3-01 Complex Numbers (3.2)

Objective: Evaluate expressions that result in complex numbers. How would you evaluate such expressions?

• By using the imaginary unit i and applying algebraic rules for complex numbers.

• By ignoring the imaginary part and solving only the real part.

• By converting all numbers to decimals before solving.

• By treating complex numbers as irrational numbers.

Answer explanation

To evaluate expressions resulting in complex numbers, use the imaginary unit i and apply algebraic rules for complex numbers. Ignoring the imaginary part or treating them as irrational numbers is incorrect.

Objective: Add, subtract, multiply, and divide complex numbers. Which of the following best describes the process for performing these operations?

• Combine like terms for addition and subtraction; use distributive property for multiplication; multiply by the conjugate for division.

• Always multiply real parts and imaginary parts separately for all operations.

• Add real parts and subtract imaginary parts for all operations.

• Use only the real part for all operations.

Answer explanation

The correct choice explains that for addition and subtraction, you combine like terms; for multiplication, you apply the distributive property; and for division, you multiply by the conjugate to simplify.

What is the value of the imaginary unit i?

• i = √-1

• i = 1

• i = -1

• i = 0

Answer explanation

The imaginary unit i is defined as the square root of -1, which is represented as i = √-1. This is the fundamental definition in complex number theory, distinguishing it from real numbers.

What is the value of i²?

• i² = -1

• i² = 1

• i² = 0

• i² = i

Answer explanation

The imaginary unit i is defined such that i² = -1. Therefore, the correct answer is i² = -1.

What is the general form of a complex number?

• a + bi

• a - b

• ab + i

Answer explanation

The general form of a complex number is expressed as a + bi, where 'a' is the real part and 'b' is the imaginary part. This format clearly represents both components, making 'a + bi' the correct choice.

In the complex number a + bi, which part is the real part?

• a is the real part

• b is the real part

• bi is the real part

• a + b is the real part

Answer explanation

In the complex number a + bi, 'a' represents the real part, while 'b' is the coefficient of the imaginary part 'i'. Therefore, the correct answer is that 'a is the real part'.

In the complex number a + bi, which part is the imaginary part?

• bi is the imaginary part

• a is the imaginary part

• a + b is the imaginary part

• a is the real part

Answer explanation

In the complex number a + bi, 'bi' represents the imaginary part, where 'b' is the coefficient and 'i' is the imaginary unit. Therefore, the correct choice is 'bi is the imaginary part'.

Any number with i is called ________.

• imaginary

• real

• rational

• integer

Answer explanation

Any number that includes the imaginary unit 'i' (where i = √-1) is classified as an imaginary number. Therefore, the correct answer is 'imaginary', distinguishing it from real, rational, or integer numbers.

Adding and Subtracting Complex Numbers: Simplify: (-1 + 2i) + (3 + 3i) = _______

• 2 + 5i

• 4 + 6i

• -4 + 5i

• 2 + i

Answer explanation

To simplify (-1 + 2i) + (3 + 3i), combine the real parts: -1 + 3 = 2, and the imaginary parts: 2i + 3i = 5i. Thus, the result is 2 + 5i, which matches the correct answer.

Simplify: (2 - 3i) - (3 - 7i) = _______

• -1 + 4i

• 1 - 10i

• 5 + 4i

• -5 + 10i

Answer explanation

To simplify (2 - 3i) - (3 - 7i), subtract the real and imaginary parts: (2 - 3) + (-3 + 7)i = -1 + 4i. Thus, the correct answer is -1 + 4i.