What properties, the following expressions show?
(i) 23+45=45+23
Answers
Step-by-step explanation:
Subtraction is not commutative. For example, 4 − 7 does not have the same difference as 7 − 4. The − sign here means subtraction.
However, recall that 4 − 7 can be rewritten as 4 + (−7), since subtracting a number is the same as adding its opposite. Applying the commutative property for addition here, you can say that 4 + (−7) is the same as (−7) + 4. Notice how this expression is very different than 7 – 4.
Now look at some multiplication examples.
Original Equation
Rewritten Equation
4.5 · 2 = 9
2 · 4.5 = 9
(−5) · 3 = -15
3 · (−5) = -15
Commutative Property of Multiplication
For any real numbers a and b, a · b = b · a.
Order does not matter as long as the two quantities are being multiplied together. This property works for real numbers and for variables that represent real numbers.
Just as subtraction is not commutative, neither is division commutative. 4 ÷ 2 does not have the same quotient as 2 ÷ 4.
Example
Problem
Write the expression (−15.5) + 35.5 in a different way, using the commutative property of addition, and show that both expressions result in the same answer.
(−15.5) + 35.5 = 20
35.5 + (−15.5)
35.5 + (−15.5)
35.5 – 15.5 = 20
Adding.
Using the commutative property, you can switch the −15.5 and the 35.5 so that they are in a different order.
Adding 35.5 and −15.5 is the same as subtracting 15.5 from 35.5. The sum is 20.
Answer (−15.5) + 35.5 = 20 and 35.5 + (−15.5) = 20
Rewrite 52 • y in a different way, using the commutative property of multiplication. Note that y represents a real number.
A) 5y • 2
B) 52y
C) 26 • 2 • y
D) y • 52
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The Associative Properties of Addition and Multiplication
The associative property of addition states that numbers in an addition expression can be grouped in different ways without changing the sum. You can remember the meaning of the associative property by remembering that when you associate with family members, friends, and co-workers, you end up forming groups with them.
Below, are two ways of simplifying the same addition problem. In the first example, 4 is grouped with 5, and 4 + 5 = 9.
4 + 5 + 6 = 9 + 6 = 15
Here, the same problem is worked by grouping 5 and 6 first, 5 + 6 = 11.
4 + 5 + 6 = 4 + 11 = 15
In both cases, the sum is the same. This illustrates that changing the grouping of numbers when adding yields the same sum.
Mathematicians often use parentheses to indicate which operation should be done first in an algebraic equation. The addition problems from above are rewritten here, this time using parentheses to indicate the associative grouping.
(4 + 5) + 6 = 9 + 6 = 15
4 + (5 + 6) = 4 + 11 = 15
It is clear that the parentheses do not affect the sum; the sum is the same regardless of where the parentheses are placed.
Associative Property of Addition
For any real numbers a, b, and c, (a + b) + c = a + (b + c).
The example below shows how the associative property can be used to simplify expressions with real numbers.
Example
Problem
Rewrite 7 + 2 + 8.5 – 3.5 in two different ways using the associative property of addition. Show that the expressions yield the same answer.
7 + 2 + 8.5 – 3.5
7 + 2 + 8.5 + (−3.5)
The associative property does not apply to expressions involving subtraction. So, re-write the expression as addition of a negative number.
(7 + 2) + 8.5 + (−3.5)
9 + 8.5 + (−3.5)
17.5 + (−3.5)
17.5 – 3.5 = 14
Group 7 and 2, and add them together. Then, add 8.5 to that sum. Finally, add −3.5, which is the same as subtracting 3.5.
Subtract 3.5. The sum is 14.
7 + 2 + (8.5 + (−3.5))
7 + 2 + 5
9 + 5
14
Group 8.5 and –3.5, and add them together to get 5. Then add 7 and 2, and add that sum to the 5.
The sum is 14.
Answer (7 + 2) + 8.5 – 3.5 = 14 and 7 + 2 + (8.5 + (−3.5)) = 14
Multiplication has an associative property that works exactly the same as the one for addition. The associative property of multiplication states that numbers in a multiplication expression can be regrouped using parentheses. For example, the expression below can be rewritten in two different ways using the associative property.
Original expression:
Expression 1:
Expression 2:
The parentheses do not affect the product, the product is the same regardless of where the parentheses are.
Associative Property of Multiplication
For any real numbers a, b, and c, (a • b) • c = a • (b • c).
Rewrite using only the associative property.
A)
B)
C)
D)
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Using the Associative and Commutative Properties
You will find that the associative and commutative properties are helpful tools in algebra, especially when you evaluate expressions. Using the commutative and associative properties, you can reorder terms in an expression so that compatible numbers are next to each other and grouped together. Compatible numbers are numbers that are easy for you to compute, such as 5 + 5, or 3 · 10, or 12 – 2, or 100 ÷ 20. (The main criteria for compatible numbers is that they “work well” together.) The two examples below show how this is done.
Example
Problem
Evaluate
Answer:
This expression is called ""Commutativity""
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