Difference between Inverse and Reciprocal

Inverse and reciprocal are two terms that often get mixed up, but they mean different things. The inverse is basically the opposite of something. For example, if you add 5 and then subtract 5, you've used inverse operations because one cancels out the other.

The reciprocal, on the other hand, is specifically about numbers. It’s the number that, when multiplied by the original number, gives you 1. For instance, the reciprocal of 2 is 1/2, because 2 times 1/2 equals 1. In this article, we will break down these terms in simple ways with examples to help you understand them better.

Reciprocal Definition

Reciprocal of the number is simply 1 divided by that number. It is often referred to as multiplicative inverse because multiplying number by its reciprocal gives 1.

For non-zero number x, the reciprocal is:

Reciprocal of x = 1/x

Reciprocal of the number is denoted as x^-1 because reciprocal of x is mathematically same as raising x to power of -1.

Examples of Reciprocal

Some examples of reciprocals are:

• Reciprocal of 5 is 1/5.
• Reciprocal of 2/3 is 3/2 , because :
  • 1/(2/3) = 3/2
• Reciprocal of -4 is 1/-4 or -1/4.

Properties of Reciprocal

• Product of number and its reciprocal is always 1.
  • x × 1/x = 1 (for any non zero x)
• Zero does not have reciprocal because division by zero is undefined.

Inverse Definition

In mathematics, the term inverse refers to an operation or value that reverses the effect of another operation or value. Essentially, an inverse undoes what the original function or number does. There are different types of inverses depending on the mathematical context:

• Inverse of a Number: The inverse of a number, often referred to as the additive inverse, is the number that, when added to the original number, results in zero.
  • For example, the inverse of 5 is -5, because 5 + (-5) = 0.
• Multiplicative Inverse (Reciprocal): This refers to a number which, when multiplied by the original number, results in 1.
  • For instance, the multiplicative inverse of 2 is 1/2, because 2 × (1/2) = 1.

Inverse of Function

For function f(x), inverse function f^-1(x) is the function that reverse action of f(x). In other words, if f(x) = y then f^-1(y) = x.

For example:

• If f(x) = 2x + 3, inverse function f^-1(x) would reverse this operation, so :
  • f^-1(x) =(x - 3)/ 2
• The sine function sin(x) has inverse function sin^-1(x) (also called arcsin), which returns angle whose sine is x.

Inverse Vs. Reciprocal

Key differences between inverse and reciprocal is given in the following table:

When to Use Reciprocal vs. Inverse?

Terms reciprocal and inverse are often confused due to their overlapping use in the multiplication. However:

• The Reciprocal is term specific to numbers and the multiplication. It refers strictly to 1/x.
• Inverse is broader term used across various mathematical operations and the contexts. It involves 'undoing' an operation whether it is addition, multiplication or function.

Conclusion

In conclusion, while both inverse and reciprocal might seem similar at first glance, they have distinct roles in mathematics. The inverse refers to reversing an operation or value, such as turning a positive number into its negative counterpart or performing an operation that undoes another. On the other hand, the reciprocal specifically refers to the multiplicative inverse, where a number is flipped, like turning 2 into 1/2​.

Read More,

• Inverse Functions
• Additive Inverse
• Multiplicative Inverse
• Reciprocal