Whole numbers are a set of numbers including all hopeful integers and also 0. Whole numbers space a part of genuine numbers that do not incorporate fractions, decimals, or negative numbers. Counting number are also considered all at once numbers. In this lesson, we will certainly learn whole numbers and also related concepts. In mathematics, the number system is composed of all types of numbers, including natural numbers and also whole numbers, prime numbers and also composite numbers, integers, genuine numbers, and imaginary numbers, etc., which space all provided to perform various calculations.

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We check out numbers everywhere about the world, for counting objects, for representing or exchanging money, for measuring the temperature, informing time, etc. Over there is practically nothing that doesn't indicate numbers, it is in it match scores, food preparation recipes, count on objects, etc.

1. | What are totality Numbers? |

2. | Whole number vs natural Numbers |

3. | Whole number on Number Line |

4. | Properties of totality Numbers |

5. | FAQs on entirety Numbers |

## What are whole Numbers?

Natural numbers refer to a set of positive integers and on the various other hand, herbal numbers in addition to zero(0) form a set, advert to as whole numbers. However, zero is an undefined identity that to represent a null collection or no result at all.

In straightforward words, whole numbers space a collection of numbers there is no fractions, decimals, or even an unfavorable integers. The is a arsenal of hopeful integers and zero. The major difference in between natural and whole number is zero.

**Whole Number Definition:**

Whole Numbers space the set of herbal numbers together with the number 0. The set of whole numbers in mathematics is the collection 0, 1,2,3,....This collection of whole numbers is denoted through the symbol** W.**

W = 0,1,2,3,4…

Here are some facts about whole numbers, i beg your pardon will assist you recognize them better:

All natural numbers are entirety numbers.All counting numbers are totality numbers.All hopeful integers including zero are whole numbers.All totality numbers are real numbers.### Whole Number Symbol

The symbol provided to represent entirety numbers is the alphabet ‘W’ in capital letters, W = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,…

### Smallest entirety Number

**Whole numbers start from 0 **(from the definition of totality numbers). Thus, 0 is the smallest totality number. The principle of zero was very first defined by a Hindu astronomer and also mathematician Brahmagupta in 628. In an easy language, zero is a number the lies in between the optimistic and an unfavorable numbers ~ above a number line. Although zero tote no value, that is offered as a placeholder. So, zero is no a confident number nor a an adverse number.

## Whole numbers vs herbal Numbers

From the over definitions, we can understand that every entirety number other than 0 is a natural number. Also, every natural number is a totality number. So, the collection of organic numbers is a part of the set of whole numbers or a subset of entirety numbers.

### Difference between Whole numbers and Natural numbers

Let us recognize the difference in between whole numbers and natural numbers through the table provided below:

Whole NumberNatural NumberThe set of whole numbers is, W=0,1,2,3,... | The set of organic numbers is, N= 1,2,3,... |

The smallest whole number is 0. | The smallest organic number is 1. |

Every herbal number is a whole number. | Every totality number is a organic number, except 0. |

## Whole numbers on Number Line

The collection of herbal numbers and also the set of whole numbers deserve to be presented on the number heat as offered below. Every the hopeful integers or the integers ~ above the right-hand side of 0, represent the organic numbers, whereas every the hopeful integers along with zero, altogether represent the entirety numbers. Both to adjust of numbers can be stood for on the number line as follows:

## Properties of totality Numbers

The simple operations on totality numbers: addition, subtraction, multiplication, and also division, lead to 4 main nature of totality numbers that are detailed below:

Closure PropertyAssociative PropertyCommutative PropertyDistributive Property**Closure Property**

The sum and also product of two entirety numbers is always a whole number. Because that example, 7 + 3 = 10 (whole number), 7 × 2 = 14 (whole number)

**Associative Property**

The amount or product of any type of three entirety numbers remains the same even if the grouping of numbers is changed. For example, when we add the following numbers we obtain the very same sum: 10 + (7 + 12) = (10 + 7) + 12 = (10 + 12) + 7 = 29. Similarly, once we main point the following numbers we gain the exact same product no matter just how the numbers space grouped: 3 × (2 × 4) = (3 × 2) × 4 = 24.

**Commutative Property**

The sum and the product of two whole numbers stay the same even after interchanging the stimulate of the numbers. This residential or commercial property states that adjust in the stimulate of enhancement does not readjust the worth of the sum. Allow 'a' and 'b' be two whole numbers, follow to the commutative property a + b = b + a. Because that example, a = 10 and b = 19 ⇒ 10 + 19 = 29 = 19 + 10. It way that the totality numbers room closed under addition. This property additionally holds true for multiplication, but not because that subtraction or division. Because that example: 7 × 9 = 63 and 9 × 7 = 63.

**Additive identity**

When a whole number is included to 0, its value continues to be unchanged, i.e., if x is a whole number climate x + 0 = 0 + x = x. Because that example, 3 + 0 = 3

**Multiplicative identity**

When a whole number is multiply by 1, the value remains unchanged, i.e., if x is a entirety number then x.1 = x = 1.x. Because that example. 4 × 1 = 4

**Distributive Property**

This residential or commercial property states that the multiplication that a entirety number is spread over the amount of the totality numbers. It method that as soon as two numbers, take for instance a and b are multiplied through the same number c and are then added, climate the amount of a and also b deserve to be multiply by c to obtain the exact same answer. This case can be stood for as: a × (b + c) = (a × b) + (a × c). Allow a = 10, b = 20 and also c = 7 ⇒ 10 × (20 + 7) = 270 and also (10 × 20) + (10 × 7) = 200 + 70 = 270. The same residential property is true because that subtraction together well. For example, we have actually a × (b − c) = (a × b) − (a × c). Allow a = 10, b = 20 and also c = 7 ⇒ 10 × (20 − 7) = 130 and also (10 × 20) − (10 × 7) = 200 − 70 = 130.

**Multiplication by zero**

When a whole number is multiply to 0, the an outcome is always 0, i.e., x.0 = 0.x = 0. For example, 4 × 0 = 0

**Division by zero**

Division of a entirety number through o is no defined, i.e., if x is a whole number then x/0 is not defined.

For an ext information around the nature of totality numbers, click here.

**Important Points**

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**Example 2: Is W closed under subtraction and also division?**

**Solution:**

Whole numbers include only the positive integers and also zero. We recognize that on subtracting one optimistic integer by another, we might not obtain their distinction as a hopeful integer, similarly, on dividing one positive number by another, we might not acquire the quotient as a defined number for example in the situation of 13/0. Thus, for any type of two entirety numbers, your difference and also quotient obtained may no be whole numbers. Therefore, W is no closed under subtraction and also division.

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**Example 3: because that the totality number worths of a, b, and also c, that is, a = 3, b = 2, c = 1, prove a × (b + c) = (a × b) + (a × c)**

**Solution:**

Substituting the values of a, b, and also c, we get: a × (b + c) = 3 × (2 + 1) = 3 × 3 = 9 and also (a × b) + (a × c) = (3 × 2) + (3 × 1) = 6 + 3 = 9. Since, LHS = RHS, 9 = 9, thus, a × (b + c) = (a × b) + (a × c), for the given whole number values. This is known as the distributive residential or commercial property of multiplication of entirety numbers.