Additive Inverse

Additive Inverse of a number is the number that when added to the original number, results in Zero. For example, Let’s have a number 5 then its additive inverse will be -5 as when 5 is added to -5 their sum is zero. In simple words, the two numbers nullify each other with respect to the addition operation. Thus, Additive Inverse is analogous to the phenomenon where a positive and a negative charge neutralize each other.

This article explores the concept of Additive Inverse including the method to find the additive inverse of a number, the additive inverse formula, and many more in detail.

What is Additive Inverse?

Additive Inverse is the opposite of a number which when added to the number yields the sum to be zero. It simply means to convert a positive number to a negative and a negative number to a positive because we know that the sum of a positive number with its negative counterpart is zero.

For Example, the sum of 2 and -2 is zero. This comes from the fundamental rule of counting if you have two chocolates and you give two chocolates to your friend then you have nothing left in your hand. Hence, on a similar line, the sum of the positive and negative versions of a number is zero.

Learn More, What are Numbers?

Additive Inverse Property

The property of additive inverse states that if the sum of any two numbers is zero then each number is said to be additive inverse of each other. Additive Inverse Property is like nullifying the effect of each other. Let’s say you have ten rupees and you gave it to your friend then you have nothing left as balance.

Another example would be let’s say you have some hot water and you add the same amount of cold water in it thus neutralizing the effect. Based on the same analogy, the additive inverse property works.

Mathematical expression for Additive Inverse Property is given as follows:

x + (-x) = x – x = 0

Where x is any real number.

Additive Inverse Formula

We know that the additive inverse of a number is the opposite in sign of any given number. Hence, the additive inverse of a positive number is the negative version of itself and the additive inverse of a negative number is the positive version of itself. This conversion of positive to negative and negative to positive is done by multiplying -1 by the number of which we need to find the additive inverse. Hence, the Additive Inverse Formula is given as

Additive Inverse of Real Numbers

Real Numbers are those numbers that can be represented on a real line. The additive inverse of a real number is the negative of the given real number. Real Number includes Natural Numbers, Whole Numbers, Integers, Fractions, Rational Numbers, and Irrational Numbers. Let’s see the additive inverse of each type of Real Number.

Additive Inverse of Natural Numbers

We know that Natural Numbers are counting numbers and start from 1. Hence, all the Natural Numbers are positive in nature. The additive inverse of the natural number is the negative version of the natural number. For Example, the additive inverse of 5 is -5.

Additive Inverse of Whole Numbers

All the natural numbers along with zero are called Whole Numbers. Hence, the additive inverse of positive whole numbers will be negative of the number. For Example, the additive inverse of 7 will be -7. The Exception to this lies in the case of zero because the additive inverse of Zero is Zero itself

Additive Inverse of Integers

Integers include all the natural numbers, zero and whole numbers i.e. it has three types of numbers positive, negative, and zero which is neither positive nor negative. The additive inverse of a positive integer is the negative version of itself, for instance, the additive inverse of 3 is -3. The additive inverse of Zero is Zero. The additive inverse of a negative integer is the positive version because when -1 is multiplied according to the formula the negative number will turn into a positive. For example, the additive inverse of -6 is -1⨯-6 = 6.

Additive Inverse of Fraction

We know that a fraction is always positive by definition. Hence, the additive inverse of a fraction will be the negative version of the given fraction i.e. if a/b is the given fraction then its additive inverse will be -a/b. For Example, the additive inverse of 2/3 will be -2/3.

Additive Inverse of a Rational Number

A Rational Number can be both positive and negative. Hence, to get the additive inverse of a Rational Number multiply by -1. Hence, we see that 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 a positive rational number p/q.

Additive Inverse of a Decimal

We know that a decimal consist of a whole part and a fractional part separated by a point which is called decimal. In the case of the additive inverse of a decimal the sign of the whole part only changes. For Example, the additive inverse of 2.35 is -2.35 and the additive inverse of -0.2 is 0.2.

Additive Inverse of Irrational Number

Irrational Numbers include mainly the non-terminating and non-repeating decimals and square root and cube roots of non-perfect squares and cubes. To find the additive the additive inverse of an irrational number simply multiply with -1.

For Example, the additive inverse of √2 is -√2.

The additive inverse of 2 – √3 is -1⨯(2 – √3) = -2 + √3.

Additive Inverse of Complex Number

A Complex number is represented in the form of Z = a ± ib where a is the real part, i is the iota and ib is the imaginary part. To find the additive inverse of a complex number we need to multiply the complex number by -1. In the additive inverse of the complex number, the symbol of both the real part and the imaginary part changes from positive to negative and vice versa. Let’s see some example

Example: Find the additive inverse of 2 + 3i

Solution:

The additive inverse of 2 + 3i is -1⨯(2 + 3i) = -2 -3i

Example: Find the additive inverse of -5 + 7i

Solution:

The additive inverse of -5 + 7i is -1⨯(-5 + 7i) = 5 – 7i

Learn more about Additive Identity and Inverse of Complex Number.

Additive Inverse and Multiplicative Inverse

The two properties additive inverse and multiplicative inverse are often confusing. As the name suggests Additive inverse is applicable in case of addition while Multiplicative Inverse is applicable for multiplication. The following table mentions the basic difference between them

Additive Inverse	Multiplicative Inverse
It is a number which when added to the original number the sum is zero	It is a number which when multiplied with the original number the product is 1
It is obtained by multiplying the number by -1	It is obtained by taking the reciprocal of the number
Example: n + (-n) = 0	Example: n ⨯ 1/n = 1

Additive Inverse of Algebraic Expression

Additive Inverse Property is not only limited to numbers it is applicable to algebraic expressions as well. To find the additive inverse of an algebraic expression we need to multiply the expression with -1. Multiplying each term of an algebraic expression with -1, the sign of the term changes from positive to negative and vice versa such that each term cancel out the other and the net sum of the algebraic expression will be zero. Let’s see some examples

Example 1: Find the additive inverse of 2x + 1

Answer:

The additive inverse of 2x + 1 is -1(2x + 1) =-2x -1

we can verify if it is true or not by taking the sum

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

Here the sum of (2x + 1) and (-2x – 1) is zero. Hence, (-2x – 1) is additive inverse of (2x + 1)

Example 2: What is the additive inverse of x³ – 3x² – 5

Answer:

The additive inverse of x³ – 3x² – 5 is -1(x³ – 3x² – 5) = -x³ + 3x² + 5

We can verify if it is true or not by taking the sum

(x³ – 3x² – 5) + (-x³ + 3x² + 5) = x³ – x³ – 3x² + 3x² – 5 + 5 = 0

Here the sum is zero. Hence, (-x³ + 3x² + 5) is additive inverse of (x³ – 3x² – 5)

Read more about Algebraic Expression.

Additive Inverse Example

Till now, we have learned to find the additive inverse of all kinds of numbers. Now under this heading, we will the additive inverse of some very commonly used numbers.

Additive Inverse of 0

We know that zero is neither positive nor negative. The additive inverse of 0 is 0. This is because when we multiply 0 with -1 then it will give zero as from the basic rule of multiplication we know that anything multiplied with zero gives zero. We can also verify that the additive inverse of 0 is 0 as 0 + 0 = 0 and by definition of additive inverse we know that sum of the number with its additive inverse yields zero.

Additive Inverse of 5

The additive inverse of 5 is -5 as -1 ⨯ 5 = -5. This can be verified by the following process:

5 + (-5) = 5 – 5 = 0

Hence, the sum of 5 and -5 is zero. Therefore, the additive inverse of 5 is -5.

Additive Inverse of 7

The additive inverse of 7 is -7 as -1 ⨯ 7 = -7. This can be verified by the following process:

7 + (-7) = 7 – 7 = 0.

Hence, the sum of 7 and -7 is zero. This confirms that the additive inverse of 7 is -7.

Read More,

• Real Numbers
• Properties of Real Numbers
• Multiplicative Identity and Multiplicative Inverse of the Complex Number

Practice Questions on Additive Inverse

Q1: Find the additive inverse of 1203.

Q2: Find the additive inverse of -125.

Q3: Find the additive inverse of 2023.

Q4: Find the sum of 234 and its additive inverse.

Q5: What is the successor of additive inverse of -916?

FAQs on Additive Inverse

1. Define Additive Inverse

Additive Inverse is the inverse of a number such that the sum of the inverse and the original number is zero.

2. What is the Meaning of Additive Inverse?

Additive Inverse means to find a number which when added to another number such that they nullify each other. This is only possible if the two numbers have the same magnitude but different signs.

3. What is the Additive Inverse of 2?

The additive inverse of 2 is -2.

4. Write the Additive Inverse of 1.

The additive inverse of 1 is -1.

5. Find the Additive Inverse of 256.

The additive inverse of 256 is -256.