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# Additive Identity|Definition & Meaning

## Definition

The additive identity of a given **number n** is an** element x**, such that: n + x = n = x + n, where the **number n** belongs to an **N** set of numbers. N can be any **number system**. N can be a system of **integers**, **rational numbers**, **real numbers,** or **complex numbers. **N can also be replaced with matrices, vectors, or 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**. This concept can be extended into **sets, matrices,** and **vectors****.**

## Representation of Additive Identity in Number Systems

As mentioned earlier, the concept of an **additive identity** can be represented in all **number systems** as its understanding is universal for all of them. The **representation **can be 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, where n can a positive or negative integer.

**In other representation media:**

Vectors: 0→ + V→ = V→ = V→ + 0→, where the → 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** regardless of which direction you add it from, will always give you back the same number. This **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** number system N**.

The **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** number system**. For example, you want to find out whether the number -2 belongs to a group of **integers**. You can have multiple methods to test this but the most straightforward method would be to simply check it through this** property**. You can perform the following test:

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

The answer would be the same no matter where you approach the **method** from. This gives us one of, if not all, the **indicators** to categorize -2 as** 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 the **entries** are** zeroes**. If you add such a **matrix** to any matrix with **non-zero entries** of the same order as the **null matrix**, the answer would be the **matrix** with the non-zero entries.

Another example of the usage of **additive identity** 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 any **set** of **numbers**, like the **set** of **whole numbers**. When you add this **null** **set** to the **set of** **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** as all **entries**:

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

The** null matrix** can be used as an **additive identity** to show the relation **O + N = N = N + O**. Where **O** is the **null matrix** and **N** is a **matrix** of the same order with **non-zero entries**.

Consider the **non-zero matrix N:**

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

Then, the relations** O + N = N** and **N + O = N**, where **O** is the **null matrix** and **N** is the **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 the **non-zero vector**. For the sake of an illustration, consider the following** non-zero vector A**.

A **null vector** is a **vector** with **zero magnitudes** and** no direction**. Consider the following **null vector** O.

Now, if we add the **vectors**, they will give back **Vector A** in the **resolution**. It can be seen in the figure.

## Use of Additive Identity for Number System Verification

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

1) 0 is the **additive identity** of -15

2) 0 is the **additive identity **of -4.5

3) 0 is the **additive identity** of 3i

### Solution

1) 0 is the **additive identity** of -15

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

We apply the** 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** 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.*