Math Doubts

Multiplicative inverse

When the product of two quantities equals to one, one quantity is called a multiplicative inverse (or reciprocal) of another quantity.

Introduction

The quantities are often involved in multiplication in mathematics. In a special case, the product of two quantities is equal to one. It is possible when a quantity is in opposite form in another quantity.

multiplicative inverse reciprocal
  1. In some cases, the meaning of opposite is expressed by a word “inverse” in mathematics.
  2. The product of two quantities equals to one is actually calculated by the multiplication.

According to these two considerable factors, one quantity is called a multiplicative inverse of the remaining quantity.

For example, $6$ and $\dfrac{1}{6}$ are two quantities and multiply them to get their product.

$6 \times \dfrac{1}{6}$

$=\,\,\,$ $\dfrac{6}{1} \times \dfrac{1}{6}$

$=\,\,\,$ $\dfrac{6 \times 1}{1 \times 6}$

$=\,\,\,$ $\dfrac{6}{6}$

$=\,\,\,$ $\dfrac{\cancel{6}}{\cancel{6}}$

$=\,\,\,$ $1$

Therefore, it is cleared that the product of a quantity and quotient of one by the same quantity is equal to one.

$\therefore\,\,\,$ $6 \times \dfrac{1}{6}$ $\,=\,$ $1$

Therefore, the number $\dfrac{1}{6}$ is called the multiplicative inverse or reciprocal of the number $6$ and vice-versa.

Algebraic form

Let $x$ be a variable and it represents a quantity. The second quantity is unknown. So, it can be denoted by a variable $y$. Assume that the product of both variables is equal to one.

$x \times y \,=\, 1$

Now, we can evaluate the variable $y$ in terms of $x$.

$\implies$ $y \,=\, \dfrac{1}{x}$

Therefore, the variable $y$ is equal to the quotient of one by $x$.

$(1).\,\,\,$ $\dfrac{1}{x}$ is called the multiplicative inverse of $x$.

$(2).\,\,\,$ $x$ is called the reciprocal of the $\dfrac{1}{x}$

Remember that, $\dfrac{1}{x}$ is simply written as $x^{-1}$ in mathematics.

Math Doubts
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more