Question

fill in the blanks.
(i) the number ……. is neither positive nor negative rational number.
(ii) there are ……... the number of rational numbers between two rational numbers.
(iii) A rational number is defined as a number which can be expressed in the form of $\dfrac{p}{q}$, where $p$ and $q$ are ……….. and $q$ is not equal to …….

Answer 1

Hint: For the first question we will use the help of number line and try to find out the position of 0 with respect to the positive and negative numbers. For the second question, we will try to find the middle value of any two rational number and try to do it as many times as possible. For the third question, we have been given the general definition of rational numbers. We will try to find the characteristics of the values of $p$ and $q$.

Complete step-by-step solution:
( i ) we know that if $x$ is a positive rational number then $x >0$.
And if $x$ is a negative rational number then $x< 0$.
From these two definitions, we can say that when $x=0$ at that time x is neither positive nor negative rational number.
So, the number 0 is neither a positive nor negative rational number.
( ii ) let’s consider two rational numbers x and y.
Also, consider that the numbers of rational numbers are finite.
But whenever we try to find the middle value of those two rational number x and y, we get $\dfrac{x+y}{2}$.
We can continue this process taking x and $\dfrac{x+y}{2}$ also.
We can find any number of rational numbers between two rational numbers.
So, there are infinite numbers of rational numbers between two rational numbers.
(iii) For the definition of rational number, the characteristics of p and q generally are integer.
We try to form the given number into its simplest form of fraction form.
Also, we know that in the fraction $0$ can’t be in the denominator.
So,$q\ne 0$.
So, A rational number is defined as a number which can be expressed in the form of $\dfrac{p}{q}$, where $p$ and $q$ are integers and $q$ is not equal to 0.

Note: when we are trying to find a non-negative and no-positive number we need to take care of that it’s been mentioned that we have been told to find a rational number as there is no such irrational number. 0 is the only possible rational number as every other rational number can be put on either side of 0 on the number line. Any number of rational can be found just by finding any ratio of two numbers, not always doing it by going through the middle number ($1:1$).