Understanding Integers

Ojective

Once we have gone through this topic we will have learnt, 

  1. What are integers.
  2. How to represent integers.
  3. How to order integers.
  4. Addition and subtraction of integers. 

Integers 

The first numbers to be discovered were natural numbers i.e. 1, 2, 3, 4,... If we include zero to the collection of natural numbers, we get a new collection of numbers known as whole numbers i.e. 0, 1, 2, 3, 4,... There are negative numbers too. If we put the whole numbers and the negative numbers together, the new collection of numbers will look like 0, 1, 2, 3, 4, 5,..., –1, – 2, – 3, –4, –5, ... and this collection of numbers is known as Integers. In this collection, 1, 2, 3, ... are said to be positive integers and – 1, – 2, – 3,.... are said to be negative integers. 

Let us understand this by the following figures. Let us suppose that the figures represent the collection of numbers written against them. 

Sample Symbols
Representation Type
Natural Numbers
Whole Numbers
Zero
Negative Numbers
Intergers
Integers - Collection of all these representations.

Representation of Integers on a Number Line 

The figure shows a line with points on it marked at equal distances. The points to the right of zero are positive integers and are marked +1, +2, +3, etc. or simply as 1, 2, 3 etc. Points to the left of zero are negative integers and are marked -1, -2, -3 etc.  So if we need to mark -6 on this line, we move 6 points to the left of zero. Same way, to mark +2 on the number line, we move two points to the right of zero. 

Ordering of Integers 

Let us once again observe the integers which are represented on the number line shown above. We know that 7 is greater than 4 and that 7 is to the right of 4. Similarly, 4 is greater than 0 and 4 is to the right of 0. Now, since 0 is to the right of -3 , 0 is greater than -3.  Again, -3 is to the right of  -8 so,  -3 is greater than -8. 

Thus, we see that on a number line the number increases as we move to the right of 0 and decreases as we move to the left of 0.  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... 

Addition of Integers on a Number Line 

Let us find the sum of 3 and 5 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. 

Let us find the sum of (-3) and (-5) on the number line.  

On the number line, we first move 3 steps to the left of 0 reaching -3, we then 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. 

Let us 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.

 

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. 

When a positive integer is added to an integer, the resulting integer becomes greater than the given integer. When a negative integer is added to an integer, the resulting integer becomes less than the given integer.  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. 

Let us add 3 and -3. We first move from 0 to +3 and then from +3, we move 3 points to the left. Where do we reach ultimately? 3 + (-3) = 0. Similarly, if we add 2 and -2, we obtain the sum as zero. Numbers such as 3 and -3, 2 and – 2, when added to each other give the sum zero. They are called additive inverse of each other.

Subtraction of Integers on a Number Line 

What would we do for 6 - (-2)? Would we move towards the left on the number line or towards the right? If we move to the left then we reach 4. Then we have to say 6 - (-2) = 4, which is not true because we know that 6 - 2 = 4 and 6 - 2 ≠ 6 - (-2). So, we have to move towards the right, i.e. 6 - (-2) = 8. 

This also means that when we subtract a negative integer we get a greater integer. Consider it in another way. We know that additive inverse of (-2) is 2. Thus, it appears that adding the additive inverse of -2 to 6 is the same as subtracting (-2) from 6.  So we write 6 - (-2) = 6 + 2. 

 Let us now find the value of -5 - (-4) using a number line. 

We can say that this is the same as -5 + (4), as the additive inverse of -4 is 4. We move 4 steps to the right on the number line starting from -5. We reach at -1. i.e. -5 + 4 = -1. Thus, -5 - (-4) = -1. 

 

Summary

  1. We have seen that there are times when we need to use numbers with a negative sign. This is when we want to go below zero on the number line. These are called negative numbers.
  2. The collection of numbers, ... -4, -3, -2, -1, 0, 1, 2, 3, 4,... are called integers. So the negative numbers,  -1, -2, -3, -4,... are negative integers and  the positive numbers, 1, 2, 3, 4,... are positive integers.
  3. We have also seen how one more than the given number gives a successor and one less than given number gives predecessor.
  4. We observed that,
    • When we have the same sign, we need to add and put the same sign.
      • When two positive integers are added, we get a positive integer, [e.g. (+3) + ( +2) = +5].
      • When two negative integers are added, we get a negative integer, [e.g. (-2) + ( -1) = -3].
    • When one positive and one negative integers are added we subtract them as whole numbers by considering the numbers without their sign and then put the sign of the bigger number with the subtraction obtained. The bigger integer is decided by ignoring the signs of the integers [e.g. (+4) + (-3) = +1 and (-4) + ( +3) = -1].
    • The subtraction of an integer is the same as the addition of its additive inverse.
  5. We have shown how addition and subtraction of integers can also be shown on a number line. 

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