7i by pure imaginary number 6i.

distribute and simplify `i^2`.
12 problems Warmup Practice Mixed Review Assessment
Problem 49

Multiply complex binomials (2+3i)(4+i).

Problem 50

Multiply complex binomials (1-2i)(3+5i).

Problem 51

Multiply complex binomials (-2+i)(6-3i).

Problem 52

Multiply complex binomials (4- i)(4+i).

Open in simulator
Problem 53

Multiply complex binomials (5+2i)(1-3i).

Problem 54

Multiply complex binomials (-3+4i)(2+i).

Problem 55

Multiply complex binomials (7-i)(-1-2i).

Problem 56

Multiply complex binomials (6i)(2-5i).

Problem 57

Multiply complex binomials (3+i)(3-i).

Problem 58

Multiply complex binomials (-1-i)(-1+i).

Problem 59

Multiply complex binomials (2-3i)(4-5i).

Problem 60

Multiply complex binomials (5-i)(2+3i).

recognize real result `a^2+b^2`.
12 problems Warmup Practice Mixed Review Assessment
Problem 61

Multiply complex conjugates (3+4i)(3-4i).

Problem 62

Multiply complex conjugates (5-2i)(5+2i).

Problem 63

Multiply complex conjugates (-1+6i)(-1-6i).

Problem 64

Multiply complex conjugates (a+bi)(a-bi).

Problem 65

Multiply complex conjugates (7i)(-7i).

Open in simulator
Problem 66

Multiply complex conjugates (10)(10).

Problem 67

Multiply complex conjugates (-2-3i)(-2+3i).

Problem 68

Multiply complex conjugates (8+5i)(8-5i).

Problem 69

Multiply complex conjugates (4-7i)(4+7i).

Problem 70

Multiply complex conjugates (1+i)(1-i).

Problem 71

Multiply complex conjugates (-3+i)(-3-i).

Problem 72

Multiply complex conjugates (6-i)(6+i).

combine operations and simplify powers of i.
12 problems Warmup Practice Mixed Review Assessment
Problem 73

Simplify complex expression (2+i)(3-4i)+5i.

Problem 74

Simplify complex expression i^5(4-2i)+3.

Problem 75

Simplify complex expression (1+2i)^2-(3-i).

Problem 76

Simplify complex expression 2(3-i)+(4+i)(1-i).

Problem 77

Simplify complex expression (5+i)/(1-i) + (2-3i).

Problem 78

Simplify complex expression (1-i)^3 - (2+i).

Problem 79

Simplify complex expression (3+2i)i^7 + (1-i)(2+i).

Open in simulator
Problem 80

Simplify complex expression (10-5i)/(1+2i) * i^2.

Problem 81

Simplify complex expression (3+i)^2 - (2-i)^2.

Problem 82

Simplify complex expression (7+i)/(2-i) + (3i-1).

Problem 83

Simplify complex expression i^3(5-i) + (2+3i)(1-i).

Problem 84

Simplify complex expression (i(3-i) + 1)^2.

change sign of imaginary part only.
12 problems Warmup Practice Mixed Review Assessment
Problem 85

Find the complex conjugate of 3+4i.

Problem 86

Find the complex conjugate of -2-5i.

Problem 87

Find the complex conjugate of 7.

Open in simulator
Problem 88

Find the complex conjugate of -6i.

Problem 89

Find the complex conjugate of a-bi.

Problem 90

Find the complex conjugate of 5-2i.

Problem 91

Find the complex conjugate of -1+3i.

Problem 92

Find the complex conjugate of 8i.

Problem 93

Find the complex conjugate of -10.

Problem 94

Find the complex conjugate of 1/2 + 3/4i.

Problem 95

Find the complex conjugate of x+yi.

Problem 96

Find the complex conjugate of sqrt(2) - i.

show results remain `a+bi`.
12 problems Warmup Practice Mixed Review Assessment
Problem 97

Verify closure of complex numbers under addition using (2+3i)+(5-i)=7+2i.

Problem 98

Verify closure of complex numbers under multiplication using (1+2i)(3-i)=5+5i.

Problem 99

Verify closure of complex numbers under addition using (-4+0i)+(0+6i)=-4+6i.

Problem 100

Verify closure of complex numbers under multiplication using (a+bi)(c+di)=(ac-bd)+(ad+bc)i.

Problem 101

Verify closure of complex numbers under addition using (1/2 + 3/4i) + (1/4 - 1/2i) = 3/4 + 1/4i.

Problem 102

Verify closure of complex numbers under addition using (√2 + i) + (3√2 - 5i) = 4√2 - 4i.

Problem 103

Verify closure of complex numbers under addition using (7-2i) + (-7+2i) = 0.

Problem 104

Verify closure of complex numbers under addition using (6i) + (3) = 3+6i.

Problem 105

Verify closure of complex numbers under multiplication using (2+i)(2-i) = 5.

Problem 106

Verify closure of complex numbers under multiplication using (3i)(4i) = -12.

Open in simulator
Problem 107

Verify closure of complex numbers under multiplication using (√3 + i)(√3 - i) = 4.

Problem 108

Verify closure of complex numbers under multiplication using (1/2 + 1/2i)(2+2i) = 2i.

interpret as vector addition on complex plane.
12 problems Warmup Practice Mixed Review Assessment
Problem 109

Represent complex addition 2+3i+4+i geometrically on the complex plane.

Problem 110

Represent complex addition -1+5i+3-2i geometrically on the complex plane.

Problem 111

Represent complex addition 5-4i+-2-3i geometrically on the complex plane.

Open in simulator
Problem 112

Represent complex addition 1+2i+3+4i geometrically on the complex plane.

Problem 113

Represent complex addition -3+i+5-4i geometrically on the complex plane.

Problem 114

Represent complex addition -2-i+-1-3i geometrically on the complex plane.

Problem 115

Represent complex addition 3+2i geometrically on the complex plane.

Problem 116

Represent complex addition -4i+2+i geometrically on the complex plane.

Problem 117

Represent complex addition 5+-2 geometrically on the complex plane.

Problem 118

Represent complex addition 6i+-3i geometrically on the complex plane.

Problem 119

Represent complex addition 1-7i+0 geometrically on the complex plane.

Problem 120

Represent complex addition 10-5i+-7+12i geometrically on the complex plane.

simplify and compare real/imaginary parts.
12 problems Warmup Practice Mixed Review Assessment
Problem 121

Compare complex arithmetic expressions (2+i)+(3-4i) and 5-3i for equivalence.

Problem 122

Compare complex arithmetic expressions (1+i)^2 and 2i for equivalence.

Problem 123

Compare complex arithmetic expressions (3+2i)(3-2i) and 13 for equivalence.

Problem 124

Compare complex arithmetic expressions (2+i)^2 and 4+i^2 for equivalence.

Problem 125

Compare complex arithmetic expressions (5-2i) - (1+3i) and 4-5i for equivalence.

Problem 126

Compare complex arithmetic expressions i(4-3i) and 3+4i for equivalence.

Problem 127

Compare complex arithmetic expressions (6+2i)/2 and 3+i for equivalence.

Problem 128

Compare complex arithmetic expressions (1+i)/(1-i) and i for equivalence.

Problem 129

Compare complex arithmetic expressions i^3 and -i for equivalence.

Problem 130

Compare complex arithmetic expressions (2-i) + (3i)(1+i) and -1+2i for equivalence.

Open in simulator
Problem 131

Compare complex arithmetic expressions (1+2i)(1-2i) and 1-4i for equivalence.

Problem 132

Compare complex arithmetic expressions (7+3i) - (2-i) and 5+2i for equivalence.

catch sign, distribution, conjugate, and `i^2` mistakes.
12 problems Warmup Practice Mixed Review Assessment
Problem 133

Correct the complex arithmetic error in (3+2i)+(4+5i)=7+7i^2.

Problem 134

Correct the complex arithmetic error in (2+i)(2-i)=4-i^2=3.

Problem 135

Correct the complex arithmetic error in The conjugate of -3+4i is 3-4i.

Problem 136

Correct the complex arithmetic error in (1+2i)^2=1+4i^2=-3.

Problem 137

Correct the complex arithmetic error in (5+3i)-(2-i) = 5+3i-2-i = 3+2i.

Problem 138

Correct the complex arithmetic error in (3i)(2i) = 6i.

Problem 139

Correct the complex arithmetic error in The conjugate of 2+7i is 2+7i.

Problem 140

Correct the complex arithmetic error in (4+2i)/2i = 2+i.

Problem 141

Correct the complex arithmetic error in (2+3i)^2 = 4+12i+9i^2.

Problem 142

Correct the complex arithmetic error in 2(3+4i) = 6+4i.

Problem 143

Correct the complex arithmetic error in i^3 = 1.

Open in simulator
Problem 144

Correct the complex arithmetic error in -(2-3i) = -2-3i.