Mastery Check Solutions

Show What You Know

Complete the statement with always, sometimes, or never.
If your answer is sometimes or never, provide an example to back up your choice.

The imaginary unit i raised to a whole number power is sometimes a pure imaginary number.

Sample: $i^{1} = i$ , $i^{2} = - 1$ , $i^{3} = - i$ , and $i^{4} = 1$ . The numbers –1 and 1 are real, not imaginary, while i and $- i$ are pure imaginary.

Complex numbers always include the sets of real and imaginary numbers.

The square root of a negative number is never a real number.

Sample: $\sqrt{- 1} = i$ , where i is an imaginary number.

Complex numbers are sometimes classified as real, complex numbers.

Sample: $a + b i$ , when $a = 0$ is pure imaginary. If an expression is in the form $a + b i$ , this is complex only. If an expression is in the form $a + b i$ , when $b = 0$ , the number is real, complex.

Note

These statements should help you think about how the imaginary numbers work within the complex number system. There are many examples you can share for A, C, and D. If you are not certain, use the Practice page to find specific examples.

Say What You Know

In your own words, talk about what you have learned using the objectives for this part of the lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

- Classify complex numbers (real, imaginary, complex).
- Define the imaginary unit as $i^{2} = - 1$ .
- Simplify roots of negative numbers using the imaginary unit.
- Use the pattern of i to simplify powers of the imaginary unit.