Imaginary Numbers – Exercises
Instructions: Try the exercises on imaginary numbers below. The answers are given in the next section.
Question 1: x and y are real numbers. ai and bi are imaginary numbers. When does ai + x = bi + y ?
A. When a = b
B. When x = y
C. When a = x and b = y
D. When ai = y and bi = x
E. When ai = bi and x = y
Question 2:For the following equation, i represents an imaginary number. Simplify the equation: (1 – i) – (3 – 2i)
A. –2 – 5i
B. –2 – i
C. –2 + i
D. –6 – 5i
E. –6 + i
Question 3: What is the best solution to √-64 = ?
A. –i
B. –8i
C. 8i
D. i
E. Cannot be represented mathematically
Question 4:
Consider the imaginary number i, where i = –1. What does i + i2 + i3 + . . . i12 equal?
A) –1
B) 0
C) 1
D) i + 1
E) i – 1
Question 5: For the following equation, i represents an imaginary number. Simplify the equation: (82 – i) + (3 – 5i)
A. –85 – 6i
B. 85 –4i
C. 85 – 6i
D. 85 – i
E. 85 –5i
Answers to the Exercises
1) The correct answer is E.
Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.
Therefore, in order for ai + x to be equal to bi + y , ai must be equal to bi and x must be equal to y.
2) The correct answer is C.
To solve this type of problem, do the operation on the parentheses first.
(1 – i) – (3 – 2i) = 1 – i – 3 + 2i
Then group the real and imaginary numbers together in order to simplify and solve the problem.
1 – i – 3 + 2i =
1 – 3 – i + 2i =
–2 + i
3) The correct answer is C.
An imaginary number is expressed as a real number multiplied by the imaginary number i .
Solve as you would for the positive square root, and then multiply by 1.
√-64 = 8i
4) The correct answer is B.
If i = –1, then i2 = 1
i3 = i2 i = –1 × 1 = –1
i4 = i2 × i2 = 1 × 1 = 1
So, a pattern emerges: the numbers with odd exponents in the series are equal to –1 and the numbers with even exponents are equal to 1.
If we complete the series up to i12, we have the following:
–1 + 1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + 1 = 0
5) The correct answer is C.
To solve this type of problem, do the operations on the parentheses first.
(82 – i) + (3 – 5i) = 82 – i + 3 – 5i
Then group the real and imaginary numbers together in order to simplify and solve the problem.
82 – i + 3 – 5i =
82 + 3 – i – 5i =
85 –6i
What are Imaginary Numbers?
You will need to understand imaginary numbers for the ACT Math Exam.
Imaginary numbers are not real numbers. So, imaginary numbers are not whole numbers, integers, decimals, or fractions.
An imaginary number is a complex number that can be expressed as a real number multiplied by the imaginary number i.