Beginning Algebra
Tutorial 5: Adding Real Numbers

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Learning Objectives

After completing this tutorial, you should be able to:

Add real numbers that have the same sign.
Add real numbers that have different signs.
Find the additive inverse or the opposite of a number.

Introduction

This tutorial reviews adding real numbers as well as finding the additive inverse or opposite of a number . I have the utmost confidence that you are familiar with addition, but sometimes the rules for negative numbers (yuck!) get a little mixed up from time to time. So, it is good to go over them to make sure you have them down.

Tutorial

Adding Real Numbers

Adding Real Numbers
with the Same Sign

Step 1: Add the absolute values.

If you need a review of absolute values, go to Tutorial 2: Symbols and Sets of Numbers.

Step 2: Attach their common sign to sum.

In other words:

If both numbers that you are adding are positive, then you will have a positive answer.

If both numbers that you are adding are negative then you will have a negative answer.

Example 1: Add -6 + (-8).

-6 + (-8) = -14

The sum of the absolute values would be 14 and their common sign is -. That is how we get the answer of -14.

You can also think of this as money. I know we can all relate to that. Think of the negative as a loss. In this example, you can think of it as having lost 6 dollars and then having lost another 8 dollars for a total loss of 14 dollars.

Example 2: Add -5.5 + (-8.7).

-5.5 + (-8.7) = -14.2

The sum of the absolute values would be 14.2 and their common sign is -. That is how we get the answer of -14.2.

You can also think of this as money - I know we can all relate to that. Think of the negative as a loss. In this example, you can think of it as having lost 5.5 dollars and then having lost another 8.7 dollars for a total loss of 14.2 dollars.

Adding Real Numbers
with Opposite Signs

Step 1: Take the difference of the absolute values.

If you need a review of absolute values, go to Tutorial 2: Symbols and Sets of Numbers.

Step 2: Attach the sign of the number that has the higher absolute value.

Which did you have more of, negative or positive?

If the number with the larger absolute value is negative, then your sum is negative. In other words you have more negative than positive.

If the number with the larger absolute value was positive, then your sum is positive. In other words you have more positive than negative.

Example 3: Add -8 + 6.

-8 + 6 = -2.

The difference between 8 and 6 is 2 and the sign of 8 (the larger absolute value) is -. That is how we get the answer of -2.

Thinking in terms of money: we lost 8 dollars and got back 6 dollars, so we are still in the hole 2 dollars.

Example 4: Add

*Mult. top and bottom of first fraction by 2 to get the LCD of 6

*Take the difference of the numerators and write over common denominator 6

*Reduce fraction

The difference between 4/6 and 1/6 is 3/6 = 1/2 and the sign of 4/6 (the larger absolute value) is +. That is how we get the answer of 1/2.

Thinking in terms of money: we had 2/3 of a dollar and lost 1/6 of a dollar, so we would come out ahead 1/2 of a dollar.

Note that if you need help on fractions, go back to Tutorial 3: Fractions.

Example 5: Add -10 + 7 + (-2) + 5.

In this example, we are needing to combine more than two numbers together, but we will still follow the same thought process we do if there are only two numbers. I’m going to go ahead and step us through it going left to right.

* -10 + 7 = -3
*-3 + (-2) = -5

Example 6: Add

In this addition problem, we have some absolute values thrown into the mix. Remember that we need to do what is inside the absolute values (grouping symbol) first and then add those numbers together. If you need a review on order of operations go to Tutorial 4: Introduction to Variable Expressions and Equations.

*Add inside the absolute values
*Evaluate the absolute values
*Add

Opposites

Opposites are two numbers that are on opposite sides of the origin (0) on the number line, but have the same absolute value. In other words, opposites are the same distance away from the origin, but in opposite directions.

The opposite of x is the number -x.

Keep in mind that the opposite of 0 is 0.

The following is an illustration of opposites using the numbers 3 and -3:

open

Double Negative Property

For every real number a,
-(-a) = a.

When you see a negative sign in front of an expression, you can think of it as taking the opposite of it. For example, if you had -(-2), you can think of it as the opposite of -2. Since a number can only have one of two signs, either a '+' or a '-', then the opposite of a negative would have to be positive. So, -(-2) = 2.

Example 7: Write the additive inverse or opposite of 1.5.

The opposite of 1.5 is -1.5, since both of these numbers have the same absolute value but are on opposite sides of the origin on the number line.

Example 8: Write the opposite of -3.

The opposite of -3 is 3, since both of these numbers have the same absolute value but are on opposite sides of the origin on the number line.

Example 9: Simplify -(-10).

When you have a negative in front of a parenthesis like this, it is another way to write that you need to find the additive inverse or opposite.

Since the opposite of a negative is a positive, our answer is 10.

Example 10: Simplify -|-5.2|.

-|-5.2| =
-(5.2) =
-5.2

*Evaluate the absolute value
*Find the opposite

Practice Problems

These are practice problems to help bring you to the next level. It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it. Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument. In fact there is no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.