 # Properties of Whole Numbers  View Notes

## What is a Whole Number In Math?

Just like any other types of numbers, whole numbers are too a part of the number system. A whole number represents all the positive integers right from 0 to infinity. These numbers are a part of the number line, therefore they are all known as real numbers. The twist is that even though all the whole numbers can be called real numbers, the real numbers cannot possibly be called as whole numbers. Not to miss it, natural numbers including ‘0’ are called whole numbers. In a whole number, there are no fractions or decimals but just positive integers with 0. For example, ¾ or 5.8 cannot be whole numbers. We can represent whole numbers as “W”. Therefore, the number set for whole numbers can be written as  {0, 1, 2, 3, 4, 5, 6, 7, 8, 9,……………}.

Given below is a picture that will give you a clear understanding of what is a whole number?

### Properties of Whole Numbers with Examples

Now that we know the whole numbers definition, it is time that we move on to understand the properties of whole numbers with examples. Given below are the properties of whole numbers while dealing with addition, subtraction, multiplication, and division. Knowing the properties of whole numbers, we can solve any problem or say the properties of whole numbers with examples will help us to solve difficult problems very easily. So let us see what they offer.

The properties of addition can be branched into four parts, i.e., the closure property, the commutative property, the associative property, and the additive identity. Let us first begin with the closure property.

The Closure Property: The closure property of a whole number says that when we add two whole numbers, the result will always be a whole number. For example, 3 + 4 = 7 (whole number).

The Commutative Property: The commutative property of a whole number says that changing the order of addition has no effect on the result. For example, 2 + 5 = 7; 5 + 2 = 7. Two plus five is seven, five plus two will also be seven.

The Associative Property: This is very similar to the closure property of whole number. The only difference is that in the closure property, we only add two whole numbers but in an associative property we add three or more numbers. For example, 10 + (5 + 12) = (10 + 5) + 12 = (10 + 12) + 5 = 27

The Additive Identity: This is also known as the property of zero. It says that when we add 0 with any whole number then the result will be the same whole number. For example, 0 + 8 = 8.

### Properties of Subtraction

Closure Property: The closure property of subtraction tells us that when we subtract two whole numbers, the result may not always be a whole number. For example, 5 - 9 = -4, the result is not a whole number.

Commutative Property: Subtraction of commutative property says that when we change the order of subtraction while dealing with two whole numbers, the result may not be the same. For example, 10 - 3 = 7 but 3 - 10 = -7.

Associative Property: An associative property means that an order of subtraction is extremely important, we cannot just group any two or more whole numbers and then subtract them first. An example will help you to understand it better. 8 − (5 − 2) = 5 is not equal to or same as (8 − 5) − 2 = 1.

Subtractive Property of Zero: On subtracting zero from a whole number, the result will be the same whole number. For example, 98 − 0 = 98.

### Properties of Multiplication

Closure Property: If we multiply two whole numbers, we get a whole number as a result. For example, 10 × 5 = 50 (whole number).

Commutative Property: If we change the order of multiplication, the product will remain the same. This is known as the commutative property of multiplication.

For example, 4 × 9 = 36 is equal to 9 × 4 = 36.

Associative Property: Changing the order of multiplication while dealing with three or more whole numbers does not affect the product.

For example, 6 × (7 × 2) is equal to (6 × 7) × 2 is also equal to (6 × 2) × 7 = 84.

Multiplicative Identity: If we multiply 1 with any whole number, we will get the same whole number as the product. For example, 1 x 5 = 5

Multiplicative Property of Zero: If we multiply 0 with a whole number, we will get 0 as the product. In other words, any number multiplied to 0 is always 0.

### Properties of Division

Closure Property: This tells us that the result of the division of two whole numbers might differ. For example, 14 ÷ 7 = 2 (whole number) but 7 ÷ 14 = ½ (not a whole number).

Commutative Property: The commutative property of division of the whole numbers is not commutative. For example, 14 ÷ 7 is not equal to 7 ÷ 14.

Associative Property: Change in the order of division changes the result. For example, 100 ÷ (25 ÷ 5) = 20 which is not equal to (100 ÷ 25) ÷ 5 = 4 ÷ 5.

### Fun Facts

1. There is no 'largest' whole number.

2. Except 0, every whole number has an immediate predecessor or a number that comes before.

3. A decimal number or a fraction lies between two whole numbers, but are not whole numbers.

### Solved Examples

Question 1: Multiply 24 × 15 by using a property.

Answer : 24 × 15 = 24 × (10 + 5) = 24 × 10 + 24 × 5 = 240 + 100 = 340.

Question 2: Solve 121 × 18 − 121 × 8 by the distributive property.

Answer : 121 × 18 − 121 × 8 = 121 × (18 − 8) = 121 × 10 = 1210.

Question 1) What is the Difference Between a Whole Number and a Natural Number?

Answer 1) While both whole numbers and natural numbers are a part of number system, there are a few differences between them. A whole number usually starts from 0 and a natural number starts with 1. The number set of a whole number can be represented as {0, 1, 2, 3, 4, 5,.........} while the number set of a natural number can be represented as {1, 2, 3, 4, 5,..........}. All whole numbers can be called natural numbers but all natural numbers cannot be called whole numbers. We represent the whole number as “W” and the natural number as “N”.

Question 2) What is a Number System?

Answer 2) A number system is a system of representing numbers. The adjective real that is used in the context was actually introduced by René Descartes in the 17th century. He made a distinction between the real numbers and the imaginary roots of the polynomials. A number system consists of different types of numbers like integers, whole numbers, natural numbers, etc. it also makes a distinction between rational numbers and irrational numbers. A number system is very important in life, especially who starts learning numbers because it helps them to understand numbers better and make a clear distinction between them.

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