Mathematics is full of identities. When we talk about the number theory, an identity is defined as something which when applied to something gives back that same thing. There are two types of identities when we talk about operations. One is additive identity and another is multiplicative identity. Multiplicative identity is the one which when multiplied to any number gives the same number. For example in case of numbers multiplicative identity is 1. For example: $5 \times 1 = 1 \times 5 = 5$. Here, we are going to discuss about additive identity in detail.

Additive identity is one which gives us the same number as a result to which it is added to. Additive identity is very useful and commonly used in various applications like number addition, in matrices etc. It is because of additive identity that we are able to find additive inverses of anything. If for any expression or number or matrix etc, the additive identity exists, then there also exists an additive inverse of that expression, number, matrices etc respectively.

In addition operation there is only one additive identity that is zero. The only difference in various applications of this identity lies in the representation - like in matrices, we write it as zero matrix, having all elements equal to zero and in number addition, simply as zero.

According to the additive identity, if M is any expression or number or matrix etc., then,

M + 0 = M

Also, the additive identity is commutative as addition is commutative so we can rewrite it as:

M + 0 = 0 + M = M

We can also derive the additive inverse using this, The additive inverse of a number is such that when added to given number gives the additive identity.

That is, if a + b = 0, this implies that b is the additive inverse of ‘a’ and vice versa.

Zero is a number such that when we add anything to it the number remains unchanged. It is this property of zero that proves the existence of additive identity. Also, this is not the case same in multiplication as when we multiply anything with zero, it absorbs it and the result is zero which contradicts the definition of identity which implies itself is back.

It is to be noted we talked about whole numbers but natural numbers is the smallest set. But we cannot talk about natural numbers when we talk about additive identity. The reason behind this is that 0 does not belong to the set of natural numbers. So the smallest set that comes into picture when we talk about additive identity is whole numbers.

0 is the additive identity of whole numbers. When 0 is added to any number belonging to even complex number set, we get the same number back.

A + 0 = 0 + A = A

Where A is any whole number, integer, rational number, irrational number, real number or complex number.

Matrices are the arrangement of elements in row and columns. A matrix M with m rows and n columns is written as $M_{m\times n} = [m_{ij}]$ where $m_{ij}$ is the element of i$^{th}$ row and j$^{th}$ column. Now, we write a zero matrix of same order as increasing zeroes in rows and columns can be done endlessly as this would not affect our zero matrix at all.

Now zero matrix = $\begin{bmatrix}0&0&...&0\\0&0&...&0\\...&...&...&...\\0&0&...&0\end{bmatrix}$

There can be any number of rows and any number of columns depending upon the requirement.

So if we have any matrix [M] and a zero matrix of same order [O], then

[M] + [O] = [O] + [M] = [M]

Clearly, again by identity property, the given operation of addition is commutative. The only thing to be kept in mind is that the order of the zero matrix should be same as that of given matrix. Again, in case of multiplication this would not hold true as when we will multiply with zero we will get zero only. Also, matrix multiplication differs from general multiplication. The identity for multiplication in matrices is a square matrix of order as that of the given square matrix with all elements equal to zero except the diagonal elements which are equal to one.

Let us see some examples that make use of additive identity properties.
Example 1:

Find the additive inverse of the following:

a) -5
b) 8
c) 19
d) -56
Solution:

a) Let x be the additive inverse of -5.
Then x + (-5) = 0
$\Rightarrow$ x – 5 = 0
$\Rightarrow$ x = 5

b) Let y be the additive inverse of 8
Then y + 8 = 0
$\Rightarrow$ y = -8

c) Let a be the additive inverse of 19
Then a + 19 = 0
$\Rightarrow$ a = -19

d) Let m be the additive inverse of -56
Then m + (-56) = 0
$\Rightarrow$ m - 56 = 0
$\Rightarrow$ m = 56