§1.2 Addition and Subtraction Real Numbers

8/7/2019 §1.2 Addition and Subtraction Real Numbers

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Chapter 1 – The R eal Numbers

§1.2 Addition and Subtraction of R eal Numbers

We are now going to learn the arithmetic of real numbers. There are only four

arithmetic operations and I’m sure you may be familiar with them: addition, subtraction,multiplication, and division.

First, recall from the last section we were told that all of the points on a number line

represent the set of real numbers. The points to the right of zero are positive real numbersand the points to the left of zero are negative real numbers. Remember, zero has no sign

itself.

Lesson 1 – The Absolute Value

Before looking at the arithmetic of real numbers our first Lesson in this section will

introduce the mathematical operation called absolute value.

Definition of the Absolute Value

The absolute value of a number is the distance on a number line between 0

and that number.

It is important to note that by definition the absolute value of a number expresses a

distance – and since distances are never negative the absolute value of a number is never

negative.

To mathematically indicate the absolute value of a number we place that number between

two vertical bars. For example 3 or 5− .

According to the above definition then for absolute value 33 = since the number 3 is 3

units away from zero and 55 =− since 5− is 5 units away from zero. Note the

direction from zero does not matter only how far from zero.



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Example 1

Find the following absolute values.

(a) 6

(b) 7−

(c)103

(d) 3.2−

(e) π −−

Solutions

(a) 66 =

(b) 77 =−

(c)103

103 =

(d) 3.23.2 =−

(e) ( ) π π π −=−=−−

Do §1.2 Lesson 1 Example 1 Quiz

Lesson 2 – Adding Real Numbers with the Same Sign

To explain the addition of real numbers we will first use a number line. We will then

apply the above definition of absolute value to formally define how to add real numbers

with the same sign.

To add two positive real numbers, say 2 + 3 we begin at zero and move two units to the

right (in a positive direction since 2 is a positive real number). This point represents thenumber 2. Next we move 3 more units to the right. This will take us to the point 5. We

conclude then that 2 + 3 = 5. See Figure 1.2.1 which follows.



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Figure 1.2.1

To add two negative real numbers, say ( )42 −+− , on the number line we begin at zero

and move 2 units to the left (in a negative direction). This point represents 2− . Next we

move 4 more units to the left. This will take us to the point 6− . We conclude that

( ) 642 −=−+− .

Figure 1.2.2

Adding real numbers with the same sign we can observe from the above two examples

that both arrows point in the same direction, building upon each other. The sum in eachcase has the same sign as the numbers being added.

This suggests the following rule.

Rule for Adding Two RealNumbers with the Same Sign

To add two real numbers with the same sign, add their absolute values and attach

their common sign to the sum.



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Example 2

Use the rule for adding two real numbers with the same sign to find

the following sums.

(a) 12 + 13

(b) ( )47 −+−

(c) 28 + 25

(d) ( )615 −+−

(e)

−+−

3

2

3

2

Solutions

(a) 12 + 13 = 25

(b) ( ) 1147 −=−+− . This is found be adding the absolute values,

7 and 4, to get 11. We then use their common sign.

(c) 28 + 25 = 53

(d) ( ) 21615 −=−+− . The sum of their absolute values, 15 and 6,

is 21. Use their common sign.

(e) 3

4

3

2

3

2−=

−+− . The sum of their absolute values is

3

4. We

then use their common sign to get the sum.


Lesson 3 – Adding Real Numbers with Different Signs

We turn our attention next to finding the sum of ( )35 −+ . Again we will use the number

line to assist us.



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To add ( )35 −+ we begin at zero and move 5 units to the right (in a positive direction).

This takes us to 5. Then we move 3 units to the left (because the 3 is a negative 3 and we

must move in the negative direction). This lands us at 2. We conclude that ( ) 235 =−+ .

See the figure below.

Figure 1.2.3

And to add 34 +− we begin at zero and move 4 units to the left (in a negative direction).

This takes us to -4. Then we move 3 units to the right (in a positive direction). This landsus at -1. We conclude that 134 −=+− . See the figure below.

Figure 1.2.4

From these two examples we can see that the arrows point in opposite directions and that

the longer arrow determines the sign of the sum.

This suggests the following rule

Rule for Adding Two RealNumbers with Different Signs

To add two real numbers with different signs, subtract their absolute values

(smaller number from larger number) and attach the sign of the number with the

larger absolute value to the sum.



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Example 3

(a) 412 +−

(b) ( )519 −+

(c) 710 +−

(d) ( )2.33.2 −+

(e)4

3

2

1+−

Solutions

(a) 8412 −=+− . Subtract their absolute values (12 minus 4) to

get 8. Use the negative sign since the arrow for -12 would be

longer.(b) ( ) 14519 =−+ . Subtract their absolute values (19 minus 5) to

get 14.

(c) 3710 −=+− . Subtract their absolute values (10 minus 7) to

get 3. Use the negative sign.

(d) ( ) 9.02.33.2 −=−+ . Subtract their absolute values (3.2 minus

2.3) to get 0.9. Use the negative sign.

(e)4

1

4

3

2

1=+− . Subtract their absolute values

4

1

2

1

4

3=−− .


Lesson 4 – Adding Several Real Numbers

There are few problems in mathematics where only two numbers need to be combinedthrough the arithmetic operation of addition. Often several numbers of varying signs will

need to be combined.

Let us look at an example of adding several real numbers.



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Example 4

Add the following real numbers.

(a) ( ) 2713 +−+ (b) ( )12314 −++−

(c) ( ) ( ) 2.29.03.15.0 +−+−+

Solutions

(a) The sum can be found by adding from left-to-right, two

numbers at a time. This will give us

( ) 2713 +−+ Original problem

= 26 + 13 + (-7) = 6

= 8 The sum of 6 + 2 = 8.

(b) Again, adding from left-to-right using the rules we have just

learned.

( )12314 −++− Original problem

= ( )1211 −+− -14 + 3 = -11

= 23− The sum of -11 + -12 = -23.

(c) Again, left-to-right, following the rules.

( ) ( ) 2.29.03.15.0 +−+−+ Original problem

= ( ) 2.29.08.0 +−+− 0.5 + (-1.3) = -0.8

= 2.27.1 +− -0.8 + (-0.9) = -1.7

= 0.5 The sum of -1.7 + 2.2 = 0.5.




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Lesson 5 – Adding Several Real Numbers Using An Alternative Method

Let us look at a second way to do these same problems.

Consider the following problem which is similar to part (c) from the example above:

( ) ( ) 229135 +−+−+ .

Instead of doing the addition from left-to-right we could reorder and then add the two

negatives numbers together, add the two positive numbers together, and then find the sumof those two sums.

In doing so we would find the following.

( ) ( ) 229135 +−+−+ Original problem

= ( ) 225913 ++−+− Reordering: putting the two negatives together and the two positives together

= 2722 +− The sum of the two negatives is -22 and the sum of the two

positives is 27.

= 5 The sum of -22 + 27 = 5.

We are allowed to reorder as we did because of a special property in arithmetic called the

commutative property of addition. Certainly you may have noticed from some of your

work in arithmetic that 3 + 4 and 4 + 3 both equal 7. This is essentially what the

commutative property says: the order of addition does not matter. You will obtain thesame result.

This property says that for any real numbers a and b, that a + b = b + a. This propertywill be formally stated in §1.4 Properties of Real Numbers. This is also one of our first

uses of algebra. Algebra allows us to state properties of arithmetic by using letters to

represent numbers.

Example 5

Use the alternative method shown above to solve the following.

(a) ( ) 19121710 +−++−

(b) ( )9.81.52.74.3 −+++−

(c)

−++

−+

3

2

4

1

4

3

2

1



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Solutions

(a) The solution steps follow.

( ) 19121710 +−++− Original problem

= ( ) 19171210 ++−+− Reordering

= 3622 +− Summing the negatives and summingthe positives

= 14 The sum of -22 + 36 = 14.

(b) The solution steps follow.

( )9.81.52.74.3 −+++− Original problem

= ( ) 1.52.79.84.3 ++−+− Reordering

= 3.123.12+−

Summing the negatives and summing

the positives

= 0 The sum of -12.3 + 12.3 = 0.

(c) The solution steps follow.

−++

−+

3

2

4

1

4

3

2

1

Original problem

=

−+

−++

3

2

4

3

4

1

2

1

Reordering

=

−+

−++ 12

8

12

9

12

3

12

6

Obtaining a common denominator

=

−+

12

17

12

9

Summing the positives and summing

the negatives

=3

2

12

8−=−

The sum is obtained and simplified.




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Lesson 6 - Subtracting Real Numbers

The subtraction problem of 5 – 3 can be thought of as beginning from the origin move 5units to the right and then 3 units to the left. These moves will land us on the point 2 as

shown below. The answer, 2 is called the difference.

Figure 1.2.5

This figure is the same as that used to illustrate the problem 5 + (-3) at the beginning of

Lesson 3 – Figure 1.2.3. The two problems, 5 – 3 and ( )35 −+ , have the same result of 2and suggest that to take subtract one real number from another we can simply change the

sign of the second number and change the problem to addition.

This suggests the following rule for subtracting real numbers.

Rule for Subtracting Two RealNumbers

If a and b represent any two real numbers, then

).( baba −+=−

Again this rule is saying that subtraction of real numbers is the same as adding the

opposite of the number we are subtracting.

Clearly this rule is not needed for every subtraction problem since 5 – 3 = 2 is obvious.

But other problems like ( )27 −−− where the difference is not obvious, this subtraction

rule of adding the opposite will come in handy.

To illustrate,

( )27 −−− Original problem

= 27 +− Add the opposite of –2, which is 2.

= 5− Perform the addition of two real numbers with different signs.



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Example 6

Subtract the following real numbers.

(a) ( )713 −−

(b) ( )1214 −−−

(c) 2.25.0 −−

Solutions

(a) We will apply the rule for subtracting.

( )713 −− Original problem

= 713 + Add the opposite of –7, which is 7

= 20 The sum is 20.

(b) Again, applying the subtraction rule.

( )1214 −−− Original problem

= 1214 +− Add the opposite of –12, which is 12.

= 2− Perform the addition of two real numbers

with different signs.

(c) Apply the subtraction rule.

2.25.0 −− Original problem

= )2.2(5.0 −+− Add the opposite of 2.2, which is –2.2.

= 7.2− Add two real numbers with the same sign.




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Lesson 7 – Subtracting Several Real Numbers

As mentioned in Lesson 4 of this section, there are few problems in mathematics where

only two numbers need to be combined through the arithmetic operation of subtraction.

Often several numbers of varying signs will need to be combined. Let’s look at anexample of subtracting several real numbers.

Example 6

Subtract the following real numbers.

(a) ( ) 19713 −−−

(b) )6.4(8.25.3 −−−−

(c)4

3

3

2

2

1−

−−

Solutions

(a) We will apply the rule for subtracting.

( ) 19713 −−− Original problem

= )19(713−++

Add the opposite of –7, which is 7, and

the opposite of 19 which is -19= 1 Adding gives us the answer.

(b) Again, applying the subtraction rule.

)6.4(8.25.3 −−−− Original problem

= 6.4)8.2()5.3( +−+− Adding opposites

= 7.1− Adding gives us the answer.



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Solutions (cont.)

(c) Apply the subtraction rule.

4

3

3

2

2

1 −

−− Original problem

=12

9

12

8

12

6−

−− Determine a common denominator

=

−++

12

9

12

8

12

6 Adding opposites

=12

5 Adding gives us the answer.


Be sure to do the Blackboard Quiz forthis entire Section §1.2.

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§1.2 Addition and Subtraction Real Numbers