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REPRESENTATION OF INTEGERS ON THE NUMBER LINE
4 INTEGERS We draw a line and fix a point almost in the middle of it. We call it 0. We set off equal distances on the right-

hand side as well as on the left-hand side of 0. On the right-hand side, we label the points of division as 1, 2, 3,
4, 5, etc., while on the left-hand side these are labelled as -1, -2, -3, -4, -5, etc., as shown below.

So far we are conversant with only two types of numbers, namely, natural numbers and whole numbers. In this
chapter, we shall extend our number system from whole numbers to integers. We shall discuss the representation
of integers on the number line, operations on integers and their properties. -5 -4 -3 -2 -1 0 1 2 3 4 5

INTRODUCTION TO INTEGERS We know that in numbers, when a smaller whole number is subtracted Clearly, 1 and -1 are at equal distances from O but in opposite directions.
from a larger one, we get a whole number. Similarly, 2 and -2 are at equal distances from 0 but in opposite directions, and so on.

But, what about 3 - 5, 5 - 8, 11 - 16, etc.? ORDERING OF INTEGERS As a consequence of the above discussion, it follows that we may represent
every integer by some point on the number line.

Clearly, there are no whole numbers to represent them. So, there is a need to extend our whole number system
so as to contain numbers to represent the above differences. If we represent two integers on the number line, we follow the convention that the number occurring to the right
ts greater than that on the left. And, the number on the left ts smaller than that on the right.
Corresponding to natural numbers 1, 2, 3, 4, 5, 6, ... , we introduce new numbers denoted by-1, Thus, we have the following examples:
-2, -3, -4, -5, -6, ... , called minus one, minus two, minus three, minus four, minus five, minus (i) 3 > 1, since 3 is to the right of 1;
six, ... , respectively such that 1 + (-1) = 0, 2 + (-2) = 0, 3 + (-3) = 0, and so on. (ii) 1 > 0, since 1 is to the right of O;
We say that -1 and 1 are the opposites of each other; (iii) 0 > -1, since 0 is to the right of -1;
-2 and 2 are the opposites of each other; (iv) -1 > -2, since -1 is to the right of -2;
-3 and 3 are the opposites of each other, and so on. (v) -2 > -3, since -2 is to the right of -3.
Thus, our new collection together with whole numbers becomes ... , -3, -2, -1, 0, 1, 2, 3, .... In general, the following results are quite obvious:
These numbers are called integers. (i) Zero ts less than every positive integer, since O ts to the left of every positive integer.
(ii)Zero ts greater than every negative integer, since O ts to the right of every negative integer.
The numbers 1, 2, 3, 4, 5, 6, ... are called positive integers, The numbers -1, -2, -3, -4, -5,-6, ... are called nega- (iii) Every positive integer ts greater than every negative integer, since every positive integer ts to the
tive integers and 0 is an integer which is neither positive nor negative. right of every negative integer.
(iv)The greater the number, the lesser ts its opposite.

In our daily life we come across statements opposite to each other. For example: (i) 4 > 3 and -4 < - 3;
(ii) 7 > 4 and -7 < - 4;
We use positive and negative integers for their representation. (ii) 7 > 4 and -7 < - 4;
EXAMPLE:1 We know that the heights of places are measured as distances from sea level. Thus, if a and bare two integers such that a > b, then - a < - b. Similarly, if a and b are
We shall represent a height of integers such that a < b, then - a > - b.
5 km above sea level as +5 km or simply 5 km;
5 km below sea level as -5 km. EXAMPLE:4 Using the number line, write the integer which ts:
EXAMPLE:2 We know that the freezing point of water is 0°c. (i) 3 more than 5 (ii) 4 more than -1
We shall represent a temperature of (iii) 5 less than 3 (iv) 2 less than -3
15°C above the freezing point of water as+ 15°C or simply 15°C; SOLUTION (i) We want to know an integer 3 more than 5.
15°C below the freezing point of water as -15°C. So, we start from 5 and proceed 3 steps to the right to obtain 8, as shown below:
EXAMPLE:3 We write:
a loss of 500 = a gain of -500;
a withdrawal of rupees 600 = a deposit of rupees -600;
a decrease of 20 = an increase of -20, etc. 0 1 2 3 4 5 6 7 8
So, 3 more than 5 is 8.