Properties of Integers

We will learn about properties of integers. This is resume of integers properti

properties of integers

Let us now study these properties in detail.

Closure Property

The System of Integers in Addition
It states that addition of two Integers always results in an Integer. For example, 7 + 4 = 11,  the result we get is an integer. Therefore, the system is closed under addition.

The System of Integers under Subtraction
It states that subtraction of two Integers always results in an Integer. For example, 7 – 4 = 3, the result we get is an integer. Also, 2 – 4 = -2. The result is also an integer. Therefore, the system is closed under subtraction.

The System of Integers under Multiplication
It states that multiplication of two integers always results in an integer. For example, 7 × 4 = 28, the result we get is an integer. Therefore, the system is closed under multiplication.

The System of Integers under Division
It states that division of two integers does not always result in an integer. For example, 7 ÷ 4 = 74, the result we get is not an integer. But, 8 ÷ 4 = 2, the result we get is an integer. Therefore, a system is not closed under division.

Commutative Property

It is a property that associates with binary operations or functions like addition, multiplication. Take any two numbers a and b and subtract them. That is a – b, say 5 – (-3). Now subtract a from b. That is b – a or -3 – 5. Are they same? No, they are not equal. So, the commutative property does not hold for subtraction. Similarly, it does not hold for division too.

Again take any two numbers a and b and add them. That is a + b. Now add b and a which comes to be b+ a. Aren’t the same? Yes, they are equal because of commutative property which says that we can swap the numbers and still we get the same answer.

Associative Property

Associative property of integers states that for any three elements(numbers) a, b and c

1) For Addition a + ( b + c ) = ( a + b ) + c
2) For Multiplication a × ( b × c ) = ( a × b ) × c
3) For Subtraction. Associative property does not hold for subtraction a – ( b – c ) != ( a – b ) – c
4) For Division. Associative property does not hold for division a ÷ ( b ÷ c ) != ( a ÷ b ) ÷ c

Multiplicative Identity for Integers
The multiplicative identity of any integer a  is a number b which when multiplied with a, leaves it unchanged, i.e. b is called as the multiplicative identity of any integer a if a× b = a. Now, when we multiply 1 with any of the integers a we get a × 1 = a = 1 × a  So, 1 is the multiplicative identity for integers.

Additive Identity for Integers

The additive identity of any integer a  is a number b which when added with a, leaves it unchanged, i.e. b is called as the additive identity of any integer a if a + b = a. Now, when we add 0 with any of the integers a we get a + 0 = a = 0 + a  So, 0 is the additive identity for integers.

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Source:
https://www.toppr.com/guides/maths/rational-numbers/properties-of-integers/