## Identity Properties / Identity Numbers

*Identity number* for a *mathematical operation* is such a number that will have no impact on the result, *identity numbers* are also known as *identity properties*. In this article we will explore what identity numbers are in context of addition and multiplication.

### Additive Identity

0 (zero) is the identity number for *addition*, adding 0 to any number will have no effect on the result. You can see in following examples that adding 0 to any other number has no effect at all.x+0=x1000+0=1000

### Multiplicative Identity

For *multiplication*, the *identity number* is 1 (one), A *number* or *expression* will retain it’s identify if we **multiply* it with 1, following examples show this behavior.x×1=x1000×1=1000

## Associative, Commutative and Distributive Properties of Addition and Multiplication

In this section we will explore the laws for *associative*, *commutative* and *distributive* properties applied to addition and multiplication operations in math.

### Associativity: Associative Property / Law

In mathematics, the associative property[1] is a property of some binary operations. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs Wikipedia. The word *associative* comes from the word *associate* or *associate*. *Associative law* states that the answer of an *algebraic expression* will remain same no matter what the *order* of it’s elements is. The *Associative law* or *Associative property* is applicable to both *addition* and *multiplication* expressions, let’s make it more clear with some practical examples.

#### Associative Property / Law of Addition

Following equations explain the associative property of addition operation.(x+y)+z=x+(y+z)(3+2)+5=3+(2+5)

#### Associative Property / Law of Multiplication

Following equations explain the associative property of multiplication operation.(x∗y)∗z=x∗(y∗z)(3∗2)∗5=3∗(2∗5)

### Commutativity: Commutative Property

According to *Commutative law* the answer of *multiplication operation* in an *algebraic expression* will remain same, even if we change the order of the members of this expression Wikipedia. The word *Commutative* is derived from *commute* and in mathematics it states that re-arranging the elements of an *algebraic* expression will not affect the resultant value. *Commutative law* is also applicable to both *addition* and *multiplication* expressions, let’s make it more clear with some practical examples.

#### Commutativity Property of Addition / Commutative Law of Addition

Following equations explain the commutative property of addition operation.x+y=y+x3+2=2+3

#### Commutativity Property of Multiplication / Commutative Law of Multiplication

Following equations explain the commutative property of multiplication operation.x∗y=y∗x3∗2=2∗3

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