## Definition

TheÂ additive identityÂ of a given number n is an element x, such that:Â  n + x = n = x + n, where the number n belongs to anÂ N set of numbers.Â N can be anyÂ number system. N can be a system ofÂ integers,Â rational numbers,Â real numbers,Â orÂ complex numbers.Â N can also be replaced withÂ matrices, vectors,Â orÂ sets.

Figure 1 – The notion of additive identity in mathematics.

An additive identity is a property of a number system such that it yields back the same number when added to any number belonging to that specific number system. This number, usually referred to as an entity, term, or identity element, is mostly 0. Â This concept can be extended into sets, matrices, and vectors.

## Representation of Additive Identity in Number Systems

As mentioned earlier, the concept of anÂ additive identityÂ can be representedÂ in allÂ number systemsÂ as its understanding is universal for all of them. TheÂ representation canÂ beÂ slightly different for each.

Following are some ways in which it is shown:

• Real numbers: 0 + a = a = a + 0, where a is any real number.
• Complex number: 0 + ai = ai = ai + 0, where ai is a complex number.
• Integers: 0 + n = n = n + 0,Â  where n can a positive or negative integer.Â

In other representation media:

Vectors:Â  0â†’ + Vâ†’ = Vâ†’ = Vâ†’ + 0â†’, where the â†’ sign represents vectors.

For natural numbers, being a subset of zero also means their additive identity is zero, as illustrated in the following figure:

Figure 2 – Additive identity of natural numbers is 0.

## Identity Element in Mathematics

Additive identity in mathematics is usually zero. It can be mathematically proven that any number when added to zeroÂ regardless of which direction you add it from, will always give you back the same number. ThisÂ property is similar to all number systems having a slightly different representation in each. It is always true for the following relation:

e + n = n = n + e

where n is a part of theÂ number system N.

TheÂ additive identity is a relatively easy concept to wrap one’s head around because of its simplicity. It is the simplest representation of the additive property.

It can be used as one of the basic criteria to judge whether a number belongs to a certainÂ number system. For example, you want to find out whether the number -2 belongs to a group ofÂ integers. You can have multiple methods to test this but the most straightforward method would be to simply check it through thisÂ property. You can perform the following test:

0 + (-2) = -2 = (-2) + 0

The answer would be the same no matter where you approach theÂ method from. This gives us one of, if not all, the indicatorsÂ to categorize -2 asÂ an integer.

As discussed, you can also extend this simple concept to other mathematical representations. For example, with matrices, you have a type of matrix called the null matrix. It is a matrix in which all theÂ entriesÂ areÂ zeroes. If you add such aÂ matrixÂ to any matrix withÂ non-zero entriesÂ of the same order as theÂ null matrix, the answer would be the matrix with the non-zero entries.

Another example of the usage ofÂ additive identityÂ is in the form of sets. For example, you can have a null set (a set with no elements). You can add the null set to anyÂ setÂ ofÂ numbers, like the setÂ ofÂ whole numbers. When you add this nullÂ setÂ to theÂ set ofÂ whole numbers, the resulting answer would be the set of whole numbers.

## Illustration of Identity Through Matrices and Sets

A null matrix is a matrix that has zeroesÂ as allÂ entries:

$O = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

The null matrixÂ can be used as anÂ additive identityÂ to show the relationÂ O + N = N = N + O. WhereÂ OÂ is theÂ null matrixÂ andÂ NÂ is aÂ matrixÂ of the same order withÂ non-zero entries.

Consider theÂ non-zero matrix N:

$N = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$

Then, the relations O + N = NÂ andÂ N + O = N, where OÂ is theÂ null matrixÂ andÂ NÂ is theÂ non-zero matrix.

$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} + \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$

$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$

Thus, if a number is added to an identity element, the result remains the same. Such a method can also be verified for vectors and sets.

When this null vector is added to a non-zero vector, it will reproduce theÂ non-zero vector. For the sake of an illustration, consider the followingÂ non-zero vector A.

Figure 3 – A vector v with a magnitude of three.

AÂ null vectorÂ is aÂ vectorÂ withÂ zero magnitudesÂ andÂ no direction. Consider the followingÂ null vectorÂ O.

Figure 4 – A null vector with zero magnitude and arbitrary direction as indicated by the lack of an arrowhead.

Now, if we add the vectors, they will give backÂ  VectorÂ A in theÂ resolution.Â  It can be seen in the figure.

Figure 5 – Addition of vector v with the null vector. the result is the vector v.

## Use of Additive Identity for Number System Verification

UseÂ additive identityÂ to verify whether the following numbers follow it or not:

1) 0 is theÂ additive identityÂ of -15

2) 0 is theÂ additive identityÂ ofÂ -4.5

3) 0 is theÂ additive identity of 3i

### Solution

1) 0 is theÂ additive identityÂ of -15

Let’s solve -15 (which isÂ an integer) forÂ additive identity, we apply theÂ additive identity test to it. By following the definition n + x = n = x + n, we can solve it as follows:

(-15) + 0 = -15

0 + (-15) = -15

So, after performing the test, it is proven.

2) 0 is theÂ additive identityÂ of -4.5

We apply theÂ additive identity test to it. By following the definition n + x = n = x + n, we can solve it as follows:

(-4.5) + 0 = -4.5

0 + (-4.5) = -4.5

So, after performing the test, it is proven.

3) 0 is the additive identity of the purely imaginary complex number 3i

We apply the additive identity test to it (0 + 0i). By following the definitionÂ n + x = n = x + n, we can solve it as follows:

3i + (0 + 0i) = 3i

(0 + 0i) + 3i = 3i

So, after performing the test, it is proven.

All images/mathematical drawings were created with GeoGebra.