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# Additive Identity|Definition & Meaning

## 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.

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 identity**Â can be representedÂ in allÂ **number systems**Â 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:

## Identity Element in Mathematics

Additive identity in mathematics is usually **zero**. It can be mathematically proven that any number when added to **zero**Â 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 the**Â 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 certain**Â 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 this**Â 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 **indicators**Â to categorize -2 as**Â 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 **entries**Â are**Â zeroes**. If you add such aÂ **matrix**Â to any matrix withÂ **non-zero entries**Â 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 identity**Â 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 **set**Â ofÂ **numbers**, like the **set**Â ofÂ **whole numbers**. When you add this **null**Â **set**Â to theÂ **set of**Â **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 **zeroes**Â as allÂ **entries**:

\[ O = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \]

The** null matrix**Â can be used as anÂ **additive identity**Â to show the relationÂ **O + N = N = N + O**. WhereÂ **O**Â is theÂ **null matrix**Â andÂ **N**Â is aÂ **matrix**Â 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 = N**Â andÂ **N + O = N**, where **O**Â is theÂ **null matrix**Â andÂ **N**Â 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 following**Â non-zero vector A**.

AÂ **null vector**Â is aÂ **vector**Â withÂ **zero magnitudes**Â and**Â no direction**. Consider the followingÂ **null vector**Â O.

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

## Use of Additive Identity for Number System Verification

UseÂ **additive identity**Â to verify whether the following numbers follow it or not:

1) 0 is theÂ **additive identity**Â 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 identity**Â of -15

Let’s solve -15 (which is**Â an integer**) forÂ **additive identity**, we apply the**Â 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 identity**Â of -4.5

We apply the**Â 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 definition**Â 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.*