Additive Inverse

When a number is added to its additive inverse, the sum is zero. We call this property additive inverse. It is often represented by n, and its additive inverse is -n. 
For any real number n, n + (-n) = 0, where 0 is the additive identity.
Let’s look at an example: The additive inverse of –80 is 80, since (–80) + 80 = 0
 

There are two types of inverse mathematical operations: additive inverse and multiplicative inverse. We will now discuss the key differences between them:

The additive inverse of any number is the opposite of the number itself. The additive inverse of a positive number is its negative.
We can convert a positive number into a negative number and vice versa by multiplying it by -1.
The formula for additive inverse is given as:

• Additive Inverse of N = (-1) × N = -N

• Additive Inverse of -N = (-1) ×  (-N) = N

Additive Inverse Property

We know that the additive inverse property states that when two numbers are additive inverses of each other if their sum equals zero.

It can be mathematically represented as:
x + (₋x) = x ₋ x = 0.
Where x is a real number.

Additive Inverse of Real Numbers

The numbers that can be represented on a number line are real numbers. The negative version of a given real number is its additive inverse. Real numbers are an umbrella term for all whole numbers, natural numbers, fractions, integers, rational numbers, and irrational numbers. 
Now, we will discuss the additive inverse of each type of real number:

Additive Inverse of Algebraic Expressions

To determine the additive inverse of an algebraic equation, we need to multiply it by – 1.
When – 1 is multiplied by each term in an expression, it results in the opposite of each term. As a result, each term cancels out, giving 0.
For example:  
The additive inverse of 3y +2 = –3y – 2
3y +2 + (–3y –2) = 0.

Additive Inverse of a Rational Number

To obtain the additive inverse, multiply the given number by -1. Therefore, the additive inverse of a positive rational number p/q is –p/q, and the additive inverse of a negative rational number – p/q is p/q.

Additive Inverse of a Decimal

Similarly, the additive inverse of a decimal number is the opposite version of the given decimal number. The additive inverse of a decimal number changes the sign of the entire number. For example, the additive inverse of 3.02 is –3.02.

Additive Inverse of Irrational Number

The square and cube roots of non-perfect squares and cubes, as well as non-terminating decimals, are classified as irrational numbers. We find the additive inverse of an irrational number by multiplying it by –1. For example, –√2 is the additive inverse of √2, as it is an irrational number and (√2) + (–√2) = 0.

Additive Inverse of a Complex Number

Let’s consider z = a ± ib to be a complex number, where:
z = a + ib
z = a - ib.
Here, a is the real part,
i is the iota, and 
ib represents the imaginary part.
We can find the additive inverse of a complex number by multiplying it by -1.
 

The additive inverse plays an important role in mathematics and beyond. We apply additive inverse in several real-life situations. Let’s look at a few:

• We use additive inverse to understand the temperature changes. For example, if the temperature is + 8o C and there is a decrease in temperature by 8oC, the temperature turns 0oC.

• Additive inverse helps in understanding bank transactions better. For example, if you deposit an amount of $1000 in your account, and you withdraw $1000, the balance becomes $0.

• It can be used in physics to mathematically understand that equal and opposite reactions cancel out.

• Businesses utilize the additive inverse to track and maintain a balance between expenses and incomes.

• Players can calculate their gains and losses in gaming. For example, if a player gains 70 points and then loses 70 points, their final score would be 0.