Contents

# Additive Inverse|Definition & Meaning

## Definition

The **additive inverse** of a number can be defined as the numeral **value**, which will result in** a zero** value when added to the number. The **two numbers** are said to be additive inverses of each other if their **addition** results in a zero value.

Consider **figure 1** for the additive inverse. If a number **n** is added with its **additive inverse** that will be **-n**, the result will be equal to** zero**. Similarly, the additive inverse of **-n** will be **n**.

## Explanation of Additive Inverse

**An additive Inverse** is a value of any number that will give a **zero** result when added to any number.

If we **add** any two numbers and the result is zero, those two numbers are the additive inverses of each other. These two numbers can be in **fractions**, **decimals**, or** imaginary numbers** like iota.

**Additive** **Inverse** is the **opposite** value of a certain number or the **negation** of the number. This is because it has the opposite **sign** of the number whose additive inverse is required.

## Use of Additive Inverse

The **addition** of the additive inverse of a number is typically used in** equations** to **eliminate** the number from one side of the **algebraic** equation.

## Additive Inverse and Its Relation With Other Properties

The **additive inverse** is closely related to all the properties of real numbers when solving algebraic equations.

The **additive identity** is illustrated to understand the concept of the **additive inverse**. Both are closely related while dealing with the properties of real numbers.

In addition to this, **multiplicative identity** and **multiplicative inverse** are also demonstrated to compare the additive inverse with the multiplicative inverse.

### Additive Inverse and Additive Identity

**Additive Identity** and additive inverse are related to each other as adding a number and its additive inverse always results in the additive identity.

They have a close **relation** and additive identity should be discussed to **understand** the additive inverse of a number.

Consider **figure 2** for the additive identity. If a number** m** is added to the **additive identity**, then the result will be the same number **m**.

### Additive Inverse and Multiplicative Identity

Just like the **additive identity**, the multiplication of a number and its multiplicative inverse also results in the **multiplicative identity**.

Consider **figure 3** for the multiplicative identity. If a number** p** is multiplied by the **multiplicative identity**, the result will be the identity of the number that is **p**.

### Additive Inverse and Multiplicative Inverse

In addition to this, the **multiplicative inverse** also has a close** resemblance** to additive identity. The additive inverse can be understood easily compared to a number’s multiplicative inverse.

Consider **figure 4** in which the number **s** is multiplied by its **multiplicative inverse** to obtain the multiplicative identity that is **one**.

The multiplicative inverse of a number **s** is **1/s** and of **1/s** is **s**.

## Explanations on Additive Identity, Multiplicative Identity and Multiplicative Inverse

The **discussions** on additive identity, multiplicative identity, and multiplicative inverse are made to have a thorough understanding of the **additive inverse** of a number.

### Additive Identity

Additive Identity is a numerical value that when added to a number, the **same number** is obtained in the **result**.

As the name suggests, the additive identity gives back the** identity** of the number after the addition of the number with it.

The number that gives the same result when **added** to any given number is zero, so the **additive identity** is **zero**.

So, adding a **number** and its **additive inverse** will always result in the additive identity that is **zero**.

### Multiplicative Identity

Multiplicative Identity is defined as the numerical **value** that when **multiplied** by a number gives the **same** number in return. It lets the number keep its **identity** even after the process of multiplication.

The concept of multiplicative identity is the same as **additive identity** with just the difference in **processes** that are addition and multiplication.

The** multiplicative identity** is **one** as it is the only number that is the same number after multiplying the number with it.

Just as the **additive inverse**, the multiplication of a number with its multiplicative inverse gives the multiplicative identity.

### Multiplicative Inverse

The **multiplicative inverse** is defined as the numerical **value** that when multiplied by a given number gives the result as the multiplicative identity that is **one**.

**Two numbers** are considered multiplicative inverses of each other if their **multiplication** yields the result as **one**.

The **multiplicative inverse **of **zero** does not exist as the multiplication of any number with zero will always **result** in zero and never one.

**Additive inverse** and multiplicative inverses are **used** essentially when solving different algebraic **equations**.

## Examples

### Example 1

What are the **additive inverse** and the **multiplicative inverse** of **-1/3**? Check the result with the help of additive and multiplicative identities.

### Solution

The **additive inverse** of** -1/3** will be such a number that when added to it gives the result equal to** zero** which is the additive identity. So, the additive inverse of **-1/3** is **1/3**. The following **solution** checks the result for this answer.

**Additive Inverse of -1/3 = 1/3**

**Check = -1/3 + 1/3 = 0**

The **multiplicative inverse** of **-1/3** will be **-3** as the result of their **multiplication** will be equal to **1** which is the multiplicative identity. The **solution** for the multiplicative identity of **-1/3** is given as follows:

**Multiplicative Inverse of -1/3 = -3**

**Check = (-1/3) Ã— (-3) = 1/3 Ã— 3 = 1**

### Example 2

What are the **additive inverse** and the **multiplicative inverse** of **7**? Also, provide the check for the answers.

### Solution

The **additive inverse** of **7** is **-7** as it is the same number but has the **opposite** sign.

It will also give the resultant as **0** if it is **added** to the given number which is **7**. The answer can be checked by the following **solution**.

**Additive Inverse of 7 = -7**

**Check = 7 + ( -7 ) = 7 – 7 = 0**

For 7, the **multiplicative inverse** is **1/7** as the multiplication of the two numbers will give the result equal to **1**. It can be confirmed by the **solution** as follows:

**Multiplicative Inverse of 7 = 1/7**

**Check = 7 Ã— (1/7) = 1**

*All the geometrical figures are created using GeoGebra.*