Cogito
Algebra 2 · Chapter 2 · Lesson 1
Complex Numbers
What happens when the discriminant is negative.
10 problems · about 20 minutes · N-CN.A.1, N-CN.A.2, N-CN.C.7
What this lesson teaches
The student defines the imaginary unit and performs arithmetic with complex numbers.
- i is defined by i² = −1.
- A complex number is a + bi.
- With complex numbers, every polynomial has its full set of roots.
Warm-Up Review
From earlier lessons. Loosen up before the new work.
2 problemsReview — Composition and Inverse Functions: f(x) = x − 7. What is f⁻¹(3)?
Answer 10
Why 10.
Review — Piecewise and Absolute Value Functions: y = |x − 7| + 3. What is the minimum value?
Answer 3
Why 3.
Warm Up
Straightforward practice. Get the method working first.
4 problemsx² = −25. What is the coefficient of i in the positive root?
Answer 5
Why 5i.
What is i defined to be?
Answer The number whose square is −1.
Why i² = −1.
(2 + 3i) + (4 + 5i). Imaginary part?
Answer 8
Why 3 + 5.
What is i⁴?
Answer 1
Why i² × i² = (−1)(−1).
Build It Up
The same ideas with more to keep track of.
2 problems(6 + 7i) − (2 + 3i). Real part?
Answer 4
Why 6 − 2.
x² = −9. What is the coefficient of i in the positive root?
Answer 3
Why 3i squared is −9.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
2 problemsThe Square: What is i²?
Answer -1
Why −1.
Adding: (3 + 2i) + (5 + 4i). What is the real part?
Answer 8
Why 8 + 6i.