### Online College Courses for Credit

#### FREE EDUCATIONAL RESOURCES PROVIDED by SOPHIA

##### Are you a student?
Free Professional Development
2 Tutorials that teach Adding and Subtracting Complex Numbers

# Adding and Subtracting Complex Numbers

##### Rating:
(0)
Author: Sophia Tutorial
##### Description:

In this lesson, students will learn how to add and subtract complex numbers with the imaginary unit i.

(more)

Sophia’s self-paced online courses are a great way to save time and money as you earn credits eligible for transfer to many different colleges and universities.*

No credit card required

29 Sophia partners guarantee credit transfer.

314 Institutions have accepted or given pre-approval for credit transfer.

* The American Council on Education's College Credit Recommendation Service (ACE Credit®) has evaluated and recommended college credit for 27 of Sophia’s online courses. Many different colleges and universities consider ACE CREDIT recommendations in determining the applicability to their course and degree programs.

Tutorial
This tutorial covers how to add and subtract imaginary and complex numbers, through the exploration of:
1. Imaginary and Complex Numbers
2. Adding and Subtracting Imaginary Numbers
3. Adding and Subtracting Complex Numbers

## 1. Imaginary and Complex Numbers

To review, the square root of a negative number is a non-real, or imaginary, number. The imaginary unit i is defined as the square root of -1.

Imaginary Number

A complex number is a value in the form below, in which a and b are real numbers, and i is the imaginary unit. In a complex number, a is the real part, and b times i is the imaginary part.

Complex Number
A value of the form a + bi, where a and b are real numbers and i is the imaginary unit
Complex numbers are used in fields such as engineering and physics.

## 2. Adding and Subtracting Imaginary Numbers

You may recall that the product property for square roots states that the square root of a times b is equal to the square root of a times the square root of b.

Product Property of Square Roots

You can apply the product property of square roots to solve equations involving the square root of a negative number, so that you are able to simplify your solution using imaginary numbers.

For example, you can rewrite the square root of -25 as follows, then apply the product property for square roots. Then you are able to simplify to arrive at your solution, which is an imaginary number.

The product property of square roots can also be applied when adding and subtracting imaginary numbers. Suppose you are solving the equation:

Applying the product property for square roots, you can simplify to:

Now, 2i, 7i, and 3i are all like terms. Therefore, you can combine them together by adding or subtracting their coefficients, to arrive at your final answer:

## 3. Adding and Subtracting Complex Numbers

Adding and subtracting complex numbers is similar to combining like terms. You can add or subtract the real parts together, and add or subtract the coefficients of the imaginary parts together. You can add or subtract complex numbers in this way because of the commutative property of addition.

Suppose you want to add the complex numbers:

You would start by adding your real parts, 4 and 2, together. Then you would add your imaginary parts, 8i and 3i, together.

Try combining like terms to subtract the complex numbers:
Start by combining and subtracting your real parts, 11 minus 7, then combine and subtract your imaginary parts, -6i minus 9i, to arrive at your final answer.

Today you reviewed imaginary numbers, recalling that the square root of a negative number is non-real, or an imaginary number; the imaginary unit i is equal to the square root of -1. You also reviewed the definition of a complex number, which is a value in the form a plus bi, where a is the real part, and b times i is the imaginary part of the complex number. You learned how to apply the product property of square roots when adding or subtracting imaginary numbers. You also learned that when adding or subtracting complex numbers, you add or subtract the real parts and add or subtract the coefficients of the imaginary parts.

Source: This work is adapted from Sophia author Colleen Atakpu.

Terms to Know
Complex Number

A value of the form a + bi, where a and b are real numbers and 'i' is the imaginary unit.

Formulas to Know
Imaginary Number

Product Property of Square Roots