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 \((–80) + 80 = 0\)

The additive inverse of a number can be found in two different ways.

Method 1: Subtraction method.

Let 'x' be any real number. To find its additive inverse, we have to subtract x from 0.

Therefore, the additive inverse of x is, 

\(\text{Additive inverse = 0 - x} \\[1em] \text{Additive inverse = - x}\)

Example, 

\(\text{The additive inverse of 9 = 0 - 9 = -9} \\[1em] \text{The additive inverse of 789 = 0 - 789 = -789}\)

Method 2: Multiplication method.

Let us take x to be any real number. To find its additive inverse, we have to multiply x by (-1).

Therefore, the additive inverse of x is, 

\(\text{Additive inverse} = x \times (-1) \\[1em] \text{Additive inverse} = -x\)

Example, 

\(\text{The additive inverse of 57} = (-1)\times57 = -57 \\[1em] \text{The additive inverse of 99} = (-1)\times99 = -99\)

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:
 

• \(\text{Additive Inverse of N = (–1) × N = – N}\)
   
• \(\text {Additive Inverse of – N = (– 1) × (– N) = N}\)

The additive inverse of a number is the value that, when added to the original number, gives zero. For example, the additive inverse of 5 is -5, because \(5 + (-5) = 0.\)

The additive inverse property definition states that if the sum of two numbers is zero, then each number is the additive inverse of the other. In symbolic form, for any real number x, \(x+(-x)=x-x=0.\)

Real numbers are the numbers that can be placed on a number line. The exact number with the opposite sign of a real number is known as its additive inverse. The set of real numbers includes natural numbers, whole numbers, integers, fractions, rational numbers, and irrational numbers. All of these have additive inverses, which are obtained by changing their signs.

Additive inverse of a fraction

A fraction is taken as positive by definition; therefore, the additive inverse is just the same fraction with a negative sign. If the fraction is a/b, then the additive inverse of the fraction is (-a/b).

For example, the additive inverse of \(\frac{2}{3}\) is \(-\frac{2}{3}. \)

We can verify this by adding the fraction and its additive inverse. 

\(\frac{2}{3} + (​​-\frac{2}{3} ) = \frac{2}{3}-\frac{2}{3} =0\)
 

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, \(-\sqrt2\) is the additive inverse of \(\sqrt2.\)

As it is an irrational number and \((\sqrt2) + (–\sqrt2) = 0.\)

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.

We can also apply the additive inverse property to algebraic expressions. To find the additive inverse of an algebraic expression, we have to multiply each term by -1. This, in turn, will change the signs of each term. The positives will become negatives, and the negatives will become positives.

Therefore, when we add the expression and its additive inverse, the result will become zero. 

For example, let us find the additive inverse of \(2x + 1.\)

Multiplying the algebraic expression by -1, we get, 

\((2x+1)(-1) = -2x-1\)

Let us check whether the additive inverse is correct by adding them.

\(2x+1+(-2x-1) = 2x+1-2x-1 = 0\)

A complex number is written as \(z=a \pm ib,\) where a stands for the real part and i represents the imaginary unit, and \(ib\) denotes the imaginary part. The additive inverse of a complex number is obtained by multiplying it by -1, which reverses the signs of both the real and imaginary parts. 

Additive inverse examples

The additive inverse of \(2+3i\) can be calculated by multiplying it by -1.

\((-1)(2+3i) = -2-3i\)

We can further verify this by adding them:

\((2+3i)+ (-2-3i) = 2+3i-2-3i = 0\)

Let us also find the additive inverse of \(3x-6.\)

Subtract the given expression with 0 to get the additive inverse:

\(\text{Additive inverse} = 0 - (3x-6)\\[1em] \text{Additive inverse} = -3x-6\)

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.
   
• Explain easy: Teachers can start teaching about additive inverses in simple, everyday language rather than giving a complex mathematical explanation. Tell them that an additive inverse is the number that brings us back to zero.
   
• Use number lines: Students can use a number line to visualize the concept of additive inverse. Try to find a number and its opposite that meet at zero. The movement helps students grasp the idea immediately.
   
• Use real-life examples: Parents can use simple real-life concepts to explain the additive inverse to their children. For example, tell them that +6 °C and -6 °C make the temperature change to zero.
   
• Teach them "What is an additive inverse?": Teachers should teach the key rule very early. Tell them in the first class that the additive inverse of any number is its opposite.

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.