Cogito
Integrated Math 2 · Chapter 3 · Lesson 2
Arithmetic With Complex Numbers
Treat i like a variable, then simplify.
12 problems · about 21 minutes · N-CN.A.2
What this lesson teaches
The student adds, subtracts, multiplies and divides complex numbers.
- Add and subtract by combining real and imaginary parts separately.
- Multiply by expanding, then replace i² with −1.
- Divide by multiplying top and bottom by the conjugate.
Warm Up
Straightforward practice. Get the method working first.
5 problems(4 + 3i)(4 − 3i). What is the result?
Answer 25
Why 25.
Why does multiplying by the conjugate give a real number?
Answer The imaginary terms cancel and i² turns the sign.
Why It gives a² + b², which is real.
(2 + 3i) + (4 + i). Real part?
Answer 6
Why 2 + 4.
Same sum. Imaginary coefficient?
Answer 4
Why 3 + 1.
(5 + 2i) − (3 + 6i). Real part?
Answer 2
Why 5 − 3.
Build It Up
The same ideas with more to keep track of.
3 problemsSame difference. Imaginary coefficient?
Answer -4
Why 2 − 6.
(1 + i)(1 − i). What is the result?
Answer 2
Why 1 + 1.
(2 + 5i)(2 − 5i). What is the result?
Answer 29
Why 4 + 25.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the division process in order.
Answer 1. Write the conjugate of the denominator. 2. Multiply top and bottom by it. 3. Replace every i squared with negative one. 4. Write the answer as a + bi.
Why The conjugate is identified before anything is multiplied.
(3 + i)(3 − i). What is the result?
Answer 10
Why 9 + 1.
The Sum: (3 + 2i) + (1 + 5i). What is the imaginary coefficient?
Answer 7
Why 7.
The Conjugate: (3 + 4i)(3 − 4i). What is the result?
Answer 25
Why 25.