Additive Inverse

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A number that is added to another number to make it zero is known as the additive inverse. For example, the result of adding -3 to 3 is zero. Therefore, -3 is the additive inverse of 3. We come across such situations in our daily lives where we nullify the value of a quantity by taking its additive inverse. Let us learn the additive inverse property of real and complex numbers in this article.

What is the Additive Inverse?

An additive inverse of a number is just like its opposite number. When a number is added to its additive inverse, a number becomes zero. Getting an additive inverse is not difficult. You need to change a positive number into a negative one and vice versa to get the respective additive inverse.

For example, 7 + (-7) = 0.

Therefore, -7 is said to be the additive inverse of 7 and 7 is the additive inverse of -7.

Additive Inverse Property

Two real numbers are said to be the additive inverse of each other when their sum equals zero. Therefore, we can write down R + (−R) = 0, where R denotes any real number. Here, R and -R are additive inverses of each other. This can be illustrated by understanding the given example:

2/7 + (−2/7) = 0.

Thus, 2/7 is the additive inverse of -2/7 and vice versa. Above-discussed example is an illustration of the additive inverse of a fraction.

Imagine you have a bucket of water at room temperature. Adding a litre of hot water raises the temperature, but when you add a litre of cold water, the overall temperature adjusts back, balancing out to room temperature. This concept is similar to the concept of the additive inverse property in mathematics, where two numbers with opposite signs cancel each other out to result in zero.

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Additive Inverse of Real Number

An additive inverse can be a whole number, a fraction, a decimal, a natural number, an integer or any of the real numbers. The additive inverse of real numbers is simply the negative of the given number.

Example	Given Number	Additive Inverse
Natural Numbers	22	-22
Whole Numbers	16	-16
Integers	-120	120
Fractions	-4/9	4/9
Decimals	2.8	-2.8

Additive Inverse of Complex Numbers

The existence of an Additive inverse is one of the properties of complex numbers. The Algebraic Properties of complex numbers define that for any Z € C there is a unique number existing, say -Z € C, such that Z +(-Z) = 0. For two real numbers x and y such that Z = (x, y), the following holds true: -z = (-x, -y).

So, for a complex number Z = x + iy. Its additive inverse is -Z = -x – iy.

For example, an additive inverse of – i – 1 equals – (- i – 1) which equals i + 1.

The Additive Inverse of a Fraction

The additive inverse of a fraction a/b is – a/b, and vice versa. This is because a/b + ( -a/b ) = 0.

For a positive fraction, the additive inverse is the same fraction with a negative sign. For example, the additive inverse of 7/9 is -7/9.
For a negative fraction, the additive inverse is the same fraction without the negative sign. For example, the additive inverse of -7/8 is 7/8.

This property ensures that the sum of a fraction and its additive inverse is always zero, maintaining balance in calculations.

Additive Inverse Formula

The general formula for the additive inverse of a number can be written in the form of the number itself. Any number when added to its negative will nullify each other and give the overall sum as zero. To find the additive inverse of the given number we need to find its negative. It means that for the given number N, we need to find -1 × (N).

Therefore,

Additive Inverse of N = -1 × (N)

Additive Inverse and Multiple Inverse

There are two properties of numbers which are additive inverse and multiplicative inverse properties that are related to addition and multiplication operation respectively. For a number x, – x is the additive inverse while 1/x is the multiplicative inverse. Below given table differentiates between additive inverse and multiplicative inverse:

Additive Inverse	Multiplicative Inverse
To find the additive inverse of the number we need to change the sign of the number.	To find the multiplicative inverse of a given number, we take the reciprocal of the number.
The Additive Inverse is added to the original number to get 0.	The Multiplicative Inverse is multiplied by the original number to get 1.
For any given number, n, the equation for additive inverse is: a + (-a) = 0	For any given number, n, the equation for multiplicative inverse is: a × 1/a = 1

Additive Inverse in Algebraic Expression

The concept of additive inverse also applies to algebraic expressions. The additive inverse of an algebraic expression is the expression that results in zero when added to the original. This is achieved by changing the sign of every term in the expression. The additive inverse of an expression is found by multiplying the entire expression by −1. Mathematically, it is expressed as:

Additive Inverse of (expression) = − (expression)

The additive inverse of x² + 4x – 7 is (-x²– 4x + 7)

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Additive Inverse Calculator

The Additive Inverse calculator is a tool that runs online to show the additive inverse of a number. An Additive Inverse Calculator is an online tool that shows you the additive inverse of some number. Just put any number not more than six digits long and within seconds you will have its additive inverse at your fingertips.

How to Use an Additive Inverse Calculator?

To use the additive inverse calculator, follow these few easy steps:

Step 1: In the input box, provide the number whose additive inverse you want.

Step 2: Select “Calculate” to calculate the number’s additive inverse.

Step 3: To enter a new number, click on “Reset” to clear the field.