Questions & Answers

Question

Answers

A. 0

B. $\dfrac{1}{\sqrt{-2}}$

C. $\dfrac{-8}{5}$

D. $\dfrac{3}{2}$

Answer
Verified

Hint: First of all check if a number is rational as not by checking that it should be of the form $\dfrac{p}{q}$ where p and q are integers and $q\ne 0$.

Complete step-by-step answer:

Then check if that number is between -1 and 1 in the number line.

Here we have to find the rational number between -1 to 1 out of given options.

Before proceeding with the question we must know what a rational number is.

A rational number is a number that can be expressed in the form of $\dfrac{p}{q}$ where p and q are integers and $q\ne 0$. Since q may be equal to 1, hence every integer is a rational number. Examples are $0,1,2,\dfrac{2}{3},-4,\dfrac{-6}{7},etc$

Now we will see what a number line is. A number line is a straight line with numbers placed at equal intervals or segments along its length. A number line contains all real numbers that are rational numbers and irrational numbers both.

We can show number line as

Now we will find the rational number between -1 and 1 out of given options.

(A) 0

Since we know that all integers are rational numbers. Therefore, 0 is also a rational number.

Also we know that 0 lies between -1 and 1 in number line as shown,

Therefore, we can say that 0 is a rational number between -1 and 1. Therefore, this option is correct.

(B) $\dfrac{1}{\sqrt{-2}}$

Since we know that in any rational number of form $\dfrac{p}{q}$, p and q must be integer but here $\sqrt{-2}$ is not an integer but an imaginary number, so $\dfrac{1}{\sqrt{-2}}$is not a rational number. Therefore, this option is incorrect.

(C) $\dfrac{-8}{5}$

$\dfrac{-8}{5}$is a rational number because it is in form of $\dfrac{p}{q}$that is $\dfrac{-8}{5}$where -8 and 5 are integers.

In decimal form we can write $\dfrac{-8}{5}=-1.6$

In number line we can show â€“ 1.6 or $\dfrac{-8}{5}$ as

Since â€“ 1.6 or $\dfrac{-8}{5}$ does not lie between â€“ 1 and 1. So this option is incorrect.

(D) $\dfrac{3}{2}$

$\dfrac{3}{2}$is a rational number because it is in form of $\dfrac{p}{q}$ that is $\dfrac{3}{2}$where 3 and 2 are integers.

In decimal form we can write $\dfrac{3}{2}=1.5$

In number line, we can show 1.5 or $\dfrac{3}{2}$ as;

Since, 1.5 or $\dfrac{3}{2}$does not lie between -1 and 1.

So this option is incorrect.

Hence option (A) is correct.

Note: Students must note that there are infinitely many rational numbers between any two numbers but here we must check from option to get rational numbers -1 and 1. Before checking if a number lies between -1 and 1 or not, students must check if it is rational or not. It is always better to convert fractional form to decimal form to judge the magnitude of a number correctly.

Complete step-by-step answer:

Then check if that number is between -1 and 1 in the number line.

Here we have to find the rational number between -1 to 1 out of given options.

Before proceeding with the question we must know what a rational number is.

A rational number is a number that can be expressed in the form of $\dfrac{p}{q}$ where p and q are integers and $q\ne 0$. Since q may be equal to 1, hence every integer is a rational number. Examples are $0,1,2,\dfrac{2}{3},-4,\dfrac{-6}{7},etc$

Now we will see what a number line is. A number line is a straight line with numbers placed at equal intervals or segments along its length. A number line contains all real numbers that are rational numbers and irrational numbers both.

We can show number line as

Now we will find the rational number between -1 and 1 out of given options.

(A) 0

Since we know that all integers are rational numbers. Therefore, 0 is also a rational number.

Also we know that 0 lies between -1 and 1 in number line as shown,

Therefore, we can say that 0 is a rational number between -1 and 1. Therefore, this option is correct.

(B) $\dfrac{1}{\sqrt{-2}}$

Since we know that in any rational number of form $\dfrac{p}{q}$, p and q must be integer but here $\sqrt{-2}$ is not an integer but an imaginary number, so $\dfrac{1}{\sqrt{-2}}$is not a rational number. Therefore, this option is incorrect.

(C) $\dfrac{-8}{5}$

$\dfrac{-8}{5}$is a rational number because it is in form of $\dfrac{p}{q}$that is $\dfrac{-8}{5}$where -8 and 5 are integers.

In decimal form we can write $\dfrac{-8}{5}=-1.6$

In number line we can show â€“ 1.6 or $\dfrac{-8}{5}$ as

Since â€“ 1.6 or $\dfrac{-8}{5}$ does not lie between â€“ 1 and 1. So this option is incorrect.

(D) $\dfrac{3}{2}$

$\dfrac{3}{2}$is a rational number because it is in form of $\dfrac{p}{q}$ that is $\dfrac{3}{2}$where 3 and 2 are integers.

In decimal form we can write $\dfrac{3}{2}=1.5$

In number line, we can show 1.5 or $\dfrac{3}{2}$ as;

Since, 1.5 or $\dfrac{3}{2}$does not lie between -1 and 1.

So this option is incorrect.

Hence option (A) is correct.

Note: Students must note that there are infinitely many rational numbers between any two numbers but here we must check from option to get rational numbers -1 and 1. Before checking if a number lies between -1 and 1 or not, students must check if it is rational or not. It is always better to convert fractional form to decimal form to judge the magnitude of a number correctly.

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