# Complex Numbers

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A series of free, online Intermediate Algebra Lessons or Algebra II lessons.
Videos, worksheets, and activities to help Algebra students.

In this lesson, we will learn

• the powers of i
• how to add and subtract complex numbers
• how to multiply complex numbers
• how to divide complex numbers

### Introduction to Imaginary Numbers

An imaginary number bi has two parts: a real number, b, and an imaginary part, i, defined as i2 = -1. Imaginary numbers are applied to square roots of negative numbers, allowing them to be simplified in terms of i. When a real number, a, is added to an imaginary number, a + bi is said to be a complex number. Beware that in some cases the letter j is used instead of i for the imaginary number.

How to define an imaginary number?

Introduction to i and imaginary numbers

### Powers of i

To determine the value of i raised to a power greater than two, we rewrite the term using exponent rules. Remember that i2 = -1 and i4 = 1. Therefore, any exponent of i that is a multiple of four will equal one; any even exponent not divisible by four will equal negative one. Also, negative exponents indicate a reciprocal of the base; if i is in the denominator, it will need to be rationalized.

Rewriting Powers of ' i ' - In this video, I take the complex number ' i ', raise it to some different powers, and simplify!

### Adding and Subtracting Complex Numbers

Subtracting and adding complex numbers is the same idea as combining like terms. In an expression, the coefficients of i can be summed together just like the coefficients of variables. If an expression has real numbers and square roots of negative numbers, rewrite using i and then combine like terms.

Adding and Subtracting Complex (Imaginary) Numbers This video explains how to add and subtract complex numbers

### Multiplying Complex Numbers

To simplify expressions by multiplying complex numbers, we use exponent rules for i and then simplify further if possible. Remember that, by definition, i2= -1, which also means that i 4= 1. If multiplying two square roots of negatives, their product is not a positive. First we rewrite the radicals using i and then multiply and simplify.

Complex Numbers: Multiplying - Ex 1. Complex Numbers: Multiplying - Ex 2
This video shows an example of multiplying three complex numbers and simplifying!

### Dividing Complex Numbers

Fractions with negative roots in the denominator or with i in the denominator must be rationalized (since i represents a square root). When dividing complex numbers with negative roots, simplify in terms of imaginary numbers and then multiply the numerator and denominator by i. When a binomial is in the denominator, rewrite using i and then multiply the numerator and denominator by the conjugate.

How to rationalize the denominator when dealing with an imaginary number?

Complex Numbers: Dividing - Ex 1
This video shows how to divide a complex number by another complex number.

Complex Numbers: Dividing - Ex 2
This video shows how to divide a complex number by another complex number. Complex Numbers: Dividing - Ex 3
This video shows how to divide a complex number by another complex number.

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