Questions: Complex Numbers Introduction

Question 1

What is sqrt(-36) expressed in terms of i?

• A. 6
• B. -6
• C. 6i
• D. sqrt(36)i²

Answer: (C)

sqrt(-36) = sqrt(36 · -1) = sqrt(36) · sqrt(-1) = 6i. The common error is writing 6 or -6, which would require sqrt(-36) to be a real number — but negative numbers have no real square roots. sqrt(-1) = i by definition.

Question 2

The imaginary unit i is just a variable that represents an unknown number, similar to x in algebra.

• T. True
• F. False

Answer: False

i is not an unknown variable — it is a defined mathematical constant: i = sqrt(-1), with the fixed property i² = -1. Unlike x, we are not solving for i; we are extending the number system to include it. Every complex number a + bi is fully determined once a and b are known.

Question 3

What is i^23? Show your reasoning using the cyclic pattern of powers of i.

Model answer: -i

The powers of i cycle with period 4: i¹=i, i²=-1, i³=-i, i⁴=1, then repeat. To find i^23, divide 23 by 4: 23 = 4×5 + 3, so the remainder is 3. Therefore i^23 = i^3 = -i. Any power of i reduces to one of {i, -1, -i, 1} based on the remainder when the exponent is divided by 4.