## 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.

\[ x + 0 = x \]

\[ 1000 + 0 = 1000 \]

### 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.

\[ x \times 1 = x \]

\[ 1000 \times 1 = 1000 \]

## 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.

\[ x + y = y + x \]

\[ 3 + 2 = 2 + 3 \]

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

Following equations explain the commutative property of multiplication operation.

\[ x * y = y * x \]

\[ 3 * 2 = 2 * 3 \]

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