Integers

Various Types of Numbers

Natural numbers :
 Counting numbers are called natural numbers. Thus, 1,2,3,4,5,6,......., etc. , are all Natural numbers.

Whole Numbers :
All natural numbers together with 0(zero) are called whole numbers. Thus, 0, 1, 2, 3, 4, ......, etc., are whole numbers. Clearly, every natural number is whole number but 0 is a whole number which is not a natural number.

Integers :
All natural numbers, 0 and negatives of counting numbers are called integers. Thus, ....., -4, -3, -2, -1, 0, 1, 2, 3, 4. ......, etc., are all integers.
a) Positive Integers : 1, 2, 3, 4, 5, ......, etc., are all positive integers.
b) Negative Integers : -1, -2, -3, -4, -5, ......, etc., are all negative integers.
c) Zero is an integer which is neither positive nor negative.

Addition of Integers

Rule 1 :
 If two positive or two negative integers are added, we add their values regardless of their signs and give the sum their common sign.
 e.g.

 i) Add 36 and 27

 Solution

 +36+27 = +63

 ii) Add -31 and -25
Solution

 -31 -25 = -56

 Rule 2 :
 To add a positive and a negative integer, we find the difference between their numerical values regardless of their signs and give the sign of the integer with the greater value to it.

Note : In order to add two integers of unlike signs, we see which is more and by how much.

 e.g.

i)                    Add -40 and 15

solution

-40 + 15 = -25

ii)                  Add -29 and 52

Solution

 -29 + 52 = +23

Properties of Addition of Integers

1)      Closure Property of Addition :

 The sum of two integers is always an integer.

 e.g.

i)                    5 + 4 = 9, which is an integer.

ii)                  3 + (-5) = -2, which is an integer.

iii)                (-2) + (-3) = -5, which is an integer.

 Hence, the sum of two integers is always an integer.

2)      Commutative Law of Addition :

 If a and b are any integers, then a+b = b + a.

 e.g.

(-8) + 9 = 1 and 9 + (-8) = 1

Therefore (-8) + 9 = 9 + (-8)

3)      Associative Law of Addition :

 If a, b, c are any three integers, then (a + b) + c = a + (b + c).

e.g.

Consider the integers -3, -2 and 6, We have
{(-3) + (-2)} +6 = 1 and (-3) + {(-2) + 6} = 1
Therefore {(-3) + (-2)} + 6 = (-3) + {(-2) + 6}

4)      Existence of Additive Identity :

 For any integer a, we have a + 0 =0 + a = a. 0 is called the additive identity of integers. e.g. 5 + 0 = 0 + 5 = 5

5)      Existence of Additive inverse :

For any integer a, we have a + (-a) = (-a) + a = 0.

The opposite of an integer a is (-a).

The sum of an integer and its opposite is 0.

Additive inverse of a is (-a).

Similarly, additive inverse of (-a) is a.

e.g.

We have 3 + (-3) = (-3) + 3 =0
So, the additive inverse of 3 is (-3). And, the additive inverse of (-3) is 3.

Subtraction of Integers

For any integers a and b, we define, a - b = a + (-b).

e.g.

Subtract
i) 9 from 5
ii) -5 from 6
iii) 4 from (-5)
iv) -9 from -8

Solution

i)                    5 - 9 = -4

ii)                  6 - (-5) = 11

iii)                -5 - 4 = -9

iv)                -8 - (-9) = 1

Properties of Subtraction

1)      Closure Property for Subtraction :

If a and b are any two integers, then (a - b) is always an integer.

2) Subtraction of integers is not commutative.

 e.g.

6 - 8 = -2 and 8 - 6 = 2

Thus, (6-8) is not equal to (8-6).

3) Subtraction of Integers is not Associative.

 e.g.

Consider the integers 5, -2 and -4, We have {5 - (-2)} - (-4) =11 and 5 - {(-2) - (-4)} = 3 Hence {5 -(-2)} - (-4) is not equal to 5 - {(-2) - (-4)}.

Multiplication of Integers

Rule 1:
To find the product of two integers with unlike signs, we find the product of their values regardless of their signs and give a minus sign to the product.

e.g.

i) 6 x (-5) = -30
ii) (-5) x 4 = -20.

Rule 2:
To find the product of two integers with the same sign, we find the product of their values regardless of their signs and give a plus sign to the product.

e.g.

i) 10 x 5 = 50
ii) (-3) x (-5) = 15

Properties of Multiplication of Integers

1. Closure Property of Multiplication
The product of two integers is always an integer.

e.g.

i) 8 x 3 = 24, which is an integer.
ii) (-8) x 4 = 32, which is an integer.
iii) (-5) x (-5) = 25, which is an integer.

2. Commutative Law of Multiplication
For any two integers a and b, we have
(a x b) = (b x a).

e.g.

i) 5 x 4 = 20 and 4 x 5 = 20
Therefore, 5 x 4 = 4 x 5.

ii) (-3) x 4 = -12 and 4 x (-3) = -12
Therefore, (-3) x 4 = 4 x (-3).

iii) (-2) x (-5) = 10 and (-5) x (-2) = 10
Therefore, (-2) x (-5) = (-5) x (-2).

3. Associative Law of Multiplication
For any integers a, b and c, we have
(a x b) x c = a x (b x c).

e.g.

i) {5 x (-2)} x 4 = -40 and 5 x {(-2) x 4} = -40
Therefore, {5 x (-2)} x 4 = 5 x {(-2) x 4}.

ii) (4 x 5) x 6 = 120 and 4 x (5 x 6) = 120
Therefore, (4 x 5) x 6 = 4 x (5 x 6).

4. Distributive Law of Multiplication over Addition
For any integers a, b, c, we have
a x (b c) = (a x b) (a x c).

e.g.

5 x (4 2) = 30 and (5 x 4) (5 x 2) = 30
Therefore, 5 x (4 2) = (5 x 4) (5 x 2).

5. Existence of Multiplicative Identity
For every integers a, we have (a x 1) = (1 x a) = a.
1 is called the multiplicative identity for integers.

6. Existence of Multiplicative Inverse
Multiplicative inverse of a nonzero integer a is the number 1/a, as a x (1/a) = (1/a) x a = 1.

7. Property of Zero
For every integer a, we have
(a x 0) = (0 x a) = 0.

e.g.

i) 7 x 0 = 0 x 7 = 0
ii) (-5) x 0 = 0 x (-5) = 0.

Division on Integers

We know that division is an inverse process of multiplication.

Rule 1:
For dividing one integer by the other, the two having unlike signs, we divide there values regardless of their signs and give a minus sign to the quotient.

e.g.

i) (-24) / 6 = -4

ii) 25 / (-5) = -5.

Rule 2:
For dividing one integer by the other having like signs, we divide their values regardless of their signs and give a plus sign to the quotient.

e.g.

i) 26 / 13 = 2
ii) (-27) / (-3) = 9.

Properties of Division on Integers

1. If a and b are integers then (a / b) is not necessarily an integer.

e.g.

i) 17 and 5 are both integers, but (17 / 5) is not an integer.

ii) (-8) an 3 are both integers, but {(-8) / 3} is not an integer.

2. If a is an integer and a is not equal to zero, then a/a = 1

3. If a is an integer, then (a / 1) = a.

4. If a is an integer and a is not equal to zero, then (0 / a) = 0 but (a / 0) is not meaningful.

5. If a, b, c are integers, then (a / b) / c is not equal to a / (b / c), unless c = 1.
Thus, division on integers is not associative.

6. If a, b, c are nonzero integers and a > b, then
i) (a / c) > (b / c), if c is positive.
ii) (a / c) < (b / c), If c is negative.