Table of Contents

# Whole numbers and its properties.

**Whole numbers**

**Whole numbers** are all positive integers along with 0. **Whole numbers** is the collection of natural numbers along with 0. Smallest whole number is 0. And there is no data till the date that which one is the last/biggest whole number. When we add 1 to a number, we will get the successor of that number and when we subtract 1 from a number, we will get the predecessor of that number.

Whole numbers are the collection of natural numbers along with zero.

**Definition of Whole number**

Whole numbers are the set of positive numbers associated with ‘0’.or we can say set of natural numbers with 0.

**Symbol used for Whole Number**

Whole numbers are set of natural numbers along with ‘0’.And we use the symbol ‘W’ to represent the set of whole numbers.

W = {0,1,2,3,4,…}

**Whole Numbers on a Number Line**

** **Let’s place **whole numbers on a line**. For that we have ‘0’ as the starting point. We don’t have any last point as there is no data of the greatest number till the date. And every number have a successor (if we add 1 to a given number).

There are certain steps to follow to draw a number line.

- Draw a line (line should be a ray as we have starting point but don’t have an end point).
- Mark a point on it and label it as ‘0’.Mark another point to the right of ‘0’ and label it as ‘1’.We have to maintain a unit distance between two numbers 0 and 1.
- Mark one more point to the right of ‘1’ and label it as ‘2’ and follow the procedure for labelling points at unit distance.

Like this we will get **a number line of whole numbers**.

**Properties of Whole Numbers**

Properties of whole Numbers helps us to understand the numbers better. If we do some operations on whole numbers, we will get several properties of whole numbers.

Some** properties of whole numbers** are listed below.

- Closure property
- Commutative property for addition and multiplication.
- Associative property for addition and multiplication.
- Distributive property for multiplication over addition.

**Now **detail explanation of these properties with examples

** Closure Property**

Whole Numbers are closed under addition and multiplication.

If we add two whole numbers, the resultant number is also a whole number.

If we multiply two whole numbers, the resultant number is also a whole number.

**Closure property under addition**.

Let’s take two whole numbers 9 and 7

Now add two numbers 9+7=16 (is also a whole number).

Let’s take another two number 45 and 78.

Now addition of 45+78=113 (is also a whole number).

So we can conclude that:

whole number1 +whole number2=whole number |

**Closure property under multiplication**

Take two whole numbers 9 and 7.

Now multiply two numbers =9*7=63 is also a whole number.

Whole number1 * whole number2 = whole number |

Check why whole numbers are not coming under subtraction and division.

**Check for subtraction**

Whole number1 | subtract | Whole number2 | result | |

9 | – | 4 | = | 5,whole number |

8 | – | 9 | = | -1,not a whole number |

7 | – | 9 | = | -2,not a whole number |

**Check for Division**

whole number 1 | division | Whole number2 | = | |

9 | / | 3 | = | 3 ,whole number |

9 | / | 4 | = | 9/4,not a whole number |

12 | / | 5 | = | 12/5,not a whole number |

**Commutative**

**Commutative for addition and multiplication**

Commutative property says that we can add and multiply two whole numbers in any order.

If we will take two variables a and b , then commutative property for addition says that

a + b = b + a |

First we will see for **addition**

Take two whole numbers 8 and 3.

Whole number1 =8

Whole number2=3

8 + 3 (whole number1+whole number2) =11 (whole number) | 3+8(whole number2+whole number1) =11(whole number) |

Now we will check for **multiplication**

** **If we will take two variables a and b, then commutative property for multiplication says that

a * b = b * a |

Same whole numbers taking for multiplication

8 * 3 (whole number1*whole number2) =24 (whole number) | 3 *8(whole number2*whole number1) =24(whole number) |

**Verify**

- We will verify what will happen if we check commutative property for
**subtraction**

Whole number1 =8

Whole number2=3

8 – 3 (whole number1-whole number2) =5 (whole number) | 3 -8(whole number2-whole number1) =-5(not a whole number) |

More examples we will see to verify

Whole number1 =7

Whole number2=4

7 – 4 (whole number1-whole number2) = 5 (whole number) | 4 -7(whole number2-whole number1) =-5(not a whole number) |

Now we cannot say that we can subtract two whole numbers in any order.

Subtraction is not commutative for whole numbers.

- Verify commutative property for
**division**.

Take two whole numbers 10 and 5.

Whole number1=10

Whole number2=5

10/5 (whole number1/whole number2) =2 (whole number) | 5/10(whole number2/whole number1) =1/2(not a whole number) |

Some more examples

Whole number1=6

Whole number2=4

6/4 (whole number1/whole number2) =3/2 (not a whole number) | 4/6(whole number2/whole number1) =2/3(not a whole number) |

Now we cannot say that we can divide two whole numbers in any order.

So, Division is not commutative for whole numbers.

## **Associative**

**Associative for Addition**

If we will take three variables a, b and c, then associative property for addition says that

(a + b) +c= a+ (b + c) |

a,b,c can be added by forming a group of any form.

To get the clear we can take

4 pens, 3 pens and 2 pens. Now we will add all.

For addition we can add 4 + 3 pens first then we will add 2 to the sum of two.

Or we can add 3+2 pens first then we will add 4 pens to the sum of two.

Like (4+3)+2 or 4+(3+2)

7+2 4+5

=9 =9

In both cases result will same.

So we can say whole numbers are associated under addition.

**Associative for Multiplication**

If we will take three variables a, b and c, then associative property for multiplication says that

(a * b) *c = a* (b * c) |

Take three whole numbers 5,4 and 7.

Whole number1=5.

Whole number2=4.

Whole number3=7.

(Wholenumber1*whole number2)*whole number3 | Wholenumber1*(whole number2*whole number3) |

(5*4)*7 20*7 =140 | 5*(4*7) 5*28 =140 |

So three whole numbers can be multiplied by in any order.

**Verify**

- Associative under
**subtraction**

What will happen if we perform associative property under subtraction?

Take three whole numbers 4, 3 and 2.

(4-3)-1 4-(3-1)

=2-1 =4-2

=1 =2

The result of both sides are different.

So we can’t say that whole numbers are associative under subtraction.

(a-b) -c ≠ a- (b-c) |

## Distributive property

**Distributive property** for multiplication over addition.

Distributive property helps to make the calculation easier to solve.it simplifies the expression.

According its name ‘distributive’, we can say multiplication will distribute over addition. The **Distributive property for multiplication** over addition says that when a number (number1) is multiplied by the sum of two whole numbers (number2 and number3), the number (number1) can be distributed as number1*number2 and number1* number3 then adding the two products(number1*number2+number1*number3) together for the same result as multiplying the number1 by the sum of number2 and number3.

For any three Whole numbers a, b and c

a*(b + c) can be distributed as a*b and a*c. This is called **distributive of multiplication over addition**.

a*(b + c) = a*b + a*c |

Example

Number1=5

Number2=3

Number3=2

5*(3+2) = 5*3+5*2

5*5=15+10

25=25

## Distributive property for multiplication over subtraction

**Distributive property** helps to simplify the expression in which we can multiply a number by the difference of two other numbers.

For any three whole numbers a, b and c

a*(b-c) can be distributed as a*b-a*c. This is called the **distributive property of multiplication **over subtraction.

a*(b-c)= a*b-a*c |

## **Distributive property for division**

The distributive property doesn’t apply to division like multiplication but yes it can be used in division to make the division easier. Example: Evaluate 236/4 using distributive property. We can break 236 into 100, 100, and 36. 100/4 + 100/4 +36/4 . 236/4 25+25+9 =59 =59 We can take more examples Evaluate 545/5 using distributive property. 545/5= 100/5 + 100/5 + 100/5 + 100/5 + 100/5 + 45/5 =20 +20 +20 +20 +20 +9 =109 |

**Identity Property (for addition and multiplication)**

When we add 0 to any whole number. We will get the same whole number again.

So Zero is known as the

**Division by zero**

Division by a number means subtracting that number repeatedly till we reach to zero(0).Result is in how many moves we reached to 0.

Let’s understand with an example;

4/2 (4 divided by 2).

Subtract 2 from 4. Result will be the number of moves to reach 0.

4-2=2 —— (move – 1)

2-2=0 —— (move 2)

Answer is 2. In 2 number of moves we got 0.

Example 2-

9/3

9-3=6 —– (move 1)

6-3=3 —– (move 2)

3-3=0 —– (move 3)

So answer is 3 .in 3 numbers of move we reached to 0.

What happen when we divide a number by 0?

Let’s check.

Take an example: 5/0

5-0=5 —– (move 1)

5-0=5 —– (move 2)

.. —– (move 3)

.. ..

.. ..

.. ..

We will never reach to 0.in every move we will get 5.

So we say 5/0 is not defined.

Division of a whole number by 0 is not defined.

**Whole Number vs. Natural Numbers**

Whole Numbers | Natural numbers |

We use ‘W’ to represent the set of whole numbers. | We use ‘N’ to represent the set of natural numbers. |

Whole numbers are the set of natural numbers along with 0. | Natural numbers are the set of counting numbers 1,2,3,…. |

Smallest whole number is 0 | Smallest natural number is 1. |

**Real life examples of Whole Numbers**

** **We are surrounded by numbers directly or indirectly. Numbers have a great role in our lives.

Some real time uses are explained below. These are some examples. If we follow each and every aspects of our lives then we will able to know that we are associated with numbers from the day we were born.

1. For preparing food we use some proportions of ingredients.

2. How many children’s do you have? The result would be a number.

3. using of numbers for counting of votes.

4. using of phone numbers –to call someone.

5. Using numbers while telling times.

6. Using numbers for calculation in business.

Few Uses are mentioned above .If you think deeply, you will get a number of uses of whole numbers.

**FAQs**

**What are whole numbers? Give examples.**

**Answer: Whole numbers are counting numbers starting with ‘0’.**

**Is ‘0’ a whole number?**

**Answer: yes, 0 is a whole number and the smallest whole number.**

**What are first 5 whole numbers?**

**Answer: The first 5 whole numbers are 0, 1, 2, 3, 4, 5.**

**Which is the smallest whole number?**

**Answer: ‘0’ is the smallest whole number.**

**How do you classify numbers?**

**Answer: numbers are classified as**

** natural numbers,**

**whole numbers,**

** integers,**

**irrational numbers,rational numbers.**

**What is called counting numbers?**

**Answer: counting numbers are natural numbers starting from 0 to infinite.**