Properties of numbers



Properties of numbers are

1. Closure property


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A set is said to be closed under an operation, if and only if the operations on two elements of the set produces another element which is also in the set. If it is not an element of the same set, then it is not closed.

2. Identity property


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A set of numbers has the identity property under an operation, if there is a particular element of the set which leaves every other element of the set unchanged under that operation.

3.Inverse property


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An inverse of an element is another element in the same set, when combines to the element under an operation, gives the identity element.

4.Commutative property


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Commutative property states that operation like addition and multiplication can be done in any order.

5.Associative property


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Distributive property is one operation distributed over the other operation. In particular we can tell that multiplication is distributive over addition.

6.Distributive property


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Distributive property is one operation distributed over the other operation. In particular we can tell that multiplication is distributive over addition.

Let us see the above properties-of-numbers in detail

Closure property:

A set is said to be closed under an operation, if and only if the operations on two elements of the set produces another element which is also in the set. If it is not an element of the same set, then it is not closed.

In other words, if a and b are two elements of set A and * is an operation then a*b must be an element in A

Examples:

a) Addition of two integers is closed.

Let us take 3 and -9 as the two integers.

Now addition of 3 and -9 is 3 + (-9)= -6 which is also an integer. So addition of two integers is closed.

b) Multiplication of two rational numbers is closed.

For example let the rational numbers be ¾ and ½

Multiplication of ¾ and ½ is 3/8 which is again a rational number.

c) Addition of two odd numbers is not an odd number. So it is not closed.

For example, let us take 3 and 5 as the odd numbers. Addition of these two numbers is 3+5=8, which is not an odd number. So it is not closed.

Identity property:

A set of numbers has the identity property under an operation, if there is a particular element of the set which leaves every other element of the set unchanged under that operation.

Usually ‘e’ stands for the identity element. Let us consider the element "b" is an element of set a,and * is the operation then,b*e=b is called as identity property.

Examples:

a) Set : Set of all real numbers

Operation : Multiplication.

Identity element : 1

If we multiply a real number by 1, then we will get the same. So 1 is the identity element under multiplication.

b) Set : Set of all whole numbers

Operation : Addition

Identity element : 0

3+0=3, 1111+0 = 1111. If we add any number with 0 we get the same number. So 0 is the identity element under addition.

c) Set : Set of whole numbers

Operation : Addition

Identity element : No identity element

As 0 is not an element of set of natural number, it is not closed under addition. But it is closed under multiplication.Because it has 1.

Apart from the above properties, let us see the remaining parts of them.

Inverse property:

An inverse of an element is another element in the same set, when combines to the element under an operation, gives the identity element.

A set has inverse property under an operation if every element of the set has the inverse element.

In other words, if a is an element of set A and * is the operation

Examples:

a) Set : Real numbers

Operation : Addition

For example if we take 3 and -3 so that if we add

3+(-3) =0 we get the identity element 0.

b) Set : Real numbers

Operation : Multiplication

For the element 7 under multiplication the inverse is its reciprocal 1/7

7 * 1/7 =1

Important point : Identity element always has its own inverse.

Commutative property:

Commutative property states that operation like addition and multiplication can be done in any order.

In other words, if a and b are elements of set A and * is the operation, then a*b = b*a is the commutative property.

Examples:

a) 3+4 = 4+3 =7

b) 5 x 8 = 8 x 5=40

Associative property:

Associative property deals with grouping of elements under any operation.

In other words, if a, b and c are elements of set A, and * is the operation, then

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

Examples:

a) 3 + (5+ 9) = (3 + 5) + 9

3 + 14 = 8 + 9

17 = 17

b) 2 x ( 3x4) = (2x3) x 4

2 x 12 = 6 x 4

24 = 24

Important point : Distributive and associative properties does not work for subtraction and division.

Distributive property:

Distributive property is one operation distributed over the other operation. In particular we can tell that multiplication is distributive over addition.

In other words, if a, b and c are elements of set A and + and x are two operations, then a x (b + c) = (a x b) + (a x c)

These are the properties of numbers which must be known by each and every student who is studying math.

properties of numbers properties of numbers


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