# What is a Multiplicative Inverse?

Omkar Phatak May 13, 2019
What is a multiplicative inverse? If that's the question bugging you, this write-up has all the answers. The examples and the definition presented here, will clear out this math concept for you.
If you enjoy solving riddles and puzzles, mathematics will interest you. It is the natural language of logic and every mathematical operation is a manipulation of symbols, according to certain rules of established logic.
There are various mathematical operations which you can carry out on numbers. Some of the simplest include adding, subtracting, dividing, and multiplying numbers.

### Definition

When you multiply any number by another number and the resultant product of multiplication is 1, the two numbers are said to be multiplicative inverses of each other. Inverse of a variable x is often denoted as '1/x' or 'x-1'. In equation form, it can be defined as follows:

a x a-1 = 1
Thus, here 'a' and 'a-1' are multiplicative inverses of each other, as their product yields 1, which is also known as 'multiplicative identity'. This property is applicable to all numbers, and they are almost always distinct numbers, except in a few cases.
It is a concept which is extended to the domain of trigonometry, matrices, as well as complex or imaginary numbers. Also known as a 'Reciprocal' of any number, the inverse of a fraction is its reciprocal fraction.

### Exceptions and Special Cases

The only number which does not have a multiplicative inverse is 0. That is because (1/0) is an undefined quantity, which is also known as a singularity. As you can see, zero is a very special number and has some peculiar properties. Also 1 is the only number, which is its own inverse, for obvious reasons.

### Examples

• The inverse of ½ is 2 as (½) x 2 = 1.
• The inverse of 6/5 is 5/6 as (6/5) x (5/6) = 1.
• The inverse of 1 is 1, as 1 x 1 = 1.
• The inverse of 0.25 is 4, as 0.25 x 4 = 1
• The inverse of 1000 is 0.001 as 0.001 x 1000 = 1.
The given examples provide a way of verifying whether two numbers are inverses of each other. Just multiply the two numbers and if the product is 1, they indeed are.
The key point to remember in this whole discussion is the following. When a number and its multiplicative inverse are multiplied, what you get is unity or 1 as the product. To find the inverse, you must divide 1 by that number. The calculation can be easily done using a scientific calculator, which usually has a sign labeled as '1/x' for it.