1.2: Adding and Subtracting Whole Numbers
- Page ID
- 137898
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The Commutative Property of Addition
The Commutative Property of Addition
Let a and b represent two whole numbers. Then,
a + b = b + a.
Grouping Symbols
In mathematics, we use grouping symbols to affect the order in which an expression is evaluated. Whether we use parentheses, brackets, or curly braces, the expression inside any pair of grouping symbols must be evaluated first. For example, note how we first evaluate the sum in the parentheses in the following calculation.
(3 + 4) + 5 = 7 + 5
= 12
The rule is simple: Whatever is inside the parentheses is evaluated first.
Writing Mathematics
When writing mathematical statements, follow the mantra:
One equal sign per line.
We can use brackets instead of parentheses.
5 + [7 + 9] = 5 + 16
= 21
Again, note how the expression inside the brackets is evaluated first.
We can also use curly braces instead of parentheses or brackets.
{2+3} + 4 = 5 + 4
= 9
Again, note how the expression inside the curly braces is evaluated first.
If grouping symbols are nested, we evaluate the innermost parentheses first.
For example,
2 + [3 + (4 + 5)] = 2 + [3 + 9]
= 2 + 12
= 14.
Grouping Symbols
Use parentheses, brackets, or curly braces to delimit the part of an expression you want evaluated first. If grouping symbols are nested, evaluate the expression in the innermost pair of grouping symbols first.
The Associative Property of Addition
Consider the evaluation of the expression (2+3)+4. We evaluate the expression in parentheses first.
(2 + 3) + 4 = 5 + 4
= 9
Now, suppose we change the order of addition to 2 + (3 + 4). Then,
2 + (3 + 4) = 2 + 7
= 9.
Although the grouping has changed, the result is the same. That is,
(2 + 3) + 4 = 2 + (3 + 4).
This property of addition of whole numbers is called the associate property of addition.
Associate Property of Addition
Let a, b, and c represent whole numbers. Then,
(a + b) + c = a + (b + c).
Because of the associate property of addition, when presented with a sum of three numbers, whether you start by adding the first two numbers or the last two numbers, the resulting sum is the same.
The Additive Identity
The Additive Identity Property
The whole number zero is called the additive identity. If a is any whole number, the
a + 0 = a.
The number zero is called the additive identity because if you add zero to any number, you get the identical number back.
Adding Larger Whole Numbers
For completeness, we include two examples of adding larger whole numbers. Hopefully, the algorithm is familiar from previous coursework.
Example 1
Simplify: 1, 234 + 498.
Solution
Align the numbers vertically, then add, starting at the furthest column to the right. Add the digits in the ones column, 4 + 8 = 12. Write the 2, then carry a 1 to the tens column. Next, add the digits in the tens column, 3 + 9 = 12, add the carry to get 13, then write the 3 and carry a 1 to the hundreds column. Continue in this manner, working from right to left
\( \begin{array}{r}{11} \\ {1234} \\ {+\quad 498} \\ \hline 1732\end{array}\)
Therefore, 1, 234 + 498 = 1, 732
Exercise
Simplify: 1,286 + 349.
- Answer
-
1635
Add three or more numbers in the same manner.
Example 2
Simplify: 256 + 322 + 418.
Solution
Align the numbers vertically, then add, starting at the furthest column to the right. Add the digits in the ones column, 6 + 2 + 8 = 16. Write the 6, then carry a 1 to the tens column. Continue in this manner, working from right to left.
\( \begin{array}{r}{256} \\ {322} \\ {+418} \\ \hline 996\end{array}\)
Therefore, 256 + 322 + 418 = 996.
Exercise
Simplify: 256 + 342 + 283
- Answer
-
881
Subtraction of Whole Numbers
The key idea is this: Subtraction is the opposite of addition. For example, consider the difference 7 - 4 depicted on the number line in Figure 1.5.
If we were adding 7 and 4, we first draw an arrow starting at zero pointing to the right with magnitude (length) seven. Then, to add 4, we would draw a second arrow of magnitude (length) 4, attached to the end of the first arrow and pointing to the right.
However, because subtraction is the opposite of addition, in Figure 1.5 we attach an arrow of magnitude (length) four to the end of the first arrow, but pointing in the opposite direction (to the left). Note that this last arrow ends at the answer, which is a shaded dot on the number line at 3. That is, 7 − 4 = 3.
Note that subtraction is not commutative; that is, it make no sense to say that 7 − 5 is the same as 5 − 7.
Subtraction is not associative. It is not the case that (9 − 5) − 2 is the same as 9 − (5 − 2). On the one hand,
(9 − 5) − 2 = 4 − 2
= 2,
but
9 − (5 − 2) = 9 − 3
= 6.
Subtracting Larger Whole Numbers
Much as we did with adding larger whole numbers, to subtract two large whole numbers, align them vertically then subtract, working from right to left. You may have to “borrow” to complete the subtraction at any step.
Example 3
Simplify: 1, 755 − 328.
Solution
Align the numbers vertically, then subtract, starting at the ones column, then working right to left. At the ones column, we cannot subtract 8 from 5, so we borrow from the previous column. Now, 8 from 15 is 7. Continue in this manner, working from right to left.
Therefore, 1, 755 − 328 = 1, 427.
Exercise
Simplify: 5,635 - 288.
- Answer
-
5,347
Order of Operations
In the absence of grouping symbols, it is important to understand that addition holds no precedence over subtraction, and vice-versa.
Perform all additions and subtractions in the order presented, moving left to right.
Let’s look at an example.
Example 4
Simplify the expression \(15 − 8 + 4\).
Solution
This example can be trickier than it seems. However, if we follow the rule (perform all additions and subtractions in the order presented, moving left to right), we should have no trouble. First comes fifteen minus eight, which is seven. Then seven plus four is eleven.
\[\begin{align*}15 − 8 + 4 &= 7 + 4 \\[4pt] &= 11. \end{align*}\]
Exercise
Simplify: \(25 − 10 + 8\).
- Answer
-
23
Caution! Incorrect answer ahead!
Note that it is possible to arrive at a different (but incorrect) answer if we favor addition over subtraction in Example 4. If we first add eight and four, then 15 − 8 + 4 becomes 15 − 12, which is 3. However, note that this is incorrect, because it violates the rule “perform all additions and subtractions in the order presented, moving left to right.”
Exercises
Determine which property of addition is depicted by the given identity
15. (51 + 66) + 88 = 51 + (66 + 88)
17. 64 + 39 = 39 + 64
21. 79 + 0 = 79
Simplify the given expression.
39. 16 − 8+2
49. 15 − 5+8
Find the sum.
75. 8583 + 592
78. 841 + 368 + 919
85. (86 + 557) + 80
Find the difference.
89. 8338 − 7366
92. 881 − 606
93. 3838 − (777 − 241)
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Answers
15. Associative property of addition
17. Commutative property of addition
21. Additive identity property of addition.
39. 10
49. 18
75. 9175
78. 2128
85. 723
89. 972
92. 275
93. 3302
105. $518 million
107. $738
109. 5 million kilometers
112. 65 people