explain closer property for rational numbers with 10 examples
Answers
Answer:
●Closure Property
●Commutative Property
●Associative Property
●Distributive Property
●Identity Property
●Inverse Property
Closure property
For two rational numbers say x and y the results of addition, subtraction and multiplication operations give a rational number. We can say that rational numbers are closed under addition, subtraction and multiplication. For example:
6 + 13 = 19
(1/3) + (5/2) = 17/6
: (-8) + 6 = 2
11 + 9 = 20
Closure property of integers under subtraction:
The difference between any two integers will always be an integer, i.e. if a and b are any two integers, a – b will be an integer.
Example: 19 – 6 = 13
-6 – (-3) = -3
Closure property of integers under multiplication:
Any two integers’ product will be an integer, i.e. if a and b are any two integers, ab will also be an integer.
Example: 3 × (-9) = -27
(–7) × (-9) = 63
Closure property of integers under division:
Division of integers doesn’t follow the closure property since the quotient of any two integers a and b, may or may not be an integer. Sometimes the quotient is undefined (when the divisor is 0).
Example: -16 ÷ 4 = -4 (an integer)
(−4) ÷ (−16) = 1/4 (not an integer)
Click here to know more about properties of integers.
Closure property of Rational Numbers
Closure property holds for addition, subtraction and multiplication of rational numbers.
Closure property of rational numbers under addition:
The sum of any two rational numbers will always be a rational number, i.e. if a and b are any two rational numbers, a + b will be a rational number.
Example: (5/6) + (2/3) = 3/2
-(1/2) + (1/4) = -1/4
Closure property of rational numbers under subtraction:
The difference between any two rational numbers will always be a rational number, i.e. if a and b are any two rational numbers, a – b will be a rational number.
Example: (7/8) – (3/8) = 1/2
(6/7) – (-3/7) = 9/7
Closure property of rational numbers under multiplication:
Closure property under multiplication states that any two rational numbers’ product will be a rational number, i.e. if a and b are any two rational numbers, ab will also be a rational number.
Example: (3/2) × (2/9) = 1/3
(-7/4) × (5/2) = -35/8
Closure property of rational numbers under division:
Division of rational numbers doesn’t follow the closure property since the quotient of any two rational numbers a and b, may or may not be a rational number. That means, it can be undefined when we take the value of b as 0.
Learn more about the properties of rational numbers here.
Closure property of Whole Numbers
Closure property holds for addition and multiplication of whole numbers.
Closure property of whole numbers under addition:
The sum of any two whole numbers will always be a whole number, i.e. if a and b are any two whole numbers, a + b will be a whole number.
Example: 12 + 0 = 12
9 + 7 = 16
Closure property of whole numbers under subtraction:
The difference between any two whole numbers may or may not be a whole number. Hence, the whole numbers are not closed under subtraction.
Example: 13 – 14 = -1 (not a whole number)
4 – 0 = 4 (whole number)
Closure property of whole numbers under multiplication:
Any two whole numbers’ product will be a whole number, i.e. if a and b are any two whole numbers, ab will also be a whole number.
Example: 4 × 6 = 24
0 × 7 = 0
Closure property of whole numbers under division:
Division of whole numbers doesn’t follow the closure property since the quotient of any two whole numbers a and b, may or may not be a whole number.
Example: 18 ÷ 4 = 9/2 (not a whole number)
22/2 = 11 (a whole number)
Closure property under Multiplication
The product of two real numbers is always a real number, that means real numbers are closed under multiplication. Thus, the closure property of multiplication holds for natural numbers, whole numbers, integers and rational numbers.
Examples:
8 × 0 = 0
(3/4) × (-1/2) = -3/8
√3 × √5 = √15
(-11) × (-3) = 33