Additive and Multiplicative Identity | Algebraic Identities with Examples

# Additive and Multiplicative Identity | Algebraic Identities with Examples

## Additive and Multiplicative Identity and Basic Properties of Identity

An additive identity is an element of a set that, when combined with any other element of the set, results in the original element. The additive identity is usually denoted by the symbol 0. For example, the additive identity of the set of integers is 0, because when combined with any other integer, it results in the original integer.

A multiplicative identity is an element of a set that, when multiplied by any other element of the set, results in the original element. The multiplicative identity is usually denoted by the symbol 1. For example, the multiplicative identity of the set of real numbers is 1, because when multiplied by any other real number, it results in the original real number.

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## The Basic Property of Identity

The basic property of identity states that an object is always the same object, regardless of where or when it is observed. This means that an object has a certain identity that is independent of the observer. For example, a particular chair in a room is always the same chair, no matter who is looking at it or when it is observed.

## The Identities of Closure Property

Closure is a term used in mathematics and computer science, particularly in the area of functional programming, to describe a property of certain functions. A function is said to have closure if it can be computed in a finite amount of time and its result depends only on the function’s arguments and not on the environment in which the function is evaluated. In other words, a function is closed if its value can be determined solely from the values of its arguments.

## The Identities of Associative Property

The identities of associative property are:

A(BC) = (AB)C
A(B+C) = AB+AC

## The Identities of Commutative Property

There are a few identities that are associated with the commutative property. These identities are:

The commutative property is associative. This means that the order of operations within a group of commutative operations does not matter.

For instance, 3 + 4 = 4 + 3 and (3 + 4) + 5 = 12.

The commutative property is distributive. This means that the commutative property applies to the addition and multiplication operations separately.

For instance, 3 + (4 + 5) = 12 and 3 × (4 + 5) = 21.

## The Identities of Distributive Property

The distributive property is a mathematical property that states that for every set A, the product of A and any one of its subsets is the same as the product of A and the whole set.

## The Identities of Additive Identity Property

The additive identity property states that the sum of any two numbers is equal to the other number. In mathematical terms, this is written as:

x + y = y + x

For example, if you add 2 and 3, you get 5. This is because 2 + 3 = 5.

## The Identities of Multiplicative Identity Property

The identities of multiplicative identity property are 1 and 1.0.

For any real numbers a, b, and c, a + b = b + a and a + c = c + a.

## Difference Between the Additive Identity and Multiplicative Identity

The additive identity is zero, while the multiplicative identity is one.

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