Additive Inverse and Multiplicative Inverse (Definition, Properties & Examples)
What is an Additive inverse?
An additive inverse of a number is defined as the value, which on adding with the original number results in zero value. It is the value we add to a number to yield zero.
Suppose, a is the original number, then its additive inverse will be minus of a i.e.,-a, such that;
a+(-a) = a – a = 0
Example:
- Additive inverse of 10 is -10, as 10 + (-10) = 0
- Additive inverse of -9 is 9, as (-9) + 9 = 0
Additive inverse is also called the opposite of the number, negation of number or changed sign of original number.
Fact: Additive inverse of zero is zero only.
How to Find the Additive inverse?
The additive inverse of any given number can be found by changing the sign of it.
The additive inverse of a positive number will be a negative, whereas the additive inverse of a negative number will be positive.
However, there will be no change in the numerical value except the sign.
For example, the additive inverse of 8 is -8, whereas the additive inverse of -6 is 6.
The addition of a number and its additive inverse is equal to the additive identity.
Additive inverse simply means changing the sign of the number and adding it to the original number to get an answer equal to 0.
The properties of additive inverse are given below, based on negation of the original number. For example, x is the original number, then its additive inverse is -x. So, here we will see the properties of -x.
- −(−x) = x
- (-x)2 = x2
- −(x + y) = (−x) + (−y)
- −(x – y) = y − x
- x − (−y) = x + y
- (−x) × y = x × (−y) = −(x × y)
- (−x) × (−y) = x × y
Additive Inverse of Rational Numbers
Suppose a/b is a rational number such that the additive inverse of a/b is -a/b and vice versa.
| Fraction | Additive Inverse | Result |
| 1/2 | -1/2 | (½) + (-½) = 0 |
| 1/4 | -1/4 | (¼) + (-¼) = 0 |
| 3/4 | -3/4 | (¾) + (-¾) = 0 |
| 2/5 | -2/5 | ⅖ + (-⅖) = 0 |
| 10/3 | -10/3 | 10/3 + (-10/3) = 0 |
What is an Multiplicative inverse?
The multiplicative inverse of a number, say, N, is represented by 1/N or N-1. It is also called reciprocal, derived from the Latin word ‘reciprocus‘. The meaning of inverse is something which is opposite.
The reciprocal of a number obtained is such that when it is multiplied by the original number, the value equals identity 1. In other words, it is a method of dividing a number by its own to generate identity 1, such as N/N = 1.
Fact:
When a number is multiplied by its own multiplicative inverse, the resultant value is equal to 1.
Consider the examples;
The multiplicative inverse of 3 is 1/3, of -1/3 is -3, of 8 is 1/8 and 4/7 is -7/4.
But the multiplicative inverse of 0 is infinite because 1/0 = infinity. So, there is no reciprocal for a number ‘0’.
Whereas the multiplication inverse of 1 is 1 only.
The multiplicative inverse of a number for any n is simply 1/n. It is denoted as:
1 / x or x-1 (Inverse of x)
It is also called as the reciprocal of a number and 1 is called the multiplicative identity.
Multiplicative Inverse Property
The product of a number and its multiplicative inverse is 1.
x. x-1 = 1
For example, consider the number 13.
The multiplicative inverse of 13 is 1/13.
According to the property,
13. (1/13) = 1
Multiplication Inverse of Fraction
If p/q is a fraction, then the multiplicative inverse of p/q should be such that, when it is multiplied to the fraction, then the result should be 1. Hence, q/p is the multiplicative inverse of fraction p/q.
p/q x q/p = 1
For example: 2/7 x 7/2 = 1
Examples:
- Mul. Inverse of 2/7 is 7/2: 2/7 x 7/2 = 1
- Mul. Inverse of ½ is 2: ½ x 2 = 1
- Mul. Inverse of ¾ is 4/3: ¾ x 4/3 = 1
- Mul. Inverse of 2/9 is 9/2: 2/9 x 9/2 = 1
Multiplication Inverse of Unit Fraction
1/2, 1/3, 1/4, 1/5, etc., are considered unit fractions because they all have numerators as 1. Hence, the multiplicative inverse of these unit fractions will be the values present in the denominator.
- 1/2 x 2 = 1
- 1/3 x 3 = 1
- 1/4 x 4 = 1
- 1/5 x 5 = 1
Multiplicative inverse of Mixed Fraction
To find the multiplicative inverse of a mixed fraction, firstly convert it into a proper fraction. Let us see some examples.
- 21/2 = 5/2: ⅖
- 32/3 = 11/3: 3/11
Difference Between Additive Inverse and Multiplicative Inverse
Additive inverse and multiplicative inverse, both have different properties. See the below table to know the differences.
| Additive Inverse | Multiplicative Inverse | |||||||||
| It is added to the original number to get 0 | It is multiplied to the original number to get 1 | |||||||||
| Results in 0 | Results in 1 | |||||||||
| Sign of the original number is changed and added | Reciprocal of the original number is multiplied | |||||||||
| Example: 66 + (-66) = 0 | Example: 66 × (1/66) = 1 |