Question 1

Which of the following is the standard form of a complex number?

Accepted Answer

$$a + bi$$

Question 2

If $$z = 3 - 4i$$, what is the complex conjugate of $$z$$?

Accepted Answer

$$3 + 4i$$

Question 3

What is the product of a complex number $$z = a + bi$$ and its conjugate $$\overline{z} = a - bi$$?

Accepted Answer

$$a^2$$ + $$b^2$$

Question 4

Which of the following correctly expresses the principal square root of $$-8$$?

Accepted Answer

$$2\sqrt{2}i$$

Question 5

Simplify and express in standard form: $$(2 + 3i) + (4 - 5i)$$

Accepted Answer

$$6 - 2i$$

Question 6

Simplify and express in standard form: $$(1 + 2i)(3 - 4i)$$

Accepted Answer

$$11 + 2i$$

Question 7

Divide and express in standard form: $$\frac{5 + 2i}{1 - i}$$

Accepted Answer

$$\displaystyle \frac{3 + 7i}{2}$$

Question 8

If $$z = 7 + 24i$$, what is $$|z|$$, the modulus of $$z$$?

Accepted Answer

$$25$$

Question 9

Which of the following statements is TRUE about the product of a complex number and its conjugate?

Accepted Answer

The product is always a real number.

Question 10

A medical imaging device uses the number $$i in $$its calculations. If $$i^2$$ appears in the computation, what is its value?

Accepted Answer

$$-1$$

Question 11

Which of the following is NOT a complex number?

Accepted Answer

$$\infty$$

Question 12

If $$z = 2 - 5i$$, what is $$z + \overline{z}$$?

Accepted Answer

$$4$$

Question 13

A chemical reaction is modeled by the equation $$x^2 + 4 = 0$. What are the solutions?

Accepted Answer

$$2i$$ and $$-2i$$

Question 14

Which operation is used to rationalize the denominator when dividing complex numbers?

Accepted Answer

Multiply numerator and denominator by the conjugate of the denominator.

Question 15

If $$z = 4 + 3i$$ and $$w = 2 - i$$, what is $$z \cdot w$$?

Accepted Answer

$$11 + 2i$$

Question 16

Which of the following best describes the set of all complex numbers?

Accepted Answer

All numbers of the form $$a + bi$$ where $$a$$ and $$b$$ are real numbers.

Question 17

A circuit engineer uses the number $$i to $$represent a phase shift. If $$i = \sqrt{-1}$$, what is $$i^4$$?

Accepted Answer

$$1$$

Question 18

Simplify: $$(5 + 2i) - (3 - 6i)$$

Accepted Answer

$$2 + 8i$$

Question 19

If $$z = a + bi$$ and $$w = c + di$$, what is the formula for $$z \cdot w$$?

Accepted Answer

$$(ac - bd) + (ad + bc)i$$

Question 20

A physicist encounters the expression $$\sqrt{-16}. $$What is its value?

Accepted Answer

$$4i$$

Complex Numbers: Definitions, Properties, and Operations

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