Mathematics: Number System: Natural, Whole Numbers and Integers

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There are four different types of numbers that represent in detail.

Pyramid of numbers

Pyramid of Numbers

Natural Numbers

  • Recall that the counting numbers 1, 2, 3, … constitute the system of natural numbers. These are the numbers which we use in our day-to-day life.

  • Recall that there is no greatest natural number, for if 1 is added to any natural number, we get the next higher natural number, called its successor.

  • We have also studied about four-fundamental operations on natural numbers.

For, example,

again, a natural number.

again, a natural number, but

is not defined in natural numbers.

Similarly, again a natural number

again, a natural number

is a natural number but is not defined in natural numbers? Thus, we can say that

  • Addition and multiplication of natural numbers again yield a natural number but Subtraction and division of two natural numbers may or may not yield a natural number

  • The natural numbers can be represented on a number line as shown below.

The natural numbers

The Natural Numbers

  • 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.

Whole Numbers

When a natural number is subtracted from itself, we cannot say what is the left-out number. To remove this difficulty, the natural numbers were extended by the number zero (0), to get what is called the system of whole numbers. Thus, the whole numbers are

Again, like before, there is no greatest whole number.

The number 0 has the following properties:

Division by zero (0) is not defined.

Four fundamental operations can be performed on whole numbers also as in the case of natural numbers (with restrictions for subtraction and division).

Whole numbers can also be represented on the number line as follows:

Whole numbers

Whole Numbers


  • 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) etc. 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 extend whole numbers by such numbers as (called negative 1), (negative 2) and so on such that

  • Thus, we have extended the whole numbers to another system of numbers, called integers.

  • The integers therefore are

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