## Need for Negative Numbers

We know that
counting numbers 1, 2, 3, 4, ... are called natural numbers, and natural numbers
together with 0 are called whole numbers. We can sort numbers in various
categories such as even numbers, odd numbers, prime numbers, composite numbers,
etc.

In many
situations, we need numbers below zero. For example, if we subtract a greater
number from a smaller number, i.e., 5 – 9 = ? and 2 – 7 = ?, etc., then we do
not get whole numbers. So, in order to find an answer in such a situation, we
need to extend our number system. Therefore, we introduce another group of
numbers called

**negative numbers**.##
Integers

In our daily life, we come
across situations like profit and loss in our business, rise and fall in the
temperature, height of a mountain, depth of a river, position of a submarine in
an ocean, etc.

Suppose the sea level is at 0
and we move upwards, then we would count as 1, 2, 3, … . If we move downwards, it
is in the opposite direction, so the numbers will all be negative and we
represent them as –1, –2, –3, … .

Here, –1 is called negative one
or minus one. 1 and –1 are opposite to each other. Similarly, 2 and –2 are
opposite to each other and so on. Thus, we develop a new system which has positive
and negative numbers separated by zero (0) called

**integers**represented by symbol**Z**or**I**.
Symbolically, we write it as

**Z**or**I**= {…, –4, –3, –2, –1, 0, 1, 2, 3, 4, …}## What are Integers?

If we include negative numbers along with whole numbers, we get a set of integers. We represent integers by I or Z and it is written as:

**Z**or**I**= {…, –4, –3, –2, –1, 0, 1, 2, 3, 4, …}**Positive Integers**

The numbers 1, 2, 3, 4 …
are greater than zero and they are called

**positive integers**.### Negative Integers

The numbers –1, –2, –3, –4 … are less than zero and they are called

**negative integers**.### Integer Zero

The number
‘0’ is an integer which is neither positive nor negative. It is greater than all negative integers and less than all positive integers.

All positive and
negative integers are called

**directed numbers**as they show direction due to their ‘+’ or ‘–’ sign. They are also known as**signed numbers**. For example, +8° C means that the temperature is above 0° C. The ‘+’ sign gives the direction and 8 gives the magnitude of the temperature.## Integers Definition

Integers are defined as the collection of whole numbers and negative numbers are called integers. These are the set of negative numbers, 0 and positive numbers.

### Integers Examples

Examples of integers are: 5, -38, 94, 0, -348, 5861, -29574, 485767, -3956567, etc.

Any number negative or positive including 0 is an example of integer. Integers do not include fractions, decimal and rational numbers.

**Are Natural Numbers, Whole Numbers, Fractions, Decimals and Rational Numbers Integers?**

- All natural numbers and whole numbers are integers.
- Only positive integers are natural numbers.
- All integers except negative numbers are whole numbers.
- All the fractions, decimals and rational numbers are not integers.
- All the integers are rational numbers.
- But only positive integers are fractions and decimals. For example, 5 can be written in a fraction form as 5/1 and in a decimal form as 5.0.

**Representation of Integers on the Number Line**

We can use a number line to show the order of numbers. The diagram
below shows a number line. An arrow is drawn at each end of the number line to
show that the line can be extended indefinitely on both ends.

If we are to mark –3 on the number line, we move 3 steps to the left of zero and reach point A. Similarly, in order to mark +5 on the number line, we move 5 steps to the right of zero and reach point B.

### Consecutive Integers

Integers which come in a proper sequence are called consecutive integers. For example, if you write the numbers written on the number line from left to right, they are consecutive integers.

1, 2, 3, 4, 5, 6, ........ are the examples of consecutive integers.

-5, -4, -3, -2, -1, 0, ....... are the examples of consecutive integers.

But 1, 2, 5, 8, ...... and -9, -6, -2, 0, ...... are not consecutive integers.

**Even Integers**

## An integer which is exactly divisible by 2 is called an **even integer**. Let A represent the setof even integers, then A = {..., –6, –4, –2, 0, 2, 4, 6, ...}.

An integer which is exactly divisible by 2 is called an

**even integer**. Let A represent the setof even integers, then A = {..., –6, –4, –2, 0, 2, 4, 6, ...}.

**Odd Integers**

## An integer which is not exactly divisible by 2 is called an **odd integer**. Let B be the setof odd integers, then B = {... –5, –3, –1, 1, 3, 5,...}.

An integer which is not exactly divisible by 2 is called an

**odd integer**. Let B be the setof odd integers, then B = {... –5, –3, –1, 1, 3, 5,...}.

**Comparing and Ordering of Integers**

On a horizontal number line:

· all the positive numbers are to the right of zero
(0).

·
all the negative numbers are to the left of zero
(0).

·
numbers are arranged in ascending (i.e., increasing)
order from left to right.

·
every number is smaller than any number on its right
and greater than any number on its left.

From the above number line, we
observe that

a. 7 > 6, since 7 is to the
right of 6.

b. 0 < 1, since 0 is to the
left of 1.

c. 0 > –1, since 0 is to the
right of –1.

d. –5 < –2, since –5 is to
the left of –2.

### Absolute Value of an Integer

Let us consider the
following number line.

To reach the point A from O, one has to take 4 steps
towards the right and to reach the point B from O, one has to take 4 steps
towards the left. The number of steps in both the cases is the same, but the
points represent different integers. [A represents + 4 and B
represents − 4.]

The absolute value of any integer
is the distance of the integer from zero. So, the absolute value of + 4 is 4 and the
absolute value of − 4 is also 4.

We denote the absolute value of −
4 using the notation │−4│ and it is read as the absolute value of − 4.

We write the absolute
value of a number

*x*as |*x*| and we define it as follows.**|**

*x***| =**

*x***if**

*x***≥ 0 and |**

*x***| = –**

*x***if**

*x***< 0**

In the above case, | –4| = – (–4) = 4

The absolute
value of a number is

**never negative**.### Solved Examples on Integers

**Example 1:**Express the following using integers.

a. 15 °C below zero

b. Profit of Rs 800

c. Withdrawal of Rs 2600

d. 1000 m above sea level

**Solution:**

a. –15 °C

b. Rs 800

c. – Rs 2600

d. 1000 m

**Example 2:**In each of the following pairs, find the integer which is to the right of the other on the number line and hence compare them using < or >.

a. 2, 7 b. –4, 4 c. 1, –8 d. 0, –3

**Solution:**

a. 7 is to the right of 2,
therefore 7 > 2 or 2 < 7.

b. 4 is to the right of –4,
therefore 4 > –4 or –4 < 4.

c. 1 is to the right of –8,
therefore 1 > –8 or –8 < 1.

d. 0 is to the right of –3,
therefore 0 > –3 or –3 < 0.

**Example 3:**Write the following integers in ascending as well as descending order:

5, –5, 0, 2, –3, –7.

**Solution:**

*Let us observe these integers on a number line.*

We know that integers on the
left are smaller than the integers on the right.

Therefore, ascending order: –7,
–5, –3, 0, 2, 5

Descending order: 5, 2, 0, –3, –5,
–7

**Example 4:**Simplify the following.

a. |–5| b. –|24| c. –|–3 | d. –|12 |

**Solution:**

a. |–5| = 5

b. –|24| = – 24

c. –|–3 | = – 3

d. –|12 | = – 12

**Example 5:**We have the integers –8, 7 and –12, arrange

a. the numbers in ascending order and

b. the absolute values of the numbers in descending order.

**Solution:**

a. Since, –12 < –8 < 7, the numbers in ascending order are: –12,
–8, 7.

b. |–8| = 8; |7| = 7; |–12| = 12

The numbers in descending order are 12, 8, 7.

∴
The absolute values of the numbers in
descending order are |–12 |, |–8|, | 7|.