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Properties of Numbers

The numbers are used for counting and measuring. A number can be defined as an arithmetic quantity that is used for denoting a particular amount of something. They act as the foundation stone for any measurement or calculation. It is anything which is the number of some class. There is different types of numbers. All types of numbers satisfy certain properties. In number system, the numbers contains various properties like associative property, additive Inverse, and closure property.

Associative, commutative and distributive are the three basic properties of numbers.

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What are the Properties of Numbers?

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Below are the properties of numbers -

  • Identity Property

Any number add to the zero the answer is the number itself.
Symbolically, if 'a' is any number a + 0 = a, 0 is called additive identity.

Example: 9 + 0 = 9

(-5) + 0 = -5

Any number multiplied by 1 the answer is that number itself.
Symbolically, if a is any number a $\times$ 1 = a, 1 is called multiplicative identity.

Example: 9 $\times$ 1 = 9
(-5) $\times$ 1 = (-5)

In both cases, 1 is the multiplicative identity.

  • Inverse Property

If the addition of any two numbers is its identity element, then one number is the additive inverse of other. Symbolically, if a is the number then (-a) is its additive inverse of a, i.e.
a + (-a) = 0

Example : 8 + (-8) = 0

Here -8 is the additive inverse of 8.

If the multiplication of any two numbers is its identity element then one number is the multiplicative inverse of other. Symbolically if a is the number then is its multiplicative inverse $a \times$ $\frac{1}{a}$ = 1

Example: 8 $\times$ $\frac{1}{8}$ = 1

Here, $\frac{1}{8}$ is the multiplicative inverse of the 8.


  • Associative Property

Modify the way three or more numbers are grouped when adding or multiplying does not change the sum or product.

ie (a + b) + c = a + (b + c) is associative property of addition

(a • b) • c = a • (b • c)
is associative property of multiplication

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

Example 2: $(4 \times 5) \times  6 = 4 \times (5 \times 6)$
$20  \times 6$ = $4 \times 30$
120 = 120

  • Commutative Property

The commutative property only change the order of numbers, it doesn’t change the result.
This means the numbers can be swapped.

ie a + b = b + a for addition

a x b = b x a for multiplication

Example 1: 3 + 2 = 2 + 3

5 = 5

Example 2: 3 $\times$ 2 = 2 $\times$ 3

6 = 6

  • Distributive Property

Distributive property allows to remove the parenthesis in an expression. Multiply the value outside the brackets with each of the terms inside the brackets.
ie a x ( b + c ) = a x b + a x c

Example: [2 • ( 3 + 6 )] = [(2 • 3)+ (2 • 6)]
2 • 9 = 6 + 12
18 = 18

  • Closure Property

The sum or product of two numbers belongs to the same set as the addends or factors.
a, b, and c are all members of the same set of numbers.

then a + b = c,

a • b = c

Example: (2 + 5) = 7 (all whole numbers)

4 • (-5) = (-20) (all integers)

Reflexive Property

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Reflexive property simply states that any number is equal to itself. Reflexive property, for all real numbers x, x = x. A number equals itself. Two numbers are only equal to each other if and only if both the numbers are same.
Example: 4 = 4 or 4 = 4

Symmetric Property

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The symmetric property states that if one number is equal to a second number, then the second number is equal to the first number. Symmetric Property, for all real numbers x and y, if x = y then y = x. If the numbers on opposite sides of the equals sign are the same, they are symmetric.
Example: If 2 = 2 then 2 = 2
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