integer

An integer (from the Latin integer meaning "whole") is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 512, and √2 are not.

Integers include all natural numbers, zero and negative numbers for example, -4, -3, -2, -1, 0, 1, 2, 3, ………. etc are all integers.

A whole number, from zero to positive or negative infinity is called Integers. I.e. it is a set of numbers which include zero, positive natural numbers and negative natural numbers. It is denoted by letter Z.

1. If we add a positive integer, we go to the right.

2. If we add a negative integer, we go to the left.

3. If we subtract a positive integer, we go to the left.

4. If we subtract a negative integer, we go to the right.

The negative of any number is the additive inverse of that number.

The additive inverse of 5 is (- 5) and additive inverse of (- 5) is 5.

Properties of Addition and Subtraction of Integers1. Closure under AdditionFor the closure property the sum of two integers must be an integer then it will be closed under addition.

Example

2 + 3 = 5

2+ (-3) = -1

(-2) + 3 = 1

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

As you can see that the addition of two integers will always be an integer, hence integers are closed under addition.

If we have two integers p and q, p + q is an integer.

2. Closure under SubtractionIf the difference between two integers is also an integer then it is said to be closed under subtraction.

Example

7 – 2 = 5

7 – (- 2) = 9

- 7 – 2 = – 9

- 7 – (- 2) = – 5

As you can see that the subtraction of two integers will always be an integer, hence integers are closed under subtraction.

For any two integers p and q, p - q is an integer.

3. Commutative Propertya. If we change the order of the integers while adding then also the result is the same then it is said that addition is commutative for integers.

For any two integers p and q

p + q = q + p

Example

23 + (-30) = – 7

(-30) + 23 = – 7

There is no difference in answer after changing the order of the numbers.

b. If we change the order of the integers while subtracting then the result is not the same so subtraction is not commutative for integers.

For any two integers p and q

p – q ≠ q – p will not always equal. 

Example

 23 - (-30) = 53

(-30) - 23 = -53

The answer is different after changing the order of the numbers.

4. Associative PropertyIf we change the grouping of the integers while adding in case of more than two integers and the result is same then we will call it that addition is associative for integers.

For any three integers, p, q and r

p + (q + r) = (p + q) + r

5. Additive IdentityIf we add zero to an integer, we get the same integer as the answer. So zero is an additive identity for integers.

For any integer p,

p + 0 = 0 + p =p

Example

2 + 0 = 2

(-7) + 0 = (-7)

Multiplication of IntegersMultiplication of two integers is the repeated addition.

Example

3 × (-2) = three times (-2) = (-2) + (-2) + (-2) = – 6 3 × 2 =  three times 2 = 2 + 2 + 2 = 6

Now let’s see how to do the multiplication of integers without the number line.

1. Multiplication of a Positive Integer and a Negative IntegerTo multiply a positive integer with a negative integer, we can multiply them as a whole number and then put the negative sign before their product.

So the product of a negative and a positive integer will always be a negative integer.

For two integers p and q, p × (-q) = (-p) × q = - (p × q) = - pqExample4 × (-10) = (- 4) × 10 = - (4 × 10) = - 40

2. Multiplication of Two Negative IntegersTo multiply two negative integers, we can multiply them as a whole number and then put the positive sign before their product.

Hence, if we multiply two negative integers then the result will always be a positive integer.

For two integers p and q,(-p) × (-q) = (-p) × (-q) = p × qExample(-10) × (-3) = 30

3. The Product of Three or More Negative IntegersIt depends upon the number of negative integers.

a. If we multiply two negative integers then their product will be positive integer

(-3) × (-7) = 21

b. If we multiply three negative integers then their product will be negative integer

(-3) × (-7) × (-10) = -210

If we multiply four negative integers then their product will be positive integer

(-3) × (-7) × (-10) × (-2) = 420

Hence, if the number of negative integers is even then the result will be a positive integer and if the number of negative integers is odd then the result will be a negative integer.

Properties of Multiplication of Integers1. Closure under MultiplicationIn case of multiplication, the product of two integers is always integer so integers are closed under multiplication.

For all the integers p and q

p×q = r, where r is an integer

Example

(-10) × (-3) = 30

(12) × (-4) = -48

2. Commutativity of MultiplicationIf we change the order of the integers while multiplying then also the result will remain the same then it is said that multiplication is commutative for integers.

For any two integers p and q

p × q = q × p

Example

20 × (-30) = – 600

(-30) × 20 = – 600

There is no difference in answer after changing the order of the numbers.

3. Multiplication by ZeroIf we multiply an integer with zero then the result will always be zero.

For any integer p,p × 0 = 0 × p = 0Example9 × 0 = 0 × 9 = 0

0 × (-15) = 0

4. Multiplicative IdentityIf we multiply an integer with 1 then the result will always the same as the integer.

For any integer q

q × 1 = 1 × q = q

Example

21 × 1 = 1 × 21 = 21

1 × (-15) = (-15)

5. Associative PropertyIf we change the grouping of the integers while multiplying in case of more than two integers and the result remains the same then it is said the associative property for multiplication of integers.

For any three integers, p, q and r

p × (q × r) = (p × q) × r

Example

If there are three integers 2, 3 and 4 and we change the grouping of numbers, then

The result remains the same. Hence, multiplication is associative for integers.

6. Distributive Propertya. Distributivity of Multiplication over Addition.

For any integers a, b and c

a × (b + c) = (a × b) + (a × c)

Example

Solve the following by distributive property.

I. 35 × (10 + 2) = 35 × 10 + 35 × 2

= 350 + 70

= 420

II. (– 4) × [(–2) + 7] = (– 4) × 5 = – 20 And

= [(– 4) × (–2)] + [(– 4) × 7]

= 8 + (–28)

= –20

So, (– 4) × [(–2) + 7] = [(– 4) × (–2)] + [(– 4) × 7]

b. Distributivity of multiplication over subtraction

For any integers a, b and c

a × (b – c) = (a × b) – (a × c)            

Example

5 × (3 – 8) = 5 × (- 5) = – 25

5 × 3 – 5 × 8 = 15 – 40 = – 25

So, 4 × (3 – 8) = 4 × 3 – 4 × 8.

Division of integers1. Division of a Negative Integer by a Positive IntegerThe division is the inverse of multiplication. So, like multiplication, we can divide them as a whole number and then place a negative sign prior to the result. Hence the answer will be in the form of a negative integer.

For any integers p and q,( – p) ÷ q = p ÷ (- q) = - (p ÷ q) where, q ≠ 0Example64 ÷ (- 8) = – 8

2. Division of Two Negative IntegersTo divide two negative integers, we can divide them as a whole number and then put the positive sign before the result.

The division of two negative integers will always be a positive integer.

For two integers p and q,

(- p) ÷ (- q) = (-p) ÷ (- q) = p ÷ q where q ≠ 0

Example

(-10) ÷ (- 2) = 5

Properties of Division of IntegersFor any integers p, q and r

PropertyGeneral formExampleConclusion Closure Propertyp ÷ q is not always an integer10 ÷ 5 = 25 ÷ 10 = 1/2(not an integer)The division is not closed under division.Commutative Propertyp ÷ q ≠ q ÷ p10 ÷ 5 = 25 ÷ 10 = 1/2The division is not commutative for integer.Division by Zerop ÷ 0 = not defined0 ÷ p = 00 ÷ 10 = 0NoDivision Identityp ÷ 1 = p10 ÷ 1 = 10Yes Associative Property(p ÷ q) ÷ r ≠ p ÷ (q ÷ r)[(–16) ÷ 4] ÷ (–2) ≠(–16) ÷ [4 ÷ (–2)](-8) ÷ (-2) ≠ (-16) ÷ (-2)4 ≠ 8Division is not Associative for integers.