There are assorted properties of totality numbers that help us to carry out operations on totality numbers. These properties define the features of operations. In this article, we space going to learn the nature of totality numbers under addition, subtraction, multiplication, and division.

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1.List of properties of totality Numbers
2.Closure Property
3.Associative Property
4.Commutative Property
5.Distributive Property
6. FAQs on nature of whole Numbers

Whole numbers room the herbal numbers in addition to the number 0. The set of entirety numbers in math is the set 0,1,2,3,.... The is denoted through the symbol, W. The four properties of entirety numbers are as follows:

Closure PropertyAssociative PropertyCommutative PropertyDistributive Property

Let's check out all four properties of totality numbers in detail.


The closure building of the entirety number claims that "Addition and also multiplication the two whole numbers is constantly a entirety number." for example: 0+2=2. Here, 2 is a entirety number. In the exact same way, multiply any kind of two whole numbers and also you will check out that the product is again a totality number. For example, 3×5=15. Here, 15 is a entirety number. Thus the collection of totality numbers, W is close up door under enhancement and multiplication.

The closure residential property of W is declared as follows:

For every a,b∈W, a+b∈W, and a×b∈W.

This property does not host true in the instance of individually and division operations on totality numbers. As, 0 and 2 are entirety numbers, but 0 - 2 = -2, i m sorry is no a entirety number. Similarly, 2/0 is no defined. Therefore, whole numbers space not closed under subtraction and division.


Associative home of whole Numbers


The associative residential or commercial property of entirety numbers claims that "The sum and the product of any kind of three totality numbers remain the exact same regardless of exactly how the numbers are grouped with each other or arranged".

Example 1: (1+2)+3 = 1+(2+3) because,

(1+2)+3 = 3+3 = 6

1+(2+3) = 1+5 = 6

Example 2: (1×2)×3 = 1×(2×3) because,

(1×2)×3 = 2×3 = 6

1×(2×3) = 1×6 = 6

Thus the set of whole numbers, W is associative under addition and multiplication. The associative home of W is stated as follows:

For all a,b,c∈W, a+(b+c)=(a+b)+c and a×(b×c)=(a×b)×c.

The associative building of entirety numbers does not hold true because that subtraction and department operations. The is because the plan of numbers is vital in this operations. Because that example, 2, 3, and 4 are entirety numbers, but 2 - (3 - 4) = 2 - (-1) = 3 and (2 - 3) - 4 = - 1 - 4 = -5. So, 3 ≠ -5. The exact same is with the division where 8 ÷ (4 ÷ 2) ≠ (8 ÷ 4) ÷ 2.


Commutative building of entirety Numbers


The commutative building of whole numbers claims that "The sum and also the product that two totality numbers continue to be the same also after interchanging the bespeak of the numbers". The is the exact same as associative property, the only distinction is that below we are only talking around two totality numbers.

Example 1: 2+3 = 3+2 because,

2+3 = 5

3+2 = 5

Example 2: 2×3 = 3×2 because,

2×3 = 6

3×2 = 6

Thus the set of totality numbers, W is commutative under enhancement and multiplication. The commutative home of W is stated as follows:

For all a,b∈W, a+b=b+a and also a×b=b×a.

The commutative property of whole numbers walk not host true under subtraction and division.

Let us summarise this three properties of entirety numbers in a table.

OperationClosure PropertyAssociative PropertyCommutative Property
Additionyesyesyes
Subtractionnonono
Multiplicationyesyesyes
Divisionnonono

Distributive residential or commercial property of totality Numbers


The distributive residential property of multiplication over addition is a×(b+c)=a×b+a×c.

Example 1: 3×(2+5) = 3×2+3×5 because,

3×(2+5) = 3×7 = 21

3×2+3×5 = 6+15 = 21

The distributive home of multiplication end subtraction is a×(b−c)=a×b−a×c.

Example 2: 3×(5−2) = 3×5−3×2 as,

3×(5−2) = 3×3 = 9

3×5-3×2 = 15-6 = 9

To conclude, let united state look in ~ the graph of nature of totality numbers given listed below to recognize which property is applicable to which operation.

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Think Tank:

Is W close up door under subtraction and division?Is W associative under subtraction and also division?Is W commutative under subtraction and also division?

Challenging concerns on properties of entirety Numbers:

Find the product using the distributive property: 28×75.The set of totality numbers is commutative under i m sorry of the operations?(a) addition (b) Subtraction(c) Multiplication (d) Division

Also Check:


Examples of nature of totality Numbers


Example 1: The collection of entirety numbers is closeup of the door under i beg your pardon of the operations?

AdditionSubtractionMultiplicationDivision

Solution: If we assume any two whole numbers, your sum and also the product are additionally the totality numbers. But their difference and quotient might or might not it is in the entirety numbers. Because that example, 1 and also 2 are entirety numbers.

1−2=−1

1÷2=0.5

Here, the difference and the quotient room NOT entirety numbers.

Therefore, together per properties of whole numbers, the collection of entirety numbers is closed only under addition and multiplication.


Example 2: uncover the adhering to product using the distributive building of totality numbers: 72×45.

Solution: By utilizing the distributive property, we deserve to write the offered product as follows:

72×45 = (70+2)×(40+5)

= 70×40+70×5+2×40+2×5

= 2800+350+80+10

= 3240

Therefore, by using the nature of totality numbers, 72×45 = 3240.


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Practice concerns on properties of entirety Numbers


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FAQs on properties of whole Numbers


What space the properties of whole Numbers?

Properties of totality numbers room a set of rules or laws that can be applied while doing basic arithmetic work on entirety numbers. For example, together per the commutative home of totality numbers, we can include 2 come 99 quite than adding 99 to 2.

What room the 4 Properties of entirety Numbers?

The 4 properties of whole numbers are given below:

Closure propertyAssociative propertyCommutative propertyDistributive property

What are the nature of totality Numbers Under enhancement and Multiplication?

The properties of whole numbers under addition are given below:

Closure property ⇒ a + b ∈ W, ∀ a,b ∈ W.Associative building ⇒ a + (b + c) = (a + b) + c, ∀ a,b,c ∈ W.Commutative residential property ⇒ a + b = b + a, ∀ a,b ∈ W.Distributive property ⇒ a × (b + c) = (a × b) + (a × c), ∀ a,b,c ∈ W.Additive identity ⇒ 0 is the identity aspect for addtion of entirety numbers as 0 + a = a + 0 = a, ∀ a ∈ W.

The properties of totality numbers under multiplication are discussed below:

Closure building ⇒ a × b ∈ W, ∀ a,b ∈ W.Associative residential or commercial property ⇒ a × (b × c) = (a × b) × c, ∀ a,b,c ∈ W.Commutative property ⇒ a × b = b × a, ∀ a,b ∈ W.Zero property ⇒ a × 0 = 0 × a = 0, ∀ a ∈ W.Multiplicative identity ⇒ 1 is the identity facet for multiplication of entirety numbers together 1 × a = a × 1 = a, ∀ a ∈ W.

What room the nature for division of entirety Numbers?

The properties of whole numbers under department are provided below:

0 split by any non-zero whole number always results in 0.0 separated by 0 is not defined.Any non-zero whole number divided by 1 constantly results in the same whole number.Any non-zero whole number split by itself constantly results in 1.The department of whole numbers satisfies the department algorithm which says "Dividend = Divisor × Quotient + Remainder".

What room the Properties because that Subtraction of totality Numbers?

The properties of entirety numbers under individually are provided below:

0 subtracted from any whole number outcomes in the very same number.Any totality number subtracted from 0 results in that is additive inverse.Closure, associative, and commutative properties carry out not hold true for subtraction.The distributive residential property of multiplication over subtraction satisfies. That implies, a × (b - c) = (a × b) - (a × c), ∀ a,b,c ∈ W.

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What is the Associative residential or commercial property of addition in entirety Numbers?

The associative property of addition of whole numbers states that the order in which three numbers space arranged walk not influence their sum. Mathematically, it can be expressed together a + (b + c) = (a + b) + c = (a + c) + b, ∀ a,b,c ∈ W.