Commutative Property
- Introduction
- What is Commutative Property?
- Commutative Property of Addition
- Commutative Property of Multiplication
- Other Properties
- Non-Commutative Property
- Summarizing Commutative Property
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Introduction
The commutative property applies to addition and multiplication, allowing us to swap numbers without affecting the outcome. For instance, adding `6` and `8` gives the same result as adding `8` and `6`. This property also holds for multiplication. The concept of the commutative property dates back to ancient times, although its formal use began in the late `18`th century. The term "commutative" originates from the French word "commute" meaning to switch or move around, indicating that swapping the positions of numbers doesn't alter the result.
What is Commutative Property?
The commutative property, also known as the commutative law, tells us that when we add or multiply numbers, changing their positions doesn't change the result. This means if we switch the order of numbers in addition or multiplication, the answer remains the same. For instance, if we add `4` and `7`, it's the same as adding `7` and `4`. Similarly, if we multiply `3` and `6`, it's equivalent to multiplying `6` and `3`. This property is helpful in addition and multiplication. However, it's important to note that this property doesn't work for subtraction or division. If you change the order of numbers in a subtraction or division problem, you'll get a different answer. For example, subtracting `10` from `15` is not the same as subtracting `15` from `10`.
Commutative Property of Addition
The commutative property of addition states that changing the order of the numbers being added doesn't affect the sum. So, whether we add `10` and `15` or `15` and `10`, we'll still get the same result. This property is represented by the formula `A + B = B + A`. For example, when we add `7` and `11`, it's the same as adding `11` and `7`, proving the commutative property of addition.
Commutative Property of Multiplication
The commutative property of multiplication tells us that no matter the order in which we multiply two numbers, the product remains the same. So, whether we multiply `3` with `7` or `7` with `3`, the product will always be unchanged. This rule applies to any two integers. For instance, if we multiply `5` and `9`, it's the same as multiplying `9` and `5`.
Other Properties
Associative Property of Addition and Multiplication
The associative property tells us that no matter how we group the numbers when adding or multiplying, the result remains the same. In simpler terms, it doesn't matter where we put the parentheses; the answer stays unchanged. For addition, this means `(A + B) + C` equals `A + (B + C)`, and for multiplication, `(A × B) × C` equals `A × (B × C)`.
Distributive Property of Multiplication
The distributive property of multiplication tells us that when we multiply a number by a sum, it's the same as multiplying each addend by that number separately and then adding the products. In simpler terms, we can distribute the multiplication across the terms inside the parentheses. For example, if we multiply `3` by the sum of `7` and `9`, it's the same as multiplying `3` by `7` and then `3` by `9`, and finally adding the two products together. This property is represented by the equation `a × (b + c) = (a × b) + (a × c)`. For instance, `2 × (6 + 8) = (2 × 6) + (2 × 8) = 12 + 16 = 28`
Non-Commutative Property
Some mathematical operations don't follow the commutative property, meaning that changing the order of the numbers gives different results. Subtraction and division are examples of non-commutative operations. Unlike addition, where switching the order doesn't change the answer, in subtraction, rearranging the terms results in different outcomes. For instance, `5 - 3` gives `2`, but `3 - 5` gives `-2`, which are two distinct integers. Similarly, the division doesn't obey the commutative law. For example, when we divide `8` by `2`, we get `4`, but when we divide `2` by `8`, we get `1/4`. Therefore, `8` divided by `2` is not the same as `2` divided by `8`.
Summarizing Commutative Property
- The commutative property asserts that altering the order of the operands doesn't alter the outcome.
- In addition, the commutative property is expressed as `A + B = B + A`.
- For multiplication, the commutative property is expressed as `A × B = B × A`.
Solved Examples
Example `1`: Utilize `8 × 6 = 48` to determine `6 × 8`.
Solution:
By applying the commutative property of multiplication, `6 × 8` equals `8 × 6`.
Given that `8 × 6` equals `48`, thus, `6 × 8` also results in `48`.
Example `2`: Complete the equations using the commutative property.
`"_________" + 52 = 52 + 19`
`72 + 36 = 36 + "_________"`
`95 ×" ______" = 81 × 95`
`124 × 67 = "________" × 124`
Solution:
`19`; by the commutative property of addition
`72`; by the commutative property of addition
`81`; by the commutative property of multiplication
`67`; by the commutative property of multiplication
Example `3`: Given `563 + 298 = 861`, determine `298 + 563`.
Solution:
Applying the commutative property of addition, we know that `298 + 563` equals `563 + 298`.
Since `563 + 298` equals `861`, therefore, `298 + 563` also equals `861`.
Example `4`: Apply the commutative property of addition to express the equation, `6 + 8 + 2 = 16`, with a different arrangement of the addends.
Solution:
`6 + 2 + 8 = 16` (because `8 + 2 = 2 + 8`)
`8 + 6 + 2 = 16` (because `6 + 8 = 8 + 6`)
`2 + 8 + 6 = 16` (because `6 + 2 = 2 + 6`)
Similarly, we can rearrange the addends and write:
`2 + 6 + 8 = 16`
`8 + 2 + 6 = 16`
Example `5`: Sarah purchased `4` packs of `5` notebooks each. Emma purchased `5` packs of `4` notebooks each. Did they buy an equal number of notebooks?
Solution:
Sarah purchased `4` packs of `5` notebooks each.
Therefore, the total number of notebooks Sarah bought `= 4 × 5`
Emma purchased `5` packs of `4` notebooks each.
Therefore, the total number of notebooks Emma bought `= 5 × 4`
According to the commutative property of multiplication, `4 × 5 = 5 × 4`.
Hence, both Sarah and Emma bought an equal number of notebooks.
Example `6`: Tom has `45` packets of `7` balloons each. Lucy has `7` packets of `45` balloons each. Do they have an equal number of balloons?
Solution:
Since Tom has `45` packets of `7` yellow balloons.
So, the total number of balloons with Tom `= 45 × 7`
Lucy has `7` packets of `45` balloons each.
So, the total number of balloons with Lucy `= 7 × 45`
By the commutative property of multiplication, `45 × 7 = 7 × 45`.
So, both Tom and Lucy have an equal number of balloons.
Practice Problems
Q`1`. Which of the following expressions demonstrates the commutative property of addition?
- `5 + 8`
- `8 - 5`
- `5 - 8`
- `8 × 5`
Answer: a
Q`2`. Which of the following expressions would exhibit the commutative property of multiplication?
- `4 × 7`
- `7 + 4`
- `4 ÷ 7`
- `7 ÷ 4`
Answer: a
Q`3`. Identify the pair of expressions that demonstrate the commutative property of addition.
- `3 + 9` and `9 + 3`
- `6 × 2` and `2 × 6`
- `10 - 4` and `4 - 10`
- `5 ÷ 2` and `2 ÷ 5`
Answer: a
Q`4`. Which of the following equations exhibits the commutative property of multiplication?
- `3 × (2 + 4) = (3 × 2) + (3 × 4)`
- `(4 × 6) + (3 × 6) = (4 + 3) × 6`
- `5 × 7 = 7 × 5`
- `8 ÷ (6 - 3) = (8 ÷ 6) - (8 ÷ 3)`
Answer: c
Q4. Which of the following equations obeys the commutative property?
- \( 8 \times 5 \)
- \( 15 + 9 \)
- \( 45 \div 9 \)
- \( 45 - 9 \)
- \( -3 \times 5 \)
Answer: a, b and e
Frequently Asked Questions
Q`1`. What is the commutative property in mathematics?
Answer: The commutative property is a fundamental principle in mathematics that applies to addition and multiplication operations. It states that changing the order of the numbers being added or multiplied doesn't change the result. For addition, it's expressed as `A + B = B + A`, and for multiplication, it's expressed as `A × B = B × A`.
Q`2`. Which operations follow the commutative property?
Answer: The commutative property applies only to addition and multiplication operations. For these operations, changing the order of the numbers being added or multiplied doesn't change the outcome. However, subtraction and division do not follow the commutative property.
Q`3`. Can the commutative property be applied to variables and algebraic expressions?
Answer: Yes, the commutative property can be applied to variables and algebraic expressions. For addition and multiplication of variables or algebraic terms, changing the order of terms does not affect the result. For example, `x + y = y + x` and `xy = yx`.
Q`4`. Why is the commutative property important in mathematics?
Answer: The commutative property is important because it simplifies calculations and helps identify equivalent expressions. It allows us to rearrange terms without changing the outcome, making mathematical operations more flexible and easier to work with.
Q`5`. Does the commutative property hold for all types of numbers?
Answer: Yes, the commutative property holds for all types of numbers, including integers, fractions, decimals, and even irrational numbers. As long as the operations are addition or multiplication, the commutative property applies, regardless of the type of numbers involved.