Let us carry out an activity. Take a jar and add coins into the jar. Suppose you add three coins to a jar, It represented by a positive value +3. Now remove three coins from a jar so this is represented by a negative value -3. But can you remove half of or a third of a coin? NO! So here the integers always represent the number of coins added or taken. Let us study this in detail.

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**What are Integers?**

An integer is any number which is included in a set of combination of all the numbers types. i.e. if we construct a set of all natural numbers, whole numbers and negative numbers then such a set is called an Integer set. Integer is represented as

(……-3, -2, -1, 0, 1, 2, 3, ……)

So, every natural number is an integer, every whole number is an integer, and every negative number is an integer.

## Properties of Integers

The three properties of integers are:

- Closure Property
- Commutativity Property
- Associative Property

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 = \( \frac{7}{4} \), 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

For example, if we take 2 , 5 , 11

2 + ( 5 + 11 ) = 18 and ( 2 + 5 ) + 11 = 18

**2) For Multiplication **a × ( b × c ) = ( a × b ) × c

For example, 2 × ( 5 × 11 ) = 110 and ( 2 × 5 ) × 11 = 110

Hence associative property is true for addition and multiplication.

**3) For Subtraction**

Associative property does not hold for subtraction a – ( b – c ) != ( a – b ) – c

For example, if we take 4, 6, 12

4 – ( 6 – 12 ) and ( 4 – 6 ) – 12

= 4 + 6 = 10 and -2 -=12 = – 14

Therefore associative property is not true for subtraction.

**4) For Division**

Associative property does not hold for division a ÷ ( b ÷ c ) != ( a ÷ b ) ÷ c

For example, again if we take 4, 6, 12

4 ÷( 6 ÷12 ) and ( 4 ÷ 6 ) ÷ 12

= 4 ÷ \( \frac{6}{12} \) and \( \frac{4}{6} \) ÷ 12

we get, = 4 × 2 = 8 and \( \frac{1}{3×6} \) = \( \frac{1}{18} \)

Therefore associative property is not true for division.

### 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.

## Solved Examples for You

Question: As per the commutative property, complete the equation for integers for the given operation: 10 × 5 =

- 500
- 5 × 10
- – 50
- None of the above

Solution: B. 10 × 5 = 5 × 10 = 50. This is a commutative property of multiplication.

Question: Positive and negative integers together are closed under …………

- addition and subtraction
- addition and multiplication
- multiplication and division
- multiplication

Solution: D. The set of positive and negative integers does not include 0. Addition of two integers may result in 0. Also, the subtraction of two integers may result in 0. We also know integers are not closed under division.

**Question: State the difference between the whole number and an integer?**

**Answer:** All natural number including 0 and 1, 2, 3, 4, 5, etc. are whole numbers. Besides, on the other hand, their negative counterpart such as -4, -3, -2, -1, 0, 1, 2, 3, etc. however, a and b are both integers.

**Question: Is there whole numbers that can’t be integers?**

**Answer:** The whole number are those number which starts with zero and end at infinity. Such as 0, 1, 2, 3, 4, etc. Whereas, integers are those number that can be both negative and positive. Such as -5, -4, -3, -2, -1, 0, 1, 2, 3, etc. Therefore, every whole number is an integer but every integer is not a whole number.

**Question: Is 16 an integer?**

**Answer:** For any number (here 16) to be an integer, firstly it needs to be a whole number. Also, you should be able to write it without a decimal or fractional component. So, 16 is in integer because it is a whole number and is not in fraction or decimal.

**Question: Is 8-bit an integer?**

**Answer:** an unsigned 8-bit has a range of 0-255, whereas an unsigned 8-bit has a range of -128 to 127. Both of these represent 256 different numbers, hence they are integers.

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