Chapter 6: Integers

Introduction
We have learnt about natural and whole numbers in the previous chapters. But when we subtract a greater whole number from a smaller whole number, then the difference will not be a whole number. Hence, there is a need to extend the number system to include such numbers. Their are many examples in hour day to day life related to such type of numbers.
(i)    During the winter season, the minimum temperature in Shimla may be 5°C below 0°C. 
So we represent this as –5°C, read as minus five degree centigrade.
(ii)     the depth of an ocean, say 300 metres below the mean sea level, can be expressed as –300 metres high. If height is considered as positive, depth is considered as negative height and vice versa. 

Integers
Their are infinite numbers with negative sign on the numebr line. All these numbers are less than zero and are called negative numbers. They are –1, –2, –3.  
Whole numbers along with the negative numbers are called integers.
Their are some important properties with integers.
(i)    There are infinite positive numbers to the right of zero and there are infinite negative numbers to the left of zero as well.
(ii)    In the number line of integers, every number on the right is greater than all the numbers on its left.
(iii)    Zero is neither negative nor positive.
The natural numbers 1, 2, 3, ... , etc., in the system of integers are called positive integers. The numbers to the left of zero are called negative integers and are always denoted as –1, –2, –3, ... , etc. . The whole number 0 is neither a positive integer nor a negative integer.

Representation of integers on a number line

First we draw a line and mark some points at equal distance on it as shown in the figure. Mark a point as zero on it. Points to the right of zero are positive integers and are marked + 1, + 2, + 3, etc. or simply 1, 2, 3 etc. Points to the left of zero are negative integers and are marked – 1, – 2, – 3 etc. 
In order to mark – 6 on this line, we move 6 points to the left of zero.

In order to mark + 2 on the number line, we move 2 points to the right of zero.

Ordering of integers
Let we observe the integers which are represented on the number line.

We know that 7 > 4 and from the number line shown above, we observe that 7 is to the right of 4. 
Similarly, 4 > 0 and 4 is to the right of 0. Now, since 0 is to the right of –3 so, 0 > – 3. Again, – 3 is to the right of – 8 so, – 3 > – 8.
Thus, we see that on a number line the number increases as we move to the right and decreases as we move to the left.
Therefore, – 3 < – 2, – 2 < – 1, – 1 < 0, 0 < 1, 1 < 2, 2 < 3 so on. 
Hence, the collection of integers can be written as..., –5, –4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5...

Ex.    Write all integers between 
         (i) – 2 and 3         (ii) – 4 and 2

(i)     The integers between – 2 and 3 are –1, 0, 1, 2.
(ii)     The integers between – 4 and 2 are – 3, – 2, –1, 0, 1.

Operation on Integers
Addition : In integers, addition can perform using following rule. 
(i)     When adding integers with like signs (both positive or both negative), add their absolute values, and place the common sign before the sum. 
(ii)     When adding integers of unlike signs, find the difference of their absolute values, and give the result the sign of the integer with the larger absolute value. 
(iii)    When the addition and subtraction signs are placed side by side without any number in between, these two opposite signs give a negative sign. 
Subtraction : Subtraction is reverse operation of addition.

Ex.1    Consider 8 – 5. Actually we have to subtract + 3 from 8. So, we need to find a number which when added to 3 gives 8. 
sol:     The answer 5, i.e., 8 – 3 = 8 + (– 3) = 5 
We can change subtraction to addition by adding the additive inverse of the second number to the first number. In the above example –3 is the additive inverse of +3 and vice versa is also possible.
Ex.2     Find the sum of (– 9) + (+ 4) + (– 6) + (+ 3)
Sol.     We can rearrange the numbers so that the positive integers and the negative integers are grouped together. We have 
(– 9) + (+ 4) + (– 6) + (+ 3) = (– 9) + (– 6) + (+ 4) + (+ 3) = (– 15) + (+ 7) = – 8

Ex.3     Find the value of (30) + (– 23) + (– 63) + (+ 55)
Sol.    (30) + (+ 55) + (– 23) + (– 63) = 85 + (– 86) = – 1 

Ex.4      Subtract (– 4) from (– 10)
Sol.:     (– 10) – (– 4) = (– 10) + (additive inverse of – 4) = –10 + 4 = – 6    

 Ex.5     Subtract (+ 3) from (– 3)
 Sol.:     (– 3) – (+ 3) = (– 3) + (additive inverse of + 3)     = (– 3) + (– 3) = – 6

Addition of integers on a number line
(i)     Let us add 3 and 5 on number line.

On the number line, we first move 3 steps to the right from 0 reaching 3, then we move 5 steps to the right of 3 and reach 8. Thus, we get 3 + 5 = 8

(ii)     Let us add – 3 and – 5 on the number line.

On the number line, we first move 3 steps to the left of 0 reaching – 3, then we move 5 steps to the left of – 3 and reach – 8. 
Thus, (– 3) + (– 5) = – 8.
We observe that when we add two positive integers, their sum is a positive integer. When we add two negative integers, their sum is a negative integer. 

(iii)     Suppose we wish to find the sum of (+ 5) and (– 3) on the number line. First we move to the right of 0 by 5 steps reaching 5. Then we move 3 steps to the left of 5 reaching 2.

Thus, (+ 5) + (– 3) = 2
(iv)     Similarly, let us find the sum of (– 5) and (+ 3) on the number line. First we move 5 steps to 
the left of 0 reaching – 5 and then from this point we move 3 steps to the right. We reach the point – 2. 
Thus, (– 5) + (+3) = – 2.    

Subtraction of Integers with the help of a Number Line
We have added positive integers on a number line. For example, consider 6+2. We start from 6 and go 2 steps to the right side. We reach at 8. So, 6 + 2 = 8. 

We also saw that to add 6 and (–2) on a number line we can start from 6 and then move 2 steps to the left of 6. We reach at 4. So, we have, 6 + (–2) = 4.

Thus, we find that, to add a positive integer we move towards the right on a number line and for adding a negative integer we move towards left. We have also seen that while using a number line for whole numbers, for subtracting 2 from 6, we would move towards left.

i.e. 6 – 2 = 4

Ex.1     Find the value of – 8 – (–10) using number line
Sol.:    – 8 – (– 10) is equal to – 8 + 10 as additive inverse of –10 is 10. 
On the number line, from – 8 we will move 10 steps towards right. 

We reach at 2. Thus, –8 – (–10) = 2
Hence, to subtract an integer from another integer it is enough to add the additive inverse of the integer that is being subtracted, to the other integer.

Ex.2 Subtract the sum of 998 and – 486 from the sum of – 290 and 732.
Sol. Sum of 998 and – 486 is 998 + (– 486) = (998 – 486) = 512
Sum of – 290 and 732 is  – 290 + 732  = 732 – 290 = 442
Now, 442 – 512 = 442 +(–512) = – 70
[We subtract 442 from 512 and give minus sign to  the result]

Ex.3  On a particular day, the temperature at Dehradun at 10 AM was 20ºC but by midnight , it fell down to 11ºC. The temperature at Bangaluru at 10AM the same day was 30º C but fell down to 18º C by the midnight . Which fall is greater ?
Sol.    Fall in temperature at Dehradun = 20ºC – 11ºC = 9ºC
Fall in temperature at Bangaluru= 30ºC – 18ºC = 12ºC
∴The fall in temperature at Bangaluru is greater and is 12ºC.