There, we covered the basic operations such as Addition, Subtraction, Multiplication and Division and their properties such as Closure, Commutativity, Associativity, Distributivity etc.

We have seen that closure property is not true for Subtraction in the set of whole numbers, with examples 3 - 7 = ? ∉ W; 20 - 23 = ? ∉ W;

i.e. when a bigger whole number is subtracted from a smaller whole number, we do not get a whole number.

So, we need to have new type of numbers which may represent the above differences.

Thus, corresponding to natural numbers 1, 2, 3, 4, 5, 6, 7, ....etc., we introduce new numbers denoted by -1, -2, -3, -4, -5, -6, -7, ....etc. called minus one, minus two, minus three, minus four, minus five, minus six, minus seven, etc., respectively such that : 1 + (-1) = 0, 2 + (-2) = 0, 3 + (-3) = 0 and so on.

The numbers -1, -2, -3, ....etc.are called opposites (or additive inverses) of 1, 2, 3, ....etc respectively and vice versa.

Thus, we get a new set of numbers given by : Z = {......., -4, -3, -2, -1, 0, 1, 2, 3, 4,...........} These numbers are known as Integers.

We denote the set of Integers by Z.

The numbers 1, 2, 3, 4, 5, 6, 7, ....etc. are known as Positive Integers.

The numbers -1, -2, -3, -4, -5, -6, -7, ....etc.are known as Negative Integers.

0 is an integer which is neither positive nor negative.

The numbers 0, 1, 2, 3, 4, 5, 6, 7, ....etc. are known as Non-Negative Integers. or Whole Numbers, as we have already seen.

We may denote the positive integers by +1, +2, +3, +4, +5, +6, +7, ....etc. Usually, we omit the "+" sign.

Use of Positive and Negative Integers in Daily Life

(i) The height of Mean Sea Level is taken as 0. The heights above Mean sea Level are taken as Positive. The heights below Mean sea Level are taken as Negative.

(ii) The freezing point of water is 0^{0}C. The temperatures above the freezing point are taken as Positive. The temperatures below the freezing point are taken as Negative.

(iii) The Atmospheric Pressure is taken as 0. The Pressures above Atmospheric Pressure are taken as Positive. The Pressures below Atmospheric Pressure are taken as Negative.

(iv) Profits in a business are represented by Positive Numbers. Losses in a business are represented by Negative Numbers.

(v) Deposits in a Bank Account are considered as Positive Numbers. Withdrawls in a Bank Account are considered as Negative Numbers.

Comparison of Integers

From the knowledge of
Decimal Number System, we can compare positive integers and know which is bigger or which is smaller.

For example, 3 < 5; 9 < 12; 99 < 100; 189 < 197

In addition to that knowledge, the following points will help in deciding which is bigger or smaller.

0 is less than every positive integer.

0 < 1 < 2 < 3 < 4 < 5 <..........etc.

0 is greater than every negative integer.

.......< -5 < -4 < -3 < -2 < -1 < 0

Every negative integer is less than every positive integer.

Rule 3 : In the two integers being added, if one is positiveand the other is negative,find the difference between their numerical values and give the sign of the integer with more numerical value, to it, which is their sum.

Examples : (i) Consider the sum (+12) + (-9) Here, one is positive and the other is nagative. Numerical value of (+12) = 12.; Numerical value of (-9) = 9. Their difference = 12 - 9 = 3 (+12) has more numerical value and its sign is +. So, the sign to 3 is +. Thus (+12) + (-9) = +(12 - 9) = +3 = 3.

(ii) Consider the sum (- 136) + (+96) Here, one is positive and the other is nagative. Numerical value of (- 136) = 136.; Numerical value of (+96) = 96. Their difference = 136 - 96 = 40 (- 136) has more numerical value and its sign is -. So, the sign to 40 is -. Thus (- 136) + (+96) = - (136 - 96) = -40 .

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What is seen to be true for in these examples, is true for any element of Z. So, we can say :

For every element a ∈ Z, a ÷ 1 = a

Division is inverse operation to Multiplication :

Look at the following Examples.

144 ÷ 18 = 8 and 8 x 18 = 144; 81 ÷ -3 = -27 and -27 x -3 = 81.

What is seen to be true for these examples, is true for any element of Z. So, we can say :

For any three elements a, b(≠0) and c ∈ Z, a ÷ b = c means a = b x c

Division by zero is not defined :

-9 ÷ 0 = ? Is it possible to assign any whole number to this quotient ? Suppose, if possible, -9 ÷ 0 = a where a is an integer. Since division is the inverse process of multiplication, we have a x 0 = -9. But, any whole number x 0 = 0 [see zero property of multiplication above.] So a x 0 = 0 and not -9. Thus there is no number which when multiplied by zero gives a non zero number.

Therefore -9 ÷ 0 can not be equal to any integer.

Thus division by zero is not defined.

Division by zero is not defined.

What about 0 ÷ 0 ?

Let 0 ÷ 0 = b Then 0 = b x 0. This is satisfied for all values of b. So 0 ÷ 0 has innumerable Answers. It can not be any one umber.

So, we can say

0 ÷ 0 is indeterminate.

Division of zero with any number gives zero :

Look at the following Examples.

0 ÷ -34 = 34 ; 0 ÷ 5617 = 0

What is seen to be true for these examples, is true for any element of Z. So, we can say :

For every element a ∈ Z, 0 ÷ a = 0

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