# Complex Numbers

### A Surprising Boost from Geometry

Student Outcomes

• Students define a complex number in the form a + bi, where a and b are real numbers and the imaginary unit i satisfies i2 = −1. Students geometrically identify i as a multiplicand effecting a 90° counterclockwise rotation of the real number line. Students locate points corresponding to complex numbers in the complex plane.
• Students understand complex numbers as a superset of the real numbers; i.e., a complex number a + bi is real when b = 0. Students learn that complex numbers share many similar properties of the real numbers: associative, commutative, distributive, addition/subtraction, multiplication, etc.

### New York State Common Core Math Algebra II, Module 1, Lesson 37

Worksheets for Algebra II, Module 1, Lesson 37

Solutions for Algebra II, Module 1, Lesson 37

Classwork

Opening Exercise
Solve each equation for 𝑥.
a. 𝑥 − 1 = 0
b. 𝑥 + 1 = 0
c. 𝑥2 − 1 = 0
d. 𝑥2 + 1 = 0

Example 1: Addition with Complex Numbers
Compute (3 + 4𝑖)+ (7 − 20𝑖).

Example 2: Subtraction with Complex Numbers
Compute (3 + 4𝑖)− (7 − 20𝑖).

Example 3: Multiplication with Complex Numbers
Compute (1 + 2𝑖)(1 − 2𝑖).

Example 4: Multiplication with Complex Numbers
Verify that −1+ 2𝑖 and −1− 2𝑖 are solutions to 𝑥2 + 2𝑥 +5 = 0

Lesson Summary

Multiplication by 𝑖 rotates every complex number in the complex plane by 90° about the origin.

Every complex number is in the form 𝑎 + 𝑏𝑖, where 𝑎 is the real part and 𝑏 is the imaginary part of the number. Real numbers are also complex numbers; the real number 𝑎 can be written as the complex number 𝑎 +0𝑖. Numbers of the form 𝑏𝑖, for real numbers 𝑏, are called imaginary numbers.

Adding two complex numbers is analogous to combining like terms in a polynomial expression.

Multiplying two complex numbers is like multiplying two binomials, except one can use 𝑖2 = −1 to further write the expression in simpler form.

Complex numbers satisfy the associative, commutative, and distributive properties.

Complex numbers allow us to find solutions to polynomial equations that have no real number solutions.

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