Practice Complex Numbers in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Numbers of the form a+bia + bi where a,ba, b are real and i=โˆ’1i = \sqrt{-1}; they extend the real numbers to solve x2=โˆ’1x^2 = -1.

Extending numbers into a second dimension to solve equations like x2=โˆ’1x^2 = -1.

Showing a random 20 of 50 problems.

Example 1

medium
Simplify i42i^{42}.

Example 2

easy
Compute i3i^3.

Example 3

medium
Solve x2+16=0x^2 + 16 = 0 over the complex numbers.

Example 4

easy
Add (4+3i)+(2โˆ’5i)(4 + 3i) + (2 - 5i).

Example 5

easy
Identify the real and imaginary parts of 4+7i4 + 7i.

Example 6

medium
Multiply (2+i)(3+2i)(2 + i)(3 + 2i).

Example 7

medium
Solve x2โˆ’2x+5=0x^2 - 2x + 5 = 0 over the complex numbers.

Example 8

medium
Compute the modulus โˆฃ5โˆ’12iโˆฃ|5 - 12i|.

Example 9

hard
If โˆฃa+biโˆฃ=13|a + bi| = 13 and a=5a = 5 with b>0b > 0, find bb.

Example 10

easy
Subtract (8โˆ’i)โˆ’(3+4i)(8 - i) - (3 + 4i).

Example 11

medium
Find the product of z=5+12iz = 5 + 12i with its conjugate zห‰\bar{z}.

Example 12

medium
Compute (2+i)+(3โˆ’4i)โˆ’(1+2i)(2 + i) + (3 - 4i) - (1 + 2i).

Example 13

medium
Multiply iโ‹…(2+3i)i\cdot(2 + 3i).

Example 14

medium
Solve x2=โˆ’25x^2 = -25 over the complex numbers.

Example 15

challenge
Compute i1+i2+i3+i4i^{1} + i^2 + i^3 + i^4.

Example 16

medium
Simplify i99i^{99}.

Example 17

easy
What is i2i^2?

Example 18

medium
Compute i10i^{10}.

Example 19

easy
Simplify i2i^2, i3i^3, and i4i^4.

Example 20

hard
Compute i+i2+i3+โ‹ฏ+i100i + i^2 + i^3 + \cdots + i^{100}.
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