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Whole Number

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A Whole number is a Set of Positive integers and 0. These are the Numbers which children learn in their initial days of schooling. The only difference between whole numbers and Natural Numbers is 0 which is added in the list of whole numbers.

There are three types of whole numbers:

  1. 0 and 1 are called identities.
  2. Prime numbers are numbers that are divisible only by itself and 1. Example: 2, 3, 5, 7, 11,.... etc.
  3. The numbers other than Prime Numbers and identities are called composite numbers.
The operations like addition, subtraction, multiplication and division can be performed on whole numbers. Like Real Numbers whole numbers also have some properties:
  1. When we add or multiply two whole numbers we always get a whole number.
  2. The real numbers holds true for commutative and Associative Property under addition and multiplication.
  3. Distributive property of multiplication holds true on real numbers under addition and subtraction.
  4. The additive and multiplicative identity property holds true on real number.

History of Whole Number

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Whole numbers are the counting Numbers we learn in our early days. To understand the history of whole numbers, one has to be aware of the discovery of number system as a whole. The early Egyptians built up a powerful structure of numerals with discrete hieroglyphs for 1, 10, and all the exponents of 10 upto over 1 million. Indeed, a place-value system with foundation fundamentally on the numerals was used by the Babylonians.

The progress with the thought that zero is regarded as a number having its own individual numeral was developed much later. Whole numbers were studied extensively by Bourbaki in 1968 and Halmos in 1974. In the Counting numbers 0 is included to make the whole numbers.


Some mathematicians add 0 in the list and some says that 0 is not a whole number. Some authors also define that whole numbers are numbers having fractional part as 0 which makes whole numbers equal to integers. Zero is neither negative nor positive.

When we see the history of Whole Number, all authors have different views about whole numbers but they distinguish them from Fractions. Whole numbers are integers which can be in form of Positive integers or Negative integers; means infinite numbers running on both direction of the number Line. Smallest whole number can either be 0 or 1 because no author has given any proved logic to clear the confusion of inclusion of 0 in whole numbers.

Properties of Whole Numbers

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A Whole Number is a Set of Positive integers and 0. These are the Numbers which children learn in their initial days of schooling. The only difference between whole numbers and Natural numbers is 0 which is added in the list of whole numbers. Whole numbers are the numbers that start from zero to infinity excluding the negative numbers. i.e, 0, 1, 2, 3, 4, 5,... etc are whole numbers value but -1, -2, -3, … are not part of whole numbers.

Like other numbers, whole number also possesses several kinds of properties called as whole number properties. Let a, b, c be three whole numbers. Here is the list of properties of whole number:

1. The closure property for whole numbers states that whenever any whole number is added or multiplied to other whole number then we get another whole number as their result. So, it can be said that, under process of addition and multiplication whole numbers are closed.

2. Commutative property of addition and multiplication for whole numbers states that when we perform addition or multiplication operation over whole number then we can perform in any order.
a + b = b + a
a * b = b * a

3. Associative property of addition and multiplication for whole number states that when we add or multiply any number we can easily associate them.
a + (b + c) = (a + b) + c
a * (b * c) = (a * b) * c

4. Distributivity of multiplication over addition and subtraction for whole numbers can be explain with help of the equation:
a * (b + c) = (a * b) + (a * c)
a * (b - c) = (a * b) - (a * c)
5. Identity property of addition and multiplication says that, when 0 or 1 is added or multiplied to any whole number it gives the same number. Hence 0, 1 are called the additive and multiplicative identity respectively.
a + 0 = a
a * 1 = a