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More Lessons for Algebra II

More Lessons for Algebra

Math Worksheets

Complex Number Worksheets

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

- about imaginary numbers
- the powers of
*i* - how to add and subtract complex numbers
- how to multiply complex numbers
- how to divide complex numbers

An imaginary number bi has two parts: a real number, b, and an imaginary part, i, defined as i^{2} = -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**

To determine the value of i raised to a power greater than two, we rewrite the term using exponent rules. Remember that i^{2} = -1 and i^{4} = 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!

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**

To simplify expressions by multiplying complex numbers, we use exponent rules for i and then simplify further if possible. Remember that, by definition, i^{2}= -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!

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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