Question

Fill in the blanks:
(i) Zero has _____ reciprocal.
(ii) The numbers _____ and _____ are their own reciprocals.
(iii) The reciprocal of $-5$ is _____.
(iv) Reciprocal of $\dfrac{1}{x}$, where $x\ne 0$ is _____.
(v) The product of two rational numbers is always a _____.
(vi) The reciprocal of a positive rational number is _____.

Answer 1

Hint: We will see the concept of a reciprocal. We will also look at the definition of a rational number. To fill in the blanks, we will use the definition of a reciprocal. We will also use the properties of the numbers given in each statement. For the fifth and sixth statement, we will look at some examples so that we can verify whether the completed statement is correct or not.

Complete answer:
Let $a$ be a number. We define the reciprocal of the number $a$ to be the fraction $\dfrac{1}{a}$. Let us look at the definition of a rational number. A rational number is a number that can be expressed as a fraction of two integers, that is in the form $\dfrac{p}{q}$, where $p,q\in \mathbb{R}$ and $q\ne 0$.
(i) We are aware that the fraction which has zero in the denominator is not defined. Hence, the number 0 does not have any reciprocal. So, we have the following statement:
Zero has no reciprocal.
(ii) We have to find two numbers that are their own reciprocal. This means that the reciprocal of this number is the number itself. Let us consider the number 1. Its reciprocal is $\dfrac{1}{1}=1$. So, 1 is its own reciprocal. Similarly, the number $-1$. Its reciprocal is $\dfrac{-1}{1}=-1$. Hence, we have the following statement,
The numbers 1 and $\underline{-1}$ are their own reciprocals.
(iii) The number given is $-5$. Its reciprocal, according to the definition, is $\dfrac{1}{-5}=-\dfrac{1}{5}$. Therefore,
The reciprocal of $-5$ is \[\underline{-\dfrac{1}{5}}\].
(iv) For this statement, we directly use the definition. Reciprocal of $\dfrac{1}{x}$, where $x\ne 0$ is $\underline{x}$.
(v) Let $\dfrac{{{p}_{1}}}{{{q}_{1}}}$ and $\dfrac{{{p}_{2}}}{{{q}_{2}}}$ be two rational numbers. The product of these two numbers is $\dfrac{{{p}_{1}}}{{{q}_{1}}}\times \dfrac{{{p}_{2}}}{{{q}_{2}}}=\dfrac{{{p}_{1}}{{p}_{2}}}{{{q}_{1}}{{q}_{2}}}$ where ${{p}_{1}}{{p}_{2}}$ and ${{q}_{1}}{{q}_{2}}$ are integers. Hence, the product is a rational number. So, we have
The product of two rational numbers is always a rational number.
(vi) Let $\dfrac{p}{q}$ be a positive rational number. Its reciprocal is $\dfrac{q}{p}$. The reciprocal is also a positive rational number. Hence,
The reciprocal of a positive rational number is positive.

Note:
For such types of questions, it is essential that we have a clear understanding of the concepts. Then we can find the answers to fill in the blanks by working with the given information in the statement. Working through the given information will help us avoid confusion. This type of question can sometimes mislead us or confuse us.