A key Notes on Properties of Numbers

Even and odd numbers, prime and composite numbers, decimals, fractions, rational and irrational numbers, natural numbers, integers, real numbers, rational and irrational numbers, and whole numbers are all examples of numbers in mathematics.

Math is nothing without numbers, which are the subject’s foundation. A number name refers to the style of writing a number in words. 1 is written as one in words, for example.

Properties of Numbers:

A property is any attribute that pertains to a given set in mathematics. Natural numbers have the following four properties:

• Closure Property
• Associative Property
• Commutative Property
• Distributive Property

Closure Property

The addition and multiplication of two or more natural numbers always results in a natural number, according to the closure property of natural numbers. Check all four arithmetic operations, as well as all p,q, N.

Addition :The resultant number, or sum, is clearly a natural number. Thus, p + q ∈ N, for all p, q ∈ N.

Multiplication: 2 × 5 = 10, 6 × 4 = 24, etc. The resultant integer, or product, is clearly a natural number. As a result, p × q ∈ N, for all p, q ∈ N.

As a result, p × q ∈ N, for all p, q ∈ N.

Subtraction: 8–5 = 3, 7–2 = -5, and so on. Clearly, the outcome might be a natural number or not. Thus ,p- q or q – p ∉ N, for all p, q ∈ N.

Division: 15 ÷ 5 = 3, 10 ÷ 3 = 3.33, etc. Clearly, the generated number could be a natural number or not. Hence we can conclude that  p ÷ q or q ÷ p ∉ N, for all p, q ∈ N.

Thus, we can conclude that the set of natural numbers is always closed while adding and multiplying, but not when subtracting and dividing.

Associative Property:

The associative property of natural numbers asserts that when the grouping of numbers is altered, the sum or product of any three natural numbers remains the same. Check for all four arithmetic operations as well as all p,q,r∈ N.

Addition: p + ( q+ r ) = ( p + q ) + r. 

For eg. ,3 + (15 + 1 ) = 19 and (3 + 15 ) + 1 = 19.

Multiplication: p × ( q × r ) = ( p × q ) × r. 

For eg. , 3 × (15 × 1 ) = 45 and ( 3 × 15 ) × 1 = 45.

Subtraction: p – ( q– r ) ≠ ( p – q ) – r. 

For eg. ,3 – (14 – 1 ) = – 10 and ( 4 – 17 ) – 1 = – 14.

Division: p ÷ ( q ÷ r ) ≠ ( p ÷ q ) ÷ r. 

For eg. , 2 ÷( 3 ÷ 6 ) = 4 and ( 2 ÷ 3 ) ÷ 6 = 0.11.

As a result, we can conclude that the set of natural numbers is associative while adding and multiplying, but not when subtracting and dividing. As a result, N’s associative property is as follows: For all p, q, r ∈ N, p + (q + r) = (p + q) + r and p × (q × r) = (p × q) × r

Commutative Property

The commutative property of natural numbers asserts that even if the order of the numbers is changed, the total or product of two natural numbers remains the same. Check all four arithmetic operations, as well as for all p, q ∈ N.

Addition: p+ q = q+ p.

Multiplication: p × q = q × p

Subtraction: p – q ≠ q – p

Division: p ÷ q ≠ q ÷ p

Therefore,we can conclude that the set of natural numbers is commutative while adding and multiplying, but not when subtracting and dividing. As a result, N’s commutative property is as follows: For all p, q ∈ N, p + q = q + p and p × q = q × p

Distributive Property

Any equation with three numbers a, b, and c written in the form p (q + r) is resolved as p (q + r) = pq + pr or p (q – r) = pq – rp, indicating that the operand an is distributed among the other two operands, q, and r.

Natural number multiplication is always distributive over addition. p × (q + r) = pq + pr

Natural number multiplication is also distributive over subtraction. p × (q – r) = pq – pr

Example: 3 × (2 + 5) = 3 × 2 + 3 × 5

3 × (2 + 5) = 3 × 7 = 21

3 × 2 + 3 × 5 = 6 + 15 = 21

3 × (2 + 5) = 3 × 7 = 21

3 × 2 + 3 × 5 = 6 + 15 = 21

Conclusion :

We use numbers to represent time, date, year, and weather.In school and at work, we use numbers for counting money, measurements, phone numbers, phone passwords, locks, reading, page numbers, and TV channels. Engineers utilise numbers in their calculations when creating structures and roadways. Doctors use it to check blood counts and administer medications.

As a result, understanding the various sorts of numbers and their properties is critical.