Introduction to Complex Numbers

Algebra 2 Tutorial

Introduction to Complex Numbers

Intro

Consider the equation x^2=-1. Since squaring any number is 0 or larger, no real number will make this equation true. Create a number i such that i^2=-1. By “adding” i to the real numbers, the complex numbers (having the form a+bi where a and b are real numbers) are created. Visually represent a point a+bi as a pair (a,b) in the x-y plane. It’s useful to associate the pair with the arrow from the origin to (a,b). The x-axis is the real number line. The complex numbers solve all polynomial equations: for example, x^5+4x=5-x^3 has 5 complex numbers as solutions (as many as the highest power).

Addition of complex numbers (a,b) and (c,d) is defined as (a,b)+(c,d)=(a+c,b+d): add the components. Visually, adding arrows is done by putting the end of the first at the start of the second; the arrow from the origin to the end of the second arrow creates the arrow representing the sum.

Subtraction of (a,b) and (c,d) is defined as (a,b)-(c,d)=(a-c,b-d): subtract the components. Visually, the difference of two arrows is an arrow from the end of the second to the end of the first (when both arrows start at the origin).

To multiply and divide complex numbers, instead of using Cartesian (horizontal/vertical) coordinates (as done when adding or subtracting), use polar coordinates (how long the arrow is and the counterclockwise angle it makes with the positive x-axis). Multiplication is defined as (a,b)*(c,d)=(a*c,b+d): to multiply arrows, multiply their lengths and add their angles. Division is defined as (a,b)/(c,d)=(a/c,b-d): to divide arrows, divide their lengths and subtract their angles.

Sample Problem

Question 1: (7,5)+(10,-6)=
Question 2: (13,6)-(7,-1)=
Question 3: (5,60)*(3,80)=
Question 4: (4,120)/(5,70)=

Solution

Question 1: Using (a,b)+(c,d)=(a+c,b+d) yields (7,5)+(10,-6)=(7+10,5+(-6))=(17,-1)

Question 2: Using (a,b)-(c,d)=(a-c,b-d) yields (13,6)-(7,-1)=(13-7,6-(-1))=(6,7)

Question 3: Using (a,b)*(c,d)=(a*c,b+d) yields (5,60)*(3,80)=(5*3,60+80)=(15,140)

Question 4: Using (a,b)/(c,d)=(a/c,b-d) yields (4,120)/(5,70)=(4/5,120-70)=(4/5,50)



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