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Math II ยท N-CN.1
Understanding `i` and Representing Complex Numbers as `a + bi`
Complex numbers let students solve equations and model rotations, waves, signals, and systems that real numbers alone cannot describe cleanly.
Concept
Number and Quantity
Domain
The Complex Number System
Read time
1 minutes
Problem Library
Problems in the App From This Objective
144 problems across 12 archetypes in the app.
use cycle `i, -1, -i, 1`.
Problem 1
Simplify i^5.
Problem 2
Simplify i^18.
Problem 3
Problem 4
Simplify i^27.
Simplify i^44.
Open in simulator
Problem 5
Simplify i^101.
Problem 6
Simplify i^0.
Problem 7
Simplify i^1.
Problem 8
Simplify i^2.
Problem 9
Simplify i^3.
Problem 10
Simplify i^4.
Problem 11
Simplify i^6.
Problem 12
Simplify i^10.
factor `sqrt(-1)`.
Problem 13
Rewrite sqrt(-25) using i.
Problem 14
Rewrite sqrt(-12) using i.
Problem 15
Rewrite sqrt(-7) using i.
Problem 16
Rewrite sqrt(-72) using i.
Problem 17
Problem 18
Rewrite sqrt(-4) using i.
Rewrite sqrt(-81) using i.
Open in simulator
Problem 19
Rewrite sqrt(-8) using i.
Problem 20
Rewrite sqrt(-18) using i.
Problem 21
Rewrite sqrt(-20) using i.
Problem 22
Rewrite sqrt(-3) using i.
Problem 23
Rewrite sqrt(-11) using i.
Problem 24
Rewrite sqrt(-50) using i.
parse `a+bi`.
Problem 25
Identify the real and imaginary parts of complex number 4+7i.
Problem 26
Identify the real and imaginary parts of complex number -3-2i.
Problem 27
Identify the real and imaginary parts of complex number 9.
Problem 28
Identify the real and imaginary parts of complex number -5i.
Problem 29
Identify the real and imaginary parts of complex number 2+3i.
Problem 30
Identify the real and imaginary parts of complex number -1+6i.
Problem 31
Identify the real and imaginary parts of complex number 8-4i.
Problem 32
Identify the real and imaginary parts of complex number -7-i.
Problem 33
Identify the real and imaginary parts of complex number 12.
Problem 34
Identify the real and imaginary parts of complex number 10i.
Problem 35
Problem 36
Identify the real and imaginary parts of complex number -i.
Identify the real and imaginary parts of complex number 1+i.
Open in simulator
order real part plus imaginary part.
Problem 37
Write expression 3i+5 in standard complex form.
Problem 38
Write expression sqrt(-16)+2 in standard complex form.
Problem 39
Write expression 7-i^2+4i in standard complex form.
Problem 40
Write expression -2i+sqrt(-9)-1 in standard complex form.
Problem 41
Problem 42
Write expression i^3+6 in standard complex form.
Write expression sqrt(-25)-3i+1 in standard complex form.
Open in simulator
Problem 43
Write expression 4-i^4+2i in standard complex form.
Problem 44
Write expression 5i-sqrt(-49)+10 in standard complex form.
Problem 45
Write expression 2i^2+3i-5 in standard complex form.
Problem 46
Write expression 1/i+7 in standard complex form.
Problem 47
Write expression sqrt(-1)+sqrt(-100)-2 in standard complex form.
Problem 48
Write expression -i^5+3i-4 in standard complex form.
understand subset relationships.
Problem 49
Classify number 7 as real, imaginary, or complex.
Problem 50
Classify number -4i as real, imaginary, or complex.
Problem 51
Classify number 3+2i as real, imaginary, or complex.
Problem 52
Classify number 0 as real, imaginary, or complex.
Problem 53
Classify number sqrt(-9) as real, imaginary, or complex.
Problem 54
Classify number 1.5 as real, imaginary, or complex.
Problem 55
Problem 56
Classify number i as real, imaginary, or complex.
Classify number -5+i as real, imaginary, or complex.
Open in simulator
Problem 57
Classify number -10 as real, imaginary, or complex.
Problem 58
Classify number 0.5i as real, imaginary, or complex.
Problem 59
Classify number -2-3i as real, imaginary, or complex.
Problem 60
Classify number sqrt(16) as real, imaginary, or complex.
plot real part horizontally and imaginary part vertically.
Problem 61
Represent complex number 3+4i on the complex plane.
Problem 62
Represent complex number -2+5i on the complex plane.
Problem 63
Represent complex number 6- i on the complex plane.
Problem 64
Represent complex number -4i on the complex plane.
Problem 65
Represent complex number 7 on the complex plane.
Problem 66
Represent complex number -5-2i on the complex plane.
Problem 67
Represent complex number 1-3i on the complex plane.
Problem 68
Represent complex number 8i on the complex plane.
Problem 69
Problem 70
Represent complex number -10 on the complex plane.
Represent complex number 0.5+2.5i on the complex plane.
Open in simulator
Problem 71
Represent complex number 100+200i on the complex plane.
Problem 72
Represent complex number -0.1-0.2i on the complex plane.
connect definition to equation solving.
Problem 73
Use i to solve equation x^2=-1.
Open in simulator
Problem 74
Use i to solve equation x^2=-9.
Problem 75
Use i to solve equation x^2+16=0.
Problem 76
Use i to solve equation x^2=-5.
Problem 77
Use i to solve equation x^2=-25.
Problem 78
Use i to solve equation x^2=-7.
Problem 79
Use i to solve equation x^2+49=0.
Problem 80
Use i to solve equation x^2+11=0.
Problem 81
Use i to solve equation 2x^2=-8.
Problem 82
Use i to solve equation 5x^2=-30.
Problem 83
Use i to solve equation x^2+100=0.
Problem 84
Use i to solve equation x^2+8=0.
simplify using `i^2=-1`.
Problem 85
Evaluate expression 3+2i-5i using i^2=-1.
Problem 86
Evaluate expression 4i^2+7 using i^2=-1.
Problem 87
Evaluate expression 2(3+i)-5i using i^2=-1.
Problem 88
Evaluate expression -6i+2i^2+1 using i^2=-1.
Problem 89
Evaluate expression -3i^2+5 using i^2=-1.
Problem 90
Evaluate expression 2(1-i^2)+3i using i^2=-1.
Problem 91
Evaluate expression 5i-i^2+2i-1 using i^2=-1.
Problem 92
Evaluate expression (7+i^2)-(3-2i) using i^2=-1.
Problem 93
Evaluate expression -4(2i^2+i)+10 using i^2=-1.
Problem 94
Evaluate expression i^2+2i^2+3i+4 using i^2=-1.
Problem 95
Problem 96
Evaluate expression (6+4i^2)/2 using i^2=-1.
Evaluate expression 10-3i+i^2-5 using i^2=-1.
Open in simulator
compare real and imaginary parts.
Problem 97
Determine whether complex expressions 3+2i+i and 3+3i are equal.
Problem 98
Determine whether complex expressions 5+i^2 and 4 are equal.
Problem 99
Determine whether complex expressions 2-3i and -3i+2 are equal.
Problem 100
Determine whether complex expressions 4+2i and 2+4i are equal.
Problem 101
Determine whether complex expressions (2+3i) + (1-i) and 3+2i are equal.
Problem 102
Determine whether complex expressions (5-2i) - (1+i) and 4+3i are equal.
Problem 103
Determine whether complex expressions (1+i)*(1-i) and 2 are equal.
Problem 104
Determine whether complex expressions (2+i)*(1+i) and 1-3i are equal.
Problem 105
Determine whether complex expressions i^3 and -i are equal.
Problem 106
Determine whether complex expressions i^4 and -1 are equal.
Problem 107
Problem 108
Determine whether complex expressions 1/(1+i) and 0.5 - 0.5i are equal.
Determine whether complex expressions 2i * i + 3 and 1+i are equal.
Open in simulator
reason that real squares are nonnegative.
Problem 109
Explain why sqrt(-4) requires complex numbers.
Problem 110
Explain why sqrt(-11) requires complex numbers.
Problem 111
Explain why sqrt(-1) requires complex numbers.
Problem 112
Explain why sqrt(-9) requires complex numbers.
Problem 113
Explain why sqrt(-25) requires complex numbers.
Problem 114
Problem 115
Explain why sqrt(-2) requires complex numbers.
Explain why sqrt(-16) requires complex numbers.
Open in simulator
Problem 116
Explain why sqrt(-100) requires complex numbers.
Problem 117
Explain why sqrt(-5) requires complex numbers.
Problem 118
Explain why sqrt(-49) requires complex numbers.
Problem 119
Explain why sqrt(-3) requires complex numbers.
Problem 120
Explain why sqrt(-1/4) requires complex numbers.
read coordinates as real and imaginary parts.
Problem 121
Convert complex-plane point 3, 4 to standard complex form.
Problem 122
Convert complex-plane point -2, 5 to standard complex form.
Problem 123
Convert complex-plane point 0, -6 to standard complex form.
Problem 124
Convert complex-plane point 7, 0 to standard complex form.
Problem 125
Convert complex-plane point 1/2, -3/4 to standard complex form.
Problem 126
Convert complex-plane point -5, -8 to standard complex form.
Problem 127
Convert complex-plane point 6, -1 to standard complex form.
Problem 128
Convert complex-plane point -9, 2 to standard complex form.
Problem 129
Problem 130
Convert complex-plane point 0, 10 to standard complex form.
Convert complex-plane point -11, 0 to standard complex form.
Open in simulator
Problem 131
Convert complex-plane point 2/3, 1/5 to standard complex form.
Problem 132
Convert complex-plane point -3/2, 1 to standard complex form.
catch power-of-i, standard-form, and real/imaginary-part mistakes.
Problem 133
Correct the complex-number representation error in i^6 is simplified as i.
Problem 134
Correct the complex-number representation error in sqrt(-36) is simplified as -6.
Problem 135
Correct the complex-number representation error in For 4-7i, the imaginary part is 7.
Problem 136
Correct the complex-number representation error in Point (2,-3) is written as -3+2i.
Problem 137
Correct the complex-number representation error in i^3 is simplified as 1.
Problem 138
Correct the complex-number representation error in i^10 is simplified as i.
Problem 139
Correct the complex-number representation error in sqrt(-4) is simplified as -2i.
Problem 140
Correct the complex-number representation error in sqrt(-9) is simplified as 3.
Problem 141
Correct the complex-number representation error in For 5+2i, the real part is 2.
Problem 142
Problem 143
Correct the complex-number representation error in For 1-i, the imaginary part is i.
Correct the complex-number representation error in Point (-1,5) is written as 5-i.
Open in simulator
Problem 144
Correct the complex-number representation error in Point (3,0) is written as 3i.