Adding Real Numbers

 

 

Learning Objective(s)

·         Add two or more real numbers with the same sign.

·         Add two or more real numbers with different signs.

·         Simplify by using the identity property of 0.

·         Solve application problems requiring the addition of real numbers.

 

Introduction

 

Adding real numbers follows the same rules as adding integers. The number 0 has some special attributes that are very important in algebra. Knowing how to add these numbers can be helpful in real-world situations as well as algebraic situations.

 

Rules for Adding Real Numbers

 

The rules for adding integers apply to other real numbers, including rational numbers.

 

To add two numbers with the same sign (both positive or both negative)

·         Add their absolute values.

·         Give the sum the same sign.

 

To add two numbers with different signs (one positive and one negative)

·         Find the difference of their absolute values. (Note that when you find the difference of the absolute values, you always subtract the lesser absolute value from the greater one.)

·        Give the sum the same sign as the number with the greater absolute value.

 

Remember—to add fractions, you need them to have the same denominator. This is still true when one or more of the fractions are negative.

 

 

Example

Problem

Find

 

 and

 

 

This problem has three addends. Add the first two, and then add the third.

 

Since the signs of the first two are the same, find the sum of the absolute values of the fractions

 

Since both addends are negative, the sum is negative.

 

 and

 

Now add the third addend. The signs are different, so find the difference of their absolute values.

 

Since , the sign of the final sum is the same as the sign of .

Answer

 

 

 

Example

Problem

Find

 

 and

 

The signs are different, so find the difference of their absolute values.

 

 

 

First rewrite  as an improper fraction, then rewrite the fraction using a common denominator.

 

 

 

Now substitute the rewritten fraction in the problem.

 

 

Subtract the numerators and keep the same denominator. Simplify to lowest terms, if possible.

Answer

Since , the sign of the final sum is the same as the sign of .

 

 

When you add decimals, remember to line up the decimal points so you are adding tenths to tenths, hundredths to hundredths, and so on.

 

 

Example

Problem

Find 27.832 + (−3.06).

 

Since the addends have different signs, subtract their absolute values.

 

                    |−3.06| = 3.06

 

The sum has the same sign as 27.832 whose absolute value is greater.

Answer

27.832 + (−3.06) = 24.772

 

 

Find −32.22 + 124.3.

 

A) 19.79

B) 44.65

C) 92.08

D) 156.52

 

Show/Hide Answer

A) 19.79

Incorrect. To find the sum of numbers with different signs, you must subtract their absolute values. When adding and subtracting decimals, you must pay attention to place value. You may have subtracted 32.22 – 12.43 instead of 124.3 – 32.22. The correct answer is 92.08.

 

B) 44.65

Incorrect. To find the sum of numbers with different signs, you must subtract their absolute values. You may have added instead. Also, when adding and subtracting decimals, you must pay attention to place value. The correct answer is 92.08.

 

C) 92.08

Correct. Since the addends have different signs, you must subtract their absolute values.

124.3 – 32.22 is 92.08. Since |124.3| > |−32.22|, the sum is positive.

 

D) 156.52

Incorrect. To find the sum of numbers with different signs, you must subtract their absolute values. You added their absolute values. The correct answer is 92.08.

 

 

 

The Additive Identity

 

The rules for adding real numbers refer to the addends being positive or negative. But 0 is neither positive nor negative.

 

It should be no surprise that you add 0 the way you always have—adding 0 doesn’t change the value.

 

7 + 0 = 7

−7 + 0 = −7

0 + 3.6 = 3.6

x + 0 = x

0 + x = x

 

Notice that x + 0 = x and 0 + x = x. This means that it doesn’t matter which addend comes first.

 

The number 0 is called the additive identity. The identity property of 0 states that adding 0 to other numbers doesn’t change their value. You can think of it in this way: adding 0 lets the other number keep its identity.

 

 

What is 0 + y, when y = 3?

 

A) −3

B) 0

C) 3

 

Show/Hide Answer

A) −3

Incorrect. The identity property says 0 + any number = that number. Substituting 3 for y gives 0 + 3, and 0 + 3 = 3.

 

B) 0

Incorrect. The sum of a number and 0 is that number, not 0. Substituting 3 for y gives 0 + 3, and 0 + 3 = 3.

 

C) 3

Correct. Substituting 3 for y gives 0 + 3, and 0 + 3 = 3.

 

 

 

Applications of Addition

 

There are many situations that use negative numbers. For example, temperatures colder than 0° are usually described using negative numbers. In golf tournaments, players’ scores are often reported as a number over or under par, instead of the total number of strokes it takes to hit the ball into the hole. (Par is the expected number of strokes needed to complete a hole.) A number under par is negative, and a number over par is positive.

 

The following examples show how addition of real numbers, including negative numbers, can be useful.

 

 

Example

Problem

Boston is, on average, 7 degrees warmer than Bangor, Maine. The low temperature on one cold winter day in Bangor was −13° F. About what low temperature would you expect Boston to have on that day?

 

If the temperature in Bangor is x, the temperature in Boston is x + 7.

The phrase "7 degrees warmer" means you add 7 degrees to Bangor's temperature to estimate Boston's temperature.

 

x = −13

On that day, Bangor's low was −13°.

 

Boston's temperature is −13 + 7

 

 

−13 + 7 = −6

Substitute −13 for x to get Boston's temperature.

 

Add the integers. Since one is positive and the other is negative, you find the difference of |−13| and |7|, which is 6. Since |−13| > |7|, the final sum is negative.

 

Answer

You would expect Boston to have a temperature of −6 degrees.

 

 

Example

Problem

Before Joanne could deposit her paycheck of $802.83, she overdrew her checking account. The balance was −$201.35. What was her balance after she deposited the paycheck?

 

−201.35 + 802.83

 

−201.35 + 802.83 = 601.83

By depositing her paycheck, Joanne is adding money to her account. The new balance is the sum of the old (−201.35) and the paycheck amount.

 

Since the numbers have different signs, find the difference of −201.35. Since |802.83| > |−201.35|, the sum is positive.

Answer

The new balance is $601.48.

 

 

 

When forces or objects are working in opposite directions, sometimes it's helpful to assign a negative value to one and a positive value to the other. This is done often in physics and engineering, but it could also be done in other contexts, such as football or a tug-of-war.

 

 

Example

Problem

Two people are in a tug-of-war contest. They are facing each other, each holding the end of a rope. They both pull on the rope, trying to move the center toward themselves.

 

Here's an illustration of this situation. The person on the right is pulling in the positive direction, and the person on the left is pulling in the negative direction.

 

tug-of-war

 

At one point in the competition, the person on the right was pulling with 122.8 pounds of force. The person on the left was pulling with 131.3 pounds of force. The forces on the center of the rope, then, were 122.8 lbs and −131.3 lbs.

a)      What was the net (total sum) force on the center of the rope?

b)     In which direction was it moving?

 

Net force = 122.8 +( −131.3)

The net force is the sum of the two forces on the rope.

 

Net force = −8.5

To find the sum, add the difference of the absolute values of the addends. Since |−131.3| > 122.8, the sum is negative.

Answer

The net force is −8.5 lbs (or 8.5 lbs to the left). The center of the rope is moving to the left (the negative direction).

Notice that it makes sense that the rope was moving to the left, since that person was pulling with more force.

 

 

After Bangor reached a low temperature of −13°, the temperature rose only 4 degrees higher for the rest of the day. What was the high temperature that day?

 

A) −17

B) −9

C) 9

D) 17

 

Show/Hide Answer

A) −17

Incorrect. Although you used the correct sign, you found the sum of the absolute values. The temperature rose (added) 4 degrees from −13, so the high temperature is −13 + 4. Since the addends have different signs, you must find the difference of the absolute values.  |−13| = 13 and |4| = 4. The difference is 13 – 4 = 9. The sign of the sum is the same as the addend with the greater absolute value. Since |−13| > |4|, the sum is negative. The correct answer is −9.

 

B) −9

Correct. The temperature rose (added) 4 degrees from −13, so the high temperature is −13 + 4. Since the addends have different signs, you must find the difference of the absolute values. |−13| = 13 and |4| = 4. The difference is 13 – 4 = 9. The sign of the sum is the same as the addend with the greater absolute value. Since |−13| > |4|, the sum is −9.

 

C) 9

Incorrect. You found the difference of the absolute values of the addends. However, because |−13| > |4|, the sign of the sum should be the same as the sign on −13. The correct answer is −9.

 

D) 17

Incorrect. You added the absolute values of the addends, and gave the sum the wrong sign. The temperature rose (added) 4 degrees from −13, so the high temperature is −13 + 4. Since the addends have different signs, you must find the difference of the absolute values.   |−13| = 13 and |4| = 4. The difference is 13 – 4 = 9. The sign of the sum is the same as the addend with the greater absolute value. Since |−13| > |4|, the sum is negative. The correct answer is −9.

 

 

 

Summary

As with integers, adding real numbers is done following two rules. When the signs are the same, you add the absolute values of the addends and use the same sign. When the signs are different, you subtract the absolute values and use the same sign as the addend with the greater absolute value. Adding 0, which is neither positive nor negative, is done using the additive identity of 0: x + 0 = x and 0 + x = x, for any value of x.