Number System: Natural Numbers, Whole Numbers, Integers

From ancient times, human beings have been trying to have a count of their belongings by using various techniques such as putting scratches on the ground or storing stones, one for each commodity kept or taken out. This was the way of having a count of their belongings without having any knowledge of counting.

One of the greatest inventions in the history of civilization is the creation of numbers. The invention of number system including zero and the rules for combining them helped people to reply questions of the type:

  • How many apples are there in the basket?
  • How many speakers have been invited for addressing the meeting?
  • What is the number of toys on the table?
  • How many bags of wheat have been the yield from the field?

The answers to all these situations and many more involve the knowledge of numbers and operations on them.

Natural Numbers

The counting numbers 1, 2, 3, ... constitute the system of natural numbers. These are the numbers which you use in our day-to-day life. There is no greatest natural number, for if 1 is added to any natural number, you get the next higher natural number, called its successor.

Addition and multiplication of natural numbers again yield a natural number.

Subtraction and division of two natural numbers may or may not yield a natural number.

Two natural numbers can be added and multiplied in any order and the result obtained is always same. This does not hold for subtraction and division of natural numbers.

The natural numbers can be represented on a number line as:

Whole Numbers

When a natural number is subtracted from itself, you can not say what is the left out number. To remove this difficulty, the natural numbers were extended by the number zero (0), to get the system of whole numbers.

Thus, the whole numbers are: 0, 1, 2, 3, and so on. There is no greatest whole number.

The number 0 has the following properties:

  • a + 0 = a = 0 + a
  • a - 0 = a but (0 - a) is not defined in whole numbers
  • a × 0 = 0 = 0 × a
  • Division by zero (0) is not defined

Whole numbers can be represented on the number line as:

Integers

While dealing with natural numbers and whole numbers we found that it is not always possible to subtract a number from another. For example, (2 - 3), (3 - 7), (9 - 20) are all not possible in the system of natural numbers and whole numbers. Thus, it needed another extension of numbers which allow such subtractions.

Thus, we have extended the whole numbers to another system of numbers, called integers. The integers are:

..., -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, ...

The integer number line is represented as:

If an integer a > b, then a will always be to the right of b, otherwise vise-versa. For finding the greater (or smaller) of the two integers a and b, follow the following rule:

  • a > b, if a is to the right of b
  • a < b, if a is to the left of b

Examples

Example 1: Identify natural numbers, whole numbers and integers from the following: 15, 22, -6, 7, -13, 0, 12, -12, 13, -31

Natural numbers are: 7, 12, 13, 15 and 22

whole numbers are: 0, 7, 12, 13, 15 and 22

Integers are: -31, -13, -12, -6, 0, 7, 12, 13, 15 and 22