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) = 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, 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

The additive inverse of a decimal is its opposite. 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 (The additive inverse is –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.

To make the learning of additive inverse easier, here are some useful tips and tricks that students can follow while practicing. 

• Think in terms of getting back to zero : Just by thinking, what should you add to this number to make the sum zero, you can easily identify an additive inverse. 
   
• Just flip the sign: The fastest trick is changing the sign of the number or each term in an expression. But simply, for a positive number, make it negative. For a negative number, make it positive and for an expression, change every sign inside it. 
   
• Zero is its own inverse: A common trick is regarding the zero. Zero is the only number that equals its own additive inverse. While all other numbers have distinct opposites, zero is neutral. 
   
• Use the bracket rule for algebraic expressions: When working with algebra, always apply the negative sign outside the bracket and distribute it to every term inside. This will ensure that you correctly find the additive inverse of complex expressions without missing any sign changes. 
   
• Connect with real life examples:  Understanding through examples will help the concept stick. For instance, a debt of - ₹200 is canceled by a credit of + ₹200.

We apply the 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 \(+ 8^{\circ}\mathrm{C} \) and there is a decrease in temperature by \(8^{\circ}\mathrm{C}\), the temperature turns \(0^{\circ}\mathrm{C} \).

• 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.