15 As A Whole Number

The whole numbers are the office of the number arrangement which includes all the positive integers from 0 to infinity. These numbers be in the number line. Hence, they are all existent numbers . We can say, all the whole numbers are real numbers, but not all the real numbers are whole numbers. Thus, nosotros can ascertain whole numbers every bit the set up of natural numbers and 0. Integers are the set of whole numbers and negative of natural numbers. Hence, integers include both positive and negative numbers including 0. Real numbers are the gear up of all these types of numbers, i.e., natural numbers, whole numbers, integers and fractions.

The complete ready of natural numbers along with '0' are called whole numbers. The examples are: 0, 11, 25, 36, 999, 1200, etc.

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Table of contents:

Definition
- Symbol
Properties
- Closure
- Commutative
- Condiment
- Multiplicative
- Associative
- Distributive
Whole Numbers and Natural numbers
Solved Examples
Practice Problems
Video Lesson
FAQs

Whole Numbers Definition

The whole numbers are the numbers without fractions and it is a collection of positive integers and nada. It is represented past the symbol "West" and the set of numbers are {0, ane, 2, three, iv, 5, 6, 7, 8, nine,……………}. Zero every bit a whole represents cipher or a null value.

Whole Numbers: W = {0, 1, 2, 3, 4, 5, 6, 7, 8, nine, ten……}
Natural Numbers: N = {1, 2, 3, 4, 5, six, 7, 8, nine,…}
Integers: Z = {….-9, -eight, -7, -6, -v, -4, -three, -2, -1, 0, 1, 2, 3, 4, v, 6, 7, 8, ix,…}
Counting Numbers: {1, 2, iii, 4, 5, 6, 7,….}

These numbers are positive integers including zero and do not include fractional or decimal parts (3/4, ii.2 and 5.three are not whole numbers). Also, arithmetics operations such as add-on, subtraction, multiplication and division are possible on whole numbers.

Symbol

The symbol to represent whole numbers is the alphabet 'West' in capital messages.

W = {0, 1, two, 3, iv, 5, half-dozen, 7, eight, 9, x,…}

Thus, the whole numbers list includes 0, 1, 2, 3, 4, 5, half dozen, 7, 8, ix, x, eleven, 12, ….

Facts:

All the natural numbers are whole numbers
All counting numbers are whole numbers
All positive integers including cypher are whole numbers
All whole numbers are real numbers

If you lot still have doubt, What is a whole number in maths? A more comprehensive understanding of the whole numbers tin can be obtained from the following chart:

Real number system

Whole Numbers and Natural Numbers
Natural Numbers
Difference Between Natural and Whole numbers
Important Questions For Course 6 Maths

Whole Numbers Properties

The properties of whole numbers are based on arithmetic operations such as addition, subtraction, sectionalisation and multiplication. Two whole numbers if added or multiplied will give a whole number itself. Subtraction of two whole numbers may non result in whole numbers, i.e. it can be an integer too. Also, the division of two whole numbers results in getting a fraction in some cases. Now, let united states of america see some more than properties of whole numbers and their proofs with the assist of examples hither.

Closure Property

They can be closed under addition and multiplication, i.e., if x and y are two whole numbers then x. y or x + y is besides a whole number.

Example:

five and 8 are whole numbers.

five + eight = 13; a whole number

5 × viii = twoscore; a whole number

Therefore, the whole numbers are closed under add-on and multiplication.

Commutative Property of Addition and Multiplication

The sum and product of two whole numbers will be the same whatever the order they are added or multiplied in, i.e., if x and y are two whole numbers, then 10 + y = y + x and x . y = y . x

Case:

Consider ii whole numbers 3 and vii.

3 + 7 = 10

7 + iii = x

Thus, 3 + seven = 7 + three .

Also,

three × 7 = 21

7 × 3 = 21

Thus, three × 7 = 7 × 3

Therefore, the whole numbers are commutative under addition and multiplication.

Condiment identity

When a whole number is added to 0, its value remains unchanged, i.due east., if ten is a whole number so x + 0 = 0 + x = ten

Case:

Consider 2 whole numbers 0 and 11.

0 + xi = 11

eleven + 0 = 11

Hither, 0 + 11 = 11 + 0 = eleven

Therefore, 0 is called the additive identity of whole numbers.

Multiplicative identity

When a whole number is multiplied by 1, its value remains unchanged, i.e., if x is a whole number then x.one = x = one.ten

Case:

Consider two whole numbers ane and 15.

1 × 15 = fifteen

15 × ane = 15

Hither, 1 × xv = 15 = 15 × i

Therefore, 1 is the multiplicative identity of whole numbers.

Associative Property

When whole numbers are being added or multiplied as a set, they can be grouped in any social club, and the result will be the same, i.e. if ten, y and z are whole numbers then ten + (y + z) = (x + y) + z and ten. (y.z)=(x.y).z

Example:

Consider three whole numbers two, 3, and 4.

two + (3 + 4) = 2 + 7 = 9

(2 + 3) + 4 = 5 + iv = 9

Thus, ii + (iii + four) = (ii + 3) + 4

2 × (3 × 4) = 2 × 12 = 24

(2 × 3) × four = 6 × 4 = 24

Here, 2 × (3 × 4) = (2 × three) × 4

Therefore, the whole numbers are associative under improver and multiplication.

Distributive Property

If 10, y and z are three whole numbers, the distributive property of multiplication over addition is x. (y + z) = (ten.y) + (x.z), similarly, the distributive belongings of multiplication over subtraction is 10. (y – z) = (x.y) – (x.z)

Case:

Allow us consider three whole numbers 9, 11 and 6.

9 × (eleven + 6) = ix × 17 = 153

(9 × 11) + (9 × 6) = 99 + 54 = 153

Hither, 9 × (11 + half-dozen) = (9 × eleven) + (9 × half dozen)

Also,

nine × (11 – 6) = 9 × 5 = 45

(ix × xi) – (9 × vi) = 99 – 54 = 45

And then, 9 × (11 – 6) = (9 × eleven) – (nine × 6)

Hence, verified the distributive holding of whole numbers.

Multiplication by zero

When a whole number is multiplied to 0, the result is always 0, i.e., x.0 = 0.x = 0

Instance:

0 × 12 = 0

12 × 0 = 0

Hither, 0 × 12 = 12 × 0 = 0

Thus, for whatsoever whole number multiplied by 0, the effect is always 0.

Division by zip

The segmentation of a whole number by o is not defined, i.eastward., if 10 is a whole number and then 10/0 is not divers.

Besides, bank check: Whole number calculator

Departure Between Whole Numbers and Natural Numbers

Difference Between Whole Numbers & Natural Numbers
Whole Numbers	Natural Numbers
Whole Numbers: {0, one, 2, iii, four, v, 6,…..}	Natural Numbers: {1, ii, three, four, 5, vi,……}
Counting starts from 0	Counting starts from 1
All whole numbers are not natural numbers	All Natural numbers are whole numbers

The beneath figure will help united states of america to sympathize the difference betwixt the whole number and natural numbers :

Whole numbers-Number line

Can Whole Numbers exist negative?

The whole number tin't be negative!

As per definition: {0, i, ii, iii, 4, 5, six, 7,……till positive infinity} are whole numbers. There is no place for negative numbers.

Is 0 a whole number?

Whole numbers are the gear up of all the natural numbers including zero. So yes, 0 (zero) is non merely a whole number but the showtime whole number.

Solved Examples

Example 1:Are 100, 227, 198, 4321 whole numbers?

Solution:Yes. 100, 227, 198, and 4321 are all whole numbers.

Instance ii: Solve 10 × (5 + 10) using the distributive property.

Solution: Distributive property of multiplication over the add-on of whole numbers is:

x × (y + z) = (10 × y) + (x × z)

10 × (five + x) = (ten × 5) + (10 × 10)

= 50 + 100

= 150

Therefore, x × (five + x) = 150

Withal, we can prove several examples of whole numbers using the backdrop of the whole numbers.

Exercise Bug

Write whole numbers between 12 and 25.
What is the additive inverse of the whole number 98?
How many whole numbers are there between -i and 14?

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Video lesson

Frequently Asked Questions on Whole Numbers

What are whole numbers?

The whole numbers are defined every bit positive integers including zip. The whole number does non contain any decimal or fractional role. It means that it represents the entire thing without pieces. The set of whole numbers is mathematically represented equally:
Due west = (0, 1, two, iii, iv, 5,……}

Can whole numbers be negative?

No, the whole numbers cannot be negative. The whole numbers offset from 0, 1, 2, 3, … and then on. All the natural numbers are considered as whole numbers, just all the whole numbers are not natural numbers. Thus, the negative numbers are not considered every bit whole numbers.

What are the properties of whole numbers?

The backdrop of whole numbers are:
Whole numbers are closed under addition and multiplication
The addition and multiplication of whole numbers is commutative
The add-on and multiplication of whole numbers is associative
It obeys the distributive property of multiplication over addition
The additive identity of whole numbers is 0
The multiplicative identity of whole numbers is 1

Is 10 a whole number?

ten is a whole also as a natural number. Information technology is written as Ten in words. Although -ten also represents a whole and not a fraction.

Which numbers are not whole numbers?

The numbers which do non exist between 0 and infinity are non whole numbers. Negative integers, fractions or rational numbers are not whole numbers. Examples are -1, -5, ½, 9/iv, pi, etc. are non whole numbers.

Are all whole numbers real numbers?

Real numbers are those numbers that include rational numbers, integers, whole numbers and natural numbers. All whole numbers are existent numbers merely not all real numbers are whole.

Are all natural numbers, whole numbers?

Natural numbers are those which outset from 1 and finish at infinity, whereas whole numbers get-go from 0 and end at infinity. All the natural numbers are whole numbers just not all whole numbers are natural.

Are natural numbers and counting numbers the same?

Natural numbers are the numbers starting from 1 and extend up to infinity. Counting numbers are used to count the objects or people or annihilation which is countable. Hence, we always start counting from ane.