## What Is the Additive Inverse of a Number?

**Additive inverse of a number is a number that, when added to the original number, gives the sum of 0. In simple words, the sum of a number and its additive inverse is always 0.**

Mathematically, the additive inverse of a real number n is denoted as – n.

Similarly, the additive inverse of – n is n.

n + ( – n) = 0 for every real number n.

In simple words, the additive inverse of a number is the opposite or negative of that number. In general, we can study the concept of additive inverse for integers, rational numbers, real numbers, etc. Note that the number 0 is known as the additive identity.

**Additive inverse examples:**

- The additive inverse of 7 is – 7, since 7 + (– 7) = 0.
- The additive inverse of –10 is 10.
- The additive inverse of – 99 is 99, since (– 99) +99 = 0.
- The additive inverse of 0 is 0 itself, because 0 + 0 = 0.

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## Additive Inverse Definition

The additive inverse of a number is the number that, when added to the given number, results in the sum of 0.

The additive inverse of a number is also called the opposite or the negation (or negative) of that number.

## Additive Inverse Property

If the sum of two real numbers is zero, then each real number is said to be the additive inverse of the other.

In simple words, if a + b = 0, where a and b are real numbers, then we can say that

- a is the additive inverse of b, and
- b is the additive inverse of a.

## Additive Inverse on a Number Line

On a number line, we can easily find the additive inverse of a number. Note down the distance of the given number from the origin. Now, mark the point to the opposite direction of 0, which lies at the same distance.

In other words, 0 is the midpoint of the line segment joining the given number and its additive inverse on a number line.

## Additive Inverse Formula

Additive inverse of x = 0 – x

Additive inverse of x = ( – 1) $×$ *x*

## How to Find the Additive Inverse of a Number

We can find the additive inverse of a number using different methods.

**Formula 1: Subtraction method**

Let *x* be any real number. To find the additive inverse of *x*, simply subtract *x* from 0.

Additive inverse of x = 0 – x

Examples:

Additive inverse of 6 = 0 – 6 = – 6

Additive inverse of – 9 = 0 – ( – 9) = 9

**Formula 2: Multiplication method**

Let *x* be any real number. To find the additive inverse of *x*, simply multiply *x* with –1.

Additive inverse of x = (– 1) $×$ *x*

Examples:

Additive inverse of 1 = ( –1 ) $×$ 1 = – 1

Additive inverse of – 78 = (– 1) × (– 78) = 78

## Additive Inverse: Properties

- – ( – x) = x
- – (x + y) = – x – y
- –(x – y) = y – x
- – ($\frac{x}{y}$)= $\frac{x}{– y}$
- $– (\frac{– x}{y})= \frac{x}{y}$
- – (x + y)] = (– x) + (– y)
- – (x – y) = ( – x) + y
- (– x) $\times$(– y)= x y
- x $\times$ (– y) = (– x) y =– (x $\times$ y)

## Additive Inverse of Different Numbers

**Additive inverse of natural numbers and whole numbers**

Natural numbers are counting numbers, defined by the set {1, 2, 3, 4, 5, 6, … }.

Whole numbers are the set of natural numbers with 0 or {0, 1, 2, 3, 4, 5, …).

Natural numbers are positive. Whole numbers (except 0) are also positive. Thus, the additive inverse of any natural numbers or a whole number (except 0) is always negative.

**Examples: **

- Additive inverse of 0 is 0.
- Additive inverse of 1 is -1.
- Additive inverse of 5783 is -5783.
- Additive inverse of 49 is -49.

**Additive inverse of integers**

The set of integers consists of positive integers, negative integers, and 0.

Integers = {…,-3,-2,-1, 0, 1, 2, 3, …}

The additive inverse of negative integers is always positive.

The additive inverse of positive integers is always negative.

Additive inverse of 0 is 0.

**Examples: **

- The additive inverse of -10 is -(-10) = 10.
- The additive inverse of 42 is -(42) = -42.
- The additive inverse of -42 is 42.

**Additive Inverse of Fractions**

A fraction is a number represented as $\frac{x}{y}$, where x and y are whole numbers, and y is not equal to 0.

The additive inverse of a fraction $\frac{x}{y}$ is $\frac{-x}{y}$.

**Examples:**

- Additive inverse of $\frac{7}{8}$ is $\frac{-7}{8}$.
- Additive inverse of $\frac{6}{7}$ is $\frac{-6}{7}$.

**Additive Inverse of Rational Numbers**

Rational numbers are of the form $\frac{p}{q}$, where p and q are integers, and q is not equal to 0.

The additive inverse of a rational number $\frac{p}{q}$ is $\frac{-p}{q}$.

**Examples:**

- Additive inverse of $\frac{5}{9}$ is $\frac{-5}{9}$.
- Additive inverse of $\frac{-1}{7}$ is $\frac{1}{7}$.

**Additive inverse of Decimals**

For any decimal x, its additive inverse is -x.

For instance, the additive inverse of 0.25 is -0.25, as 0.25 + (- 0.25) = 0.

## Additive Inverse and Multiplicative Inverse

Additive Inverse | Multiplicative Inverse |
---|---|

Additive inverse is a number that, when added to the original number, the sum is 0. | Multiplicative inverse is a number that, when multiplied with the original number, the product is 1. |

The additive inverse of a number is the negative or opposite of that number. | The multiplicative inverse of a number is the reciprocal of that number. |

The sum of a number and its additive inverse is always 0. | The product of a number and its multiplicative inverse is always 1. |

Additive inverse of 0 is 0. | 0 does not have a multiplicative inverse. |

For any real number n, its additive inverse is -n. | For any non-zero real number n, its additive multiplicative inverse is $\frac{1}{n}$. |

## Facts about Additive Inverse

- The additive inverse of the additive inverse is the original number.
- The additive inverse of 0 is 0 itself.

## Conclusion

In this article, we learned about the concept of additive inverses and how they play a crucial role in mathematics. Understanding the additive inverse allows us to work with negative numbers more effectively and solve a variety of mathematical problems. Let’s solidify our understanding by practicing a few examples and attempting MCQs for better comprehension.

## Solved Examples on Additive Inverse

**1. What is the additive inverse of 12345?**

**Solution: **

Given number = 12345

Additive inverse of 12345 = 0 – 12345 = -12345

**2. What is the additive inverse of **$\frac{-10}{13}$**?**

**Solution: **

The additive inverse of $\frac{x}{y}$ is $\frac{-x}{y}$.

Additive inverse of $\frac{-10}{13} = \;-\; (\frac{-10}{13}) = \frac{10}{13}$

Thus, the additive inverse of $\frac{-10}{13}$ is $\frac{10}{13}$**.**

**3. The additive inverse of a number is ****-6****. Find the number.**

**Solution:**

Let x be the number.

Additive inverse of x = -x

It is given that the additive inverse is **-6****.**

Thus, – x = -6

x = 6

The required number is 6.

**4. Is ****3x + 5**** additive inverse of ****– 3x – 5****?**

**Solution:**

The sum of a number and its additive inverse is 0.

Let’s find the sum of **3x + 5**** and ****– 3x – 5****.**

(3x + 5) + (- 3x – 5)= 3x + 5 – 3x – 5 = 0

So, 3x + 5 is the additive inverse of -3x – 5.

## Practice Problems on Additive Inverse

## Additive Inverse: Definition, Formula, Properties, Facts, Examples

### What is the additive inverse of the additive inverse of 3?

Additive inverse of $3 = \;-\;3$

Additive inverse of $\;-\;3 = 3$

Thus, the additive inverse of the additive inverse of 3 is 3.

### What is the additive inverse of 0?

Additive inverse of 0 is 0.

### To find the additive inverse of a non-zero real number,

All options are different ways to find the additive inverse of a number.

### The sum of a number and its additive inverse is _______.

The sum of a number and its additive inverse is 0.

### What is the multiplicative inverse of a number whose additive inverse -0.6?

Additive inverse of a number $= \;-\;0.6$

Thus, the number $= 0.6$

Multiplicative inverse of $0.6 = \frac{1}{0.6} = \frac{10}{6} = \frac{5}{3}$

## Frequently Asked Questions on Additive Identity

**What is the additive inverse of 0?**

The additive inverse of 0 is 0 itself.

**Is the additive inverse all whole numbers negative?**

Except for 0, the additive inverse of all other whole numbers (natural numbers) is always negative, since all the natural numbers are positive.

**Is additive inverse the same as additive identity?**

The sum of a number and its additive identity is the number itself, whereas the sum of a number and its additive inverse is 0.