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Properties of Rational Numbers - Commutativity

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notes

1) Whole numbers -

Operation Numbers Remarks

 

Addition

0 + 7 = 7 + 0 = 7 ,2 + 3 = 3 + 2 = 5
For any two whole numbers a and b, a + b = b + a

Addition is commutative .
Subtraction 3 - 2 = 1  but
2 - 3 = -1 
For any teo whole numbers a and b , a-b ≠ b - a 
Subtraction is not commutative.
Multiplication 2 × 3 = 6 and
3 × 2 = 6
For any two whole numbers a and b, a × b = b × a
Multiplication is commutative . 
Division 4 ÷ 2 = 2 but
2 ÷ 4 = 0.5
For any two whole numbers a and b, a ÷ b ≠ b ÷ a
Division is not commutative .

2 ) Integers :

Operation Numbers  Remarks 

Addition

-2 + 3 = 1 and
3 + (-2) = 1  
Addiion is commutative.
Subtraction (-2)-(1) = -3  but  (1) - (-2) = 3    Subtraction is not commutative.
Multiplication -2 * (-7) = 14 and (-7) * (-2) = 14 Multiplication is commutative.
Division -2 ÷ 3 = - 1.666... but 3 ÷ -2 = -1.5  Division  is not commutative .

3) Rational numbers : 
a. Addition
example- `-2/3 + 5/7 = 1/21` and `5/7+(-2/3)=1/21`
We say that two rational numbers can be added in any order. We say that addition is commutative for rational numbers. That is, for any two rational numbers a and b, 
a + b = b + a.

b. Subtraction :
Example - `2/3 - 5/4 =-7/12` and `5/4 - 2/3 = 7/12`
Note  that subtraction is not commutative for integers and integers are also rational numbers. So, subtraction will not be commutative for rational numbers too.

c. Multiplication :
Example - `-7/3 xx 6/5 = -42/15 = 6/5 xx -7/3`
We say that  multiplication is commutative for rational numbers. In general, a × b = b × a for any two rational numbers a and b. 

d. Division :
Example - `-5/4 ÷3/7 = -35/12`, `3/7 ÷ (-5/4) = (12/-35)`
The expressions on both sides are not equal. So division is not commutative for rational numbers.

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