Are you one of those candidates looking eagerly to learn about the concept of Integers? If yes, you must check this page to know the complete details about Integers. Integers is a basic and important concept that lays a stronger foundation for your maths. Know definition, rules, numbers, solved questions, symbols, etc.  Go through the below sections to find various methods and formulae.

## Integers – Definition

The integer word is derived from the Latin word “Integer” which represents the whole. Integers are the positive, negative numbers, or zero. Integer values cannot be in decimals, fractions, percents, and we can perform various operations(arithmetic operations) like subtraction, addition, multiplication, division, etc. Examples of integers are 1,2,3,-4,-5, etc.  Integers also include various sets like

Integers also include various sets like zero, whole numbers, natural numbers, additive inverses, etc. These are the subset of real numbers.
Example of integer set: -5,-3, -1, 0, 2, 5

### Representation of Integers

As integers contain various numbers and sets and are the subset of real numbers, they are represented with the letter “Z”.

Example:
Z= {5,-3, -1, 0, 2, 5}

Types of numbers in Integers

• Natural Numbers
• Whole Numbers
• Real Numbers
• Rational Numbers
• Irrational Numbers
• Odd Numbers
• Even Numbers

### Integers Rules

• The Sum of 2 positive integer numbers is an integer number
• The Sum of 2 negative integer numbers is an integer number
• Product of 2 positive integer numbers is an integer number
• Product of 2 negative integer numbers is an integer number
• Sum of an integer number and its inverse equals zero
• Product of an integer number and its reciprocal equals 1

While adding 2 positive or negative integers(with the same sign), add the absolute values and note down the sum of those numbers with the sign provided with numbers.

Example:

(+6)+(+5) = +11
(-5)+(-5)= -10

While adding 2 integers with a different sign, subtract the absolute values and note down the difference of those numbers with the sign provided with numbers.

Example:

(-5)+(+2)= -3
(+6)+(-3)= -3

### Subtraction of Integer Numbers

While subtracting we follow the rules of addition but change the 2nd number which is being subtracted.

Example:

(-4)+(-3)= (-4)-(+3) = -11
(+5)-(+4)=(+5)+(-4)= +1

### Division and Multiplication of Integer Numbers

The rule is simple while dividing and multiplying 2 integer numbers.

• If both the integers have the same sign, the result is positive.
• If both the integers have a different sign, the result is negative.

Example:

(+3)*(-4) = -12
(+4)*(+3) = 12
(+16)/(+4) = +4
(-6)/(+2) = -3

### Integer Properties

There are 7 properties of integers. The major properties are

1. Associative Property
2. Distributive Property
3. Closure Property
4. Commutative Property
5. Identity Property
6. Multiplicative Inverse Property

#### 1. Associative Property

This property refers to grouping and rules can be applied for addition and multiplication.

Associative property enables the special feature of grouping the numbers in your own way and still, you get the same answer.

(a+b)+c = a+(b+c)

Example:

(-4+2)+3= -2+(3+4)

In the above example, if we consider the first equation you can solve it in either way i.e., First you take the difference of 4 and 2 and then add 3 to it or you can first add 2 and 3 and then subtract 4 from it. In both ways, you get a constant answer.

Associative Property of Multiplication

This property also refers to the same as the addition property. In whatever way you group numbers, you still get the same answer.

(ab)c= a(bc)

Example:

-2(4)*3=-2(4*3)

In the above example, you can solve it 2 ways and still find the same answer. First, you can multiply 2,4 and then multiply that with 3 or you can first multiply 4,3 and then multiply it with 4.

#### 2. Distributive Property

The distributive property is used when the expression involving addition is then multiplied by a number. This property tells us that we can multiply first and then add or add first and multiply then. In both ways, the multiplication is distributed for all the terms in parentheses.

a(b+c) = ab+ac

Example:

-4(2+3)= (-4*2)+(-4*3)

In the above example, we can first add 2 and 3, then multiply it with 4 or we can multiply 4 with 2 and 3 separately and then add it, still you get the same answer.

#### 3. Closure Property

Closure property for addition or subtraction states that the sum or difference of any 2 integers will be an integer value.

a + b = integer
a x b = integer

Example:

6-3= 3
6+(-3)= 3

The closure property for multiplication also states that the product of any two integer numbers is an integer number.

Example:

5*5=25
(-5)*(-5)=25

The closure property for division does not hold true that the division of two integers is an integer value.

Example:

(-3)/(-12)=1/4, which is not an integer

#### 4. Commutative Property

The commutative property for addition states that when two integer numbers undergo swapping, the result remains unchanged.

a+b=b+a
a*b=b*a

Example:

28+5+43=5+43+28=76

The commutative property for multiplication also states the same that if two integers are swapped, the result remains unchanged.

Example:

5*4*2=2*4*5=40

The commutative property doesn’t hold true for subtraction.

#### 5. Identity Property

Identity Property states that any number that is added with zero will give the same number. Zero is called additive identity.

a+0=a
a*1=a

Example:

5+0=5

The identity property for multiplication also states the same that the integer number multiplied by 1 will give the same number. 1 is called the additive identity.

Example:

5*1=5

#### 6. Multiplicative Inverse Property

Consider “a” as an integer, then as per the multiplicative inverse property of integers,

a*(1/a)=1

Here, 1/a is the multiplicative inverse of integer a.