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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

  • about imaginary numbers
  • 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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