Cogito
Algebra 2 · Chapter 2 · Lesson 2
Arithmetic with Complex Numbers
Treat i like a letter, then replace i squared.
11 problems · about 23 minutes · N-CN.A.2
What this lesson teaches
The student adds, multiplies, and divides complex numbers.
- Add and subtract by collecting real and imaginary parts separately.
- Multiply as usual, then replace i² with −1.
- The conjugate turns a complex denominator into a real one.
Warm Up
Straightforward practice. Get the method working first.
4 problems(4 + 3i)(4 − 3i). Result?
Answer 25
Why 25.
(1 + i) + (5 + 3i). Imaginary part?
Answer 4
Why 1 + 3.
(6 + 2i) − (2 + 5i). Real part?
Answer 4
Why 6 − 2.
What is i⁴?
Answer 1
Why (i²)².
Build It Up
The same ideas with more to keep track of.
3 problems(2 + i)(2 − i). What is the result? It is real.
Answer 5
Why 4 + 1.
Conjugate of 3 + 7i. What is its imaginary part?
Answer -7
Why The sign flips.
(1 + 2i)(1 − 2i). Result?
Answer 5
Why 1 + 4.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsOrder the steps for dividing by a complex number.
Answer 1. Find the conjugate of the denominator 2. Multiply top and bottom by it 3. Replace every i squared with −1 4. Split into real and imaginary parts
Why The conjugate is what makes the denominator real.
Is 5 a complex number? 1 for yes, 0 for no.
Answer 1
Why Its imaginary part is zero.
Adding: (3 + 2i) + (4 + 5i). What is the real part?
Answer 7
Why 7.
The Square: What is i²?
Answer -1
Why −1.