Transcript Adding and Subtracting Real Numbers

Adding and Subtracting Real
Numbers
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Add two numbers with the same sign.
Add positive and negative numbers.
Use the definition of subtraction.
Use the rules for order of operations with real
numbers.
Interpret words and phrases involving
addition and subtraction.
Use signed numbers to interpret data.
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Objective 1
Add two numbers with the same
sign.
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Add two numbers with the same sign.
The sum of two negative numbers is a negative number whose
distance from 0 is the sum of the distance of each number from 0.
That is, the sum of two negative numbers is the negative sum
of the sum of their absolute values.
To add two numbers with the same sign, add the absolute
values of the numbers. The sum has the same sign as the given
numbers.
Example: 4   3  7
To avoid confusion, two operation symbols should not be written
successively without a parenthesis between them.
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EXAMPLE 1
Adding Numbers on a Number
Line
Use a number line to find each sum.
Solution:
1 4
5
2   5
−7
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EXAMPLE 2
Adding Two Negative Numbers
Find the sum.
15   4
Solution:
   15  4 
  15  4
 19
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Objective 2
Add positive and negative
numbers.
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Add two numbers with the different signs.
To add two numbers with different signs, find the absolute
values of the numbers and subtract the smaller absolute value
from the larger. Give the answer the sign of the number having
the larger absolute value.
For instance, to add −12 and 5, find their absolute values:
12  12 and 5  5
Then find the difference between these absolute values:
12  5  7
The sum will be negative, since 12
5,
so the final answer is 12  5  7.
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EXAMPLE 3
Adding Numbers with Different
Signs
Use a number line to find the sum.
Solution:
6   3
3
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EXAMPLE 4
Adding Mentally
Check each answer.
Solution:
3  11 
5
   
4  8
8
Correct
3.8  9.5  5.7
Correct
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Objective 3
Use the definition of subtraction.
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Use the definition of subtraction.
We can illustrate the subtraction of 4 from 7, written
7  4 , with a number line.
The procedure to find the difference 7  4 is exactly
the same procedure that would be used to find the sum
7  (4) , so 7  4  7  (4)
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Use the definition of subtraction. (cont’d)
The previous equation suggests that subtracting a
positive number from a larger positive number is the
same as adding the additive inverse of the smaller
number to the larger.
From this comes the definition of subtraction, for
any real numbers x and y,
x  y  x  ( y).
That is, to subtract y from x, add the additive inverse
(or opposite) of y to x.
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EXAMPLE 5
Use the Definition of Subtraction
Subtract.
Solution:
8  5
 8  (5)
 13
8  (12)  8  (12)
4
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Uses of the Symbol −
We use the symbol − for three purposes:
1. to represent subtraction, as in 9  5  4;
2. to represent negative numbers, such as −10, −2, and −3;
3. to represent the opposite (or negative) of a number, as in
“the opposite (or negative) of 8 is −8.”
We may see more than one use of − in the same problem, such
as −6 − (−9) where −9 is subtracted from −6. The meaning of the
symbol depends on its position in the algebraic expression.
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Objective 4
Use the rules for order of
operations with real numbers.
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EXAMPLE 6
Adding and Subtracting with
Grouping Symbols
Perform the indicated operations.
Solution:
6   1  4   2 
 6   1   4    2 
 6  5  2
 6   5   2  
 6   7 
 1
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Objective 5
Interpret words and phrases
involving addition and subtraction.
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Interpret words and phrases involving
addition.
The word sum indicates addition. The table lists other words
and phrases that indicate addition in problem solving.
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EXAMPLE 7
Interpreting Words and Phrases
Involving Addition
Write a numerical expression for the phrase and
simplify the expression.
7 increased by the sum of 8 and −3
Solution:
7  8   3 
 7   5
 12
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Interpret words and phrases involving
subtraction.
The word difference indicates subtraction. Other words and
phrases that indicate subtraction in problem solving are given in
the table.
In subtracting two numbers, be careful to write them in the correct order,
because in general, a  b  b  a. For example, 5  3  3  5.
Think carefully before interpreting an expression involving subtraction.
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EXAMPLE 8
Interpreting Words and Phrases
Involving Subtraction
Write a numerical expression for the phrase and
simplify the expression.
−2 subtracted from the sum of 4 and −4
Solution:
4  (4)  (2)
  0  2
2
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EXAMPLE 9
Solving a Problem Involving
Subtraction
The highest Fahrenheit temperature ever recorded in
Barrow, Alaska, was 79°F, while the lowest was −56°F.
What is difference between these highest and lowest
temperatures? (Source: World Almanac and Book of
Facts 2006.)
Solution:
79  (56)
 79  56
 135 F
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Objective 6
Use signed numbers to interpret
data.
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EXAMPLE 10
Using a Signed Number to
Interpret Data
Refer to Figure 17 and use a signed number to represent the
change in the PPI from 2002 to 2003.
Solution:
135.3  108.1
 135.3  (108.1)
 27.2
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