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Question

Answers

a)0

b)- 4.3

c)- 3.4

d)1.101100110001……..

Answer
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Hint: A rational number is a number that can be expressed as a function $\dfrac{p}{q}$ where p, q are two integers and$q\ne 0$. Rational numbers will be terminating or of recurring type. Any number which is non-recurring is termed as an irrational number or that is not a rational number. Use this definition and check all four given answers.

Complete step by step answer:

In mathematics, rational numbers are the numbers which can be expressed in the form of$\dfrac{p}{q}$, where p and q are integers and q will never be 0. As any integer (5, 4, -3) can be written in the form of $\dfrac{p}{q}$ by putting the value of q as 1, It means integers are rational numbers as well.

Irrational numbers are the numbers which can not be represented in form of ‘$\dfrac{p}{q}$’ i.e. just opposite to rational numbers. Another properties of rational number can be given as

i) Terminating: It means the rational number has some end. Ex: 0.1, 0.00023, 5, 7.88996 etc.

ii) Recurring: Numbers which digits get repetitive are termed as recurring numbers and they also termed as rational numbers. Example: 5.55555……or$5.\overline{5}$ , 7.366666……or$7.3\overline{6}$, 8.45454545…….or $8.\overline{45}$ etc.

As these numbers are getting repeated after decimal. So, these are rational as well and we can represent them in $\dfrac{p}{q}$ form as well. $Ex:\dfrac{1}{6}=0.16666,0.1\overline{6}$. Hence, we get that rational numbers can be terminating or recurring but irrational numbers are non-terminating and non-recurring.

Now, we can observe the given options to get a rational number between - 3 and 3.

Option (a): 0

As ‘0’ can be represented in the form of $\dfrac{p}{q}$ as $\dfrac{0}{1},q\ne 0$. Hence 0 is a rational number.

And we can observe that ‘0’ is greater than ‘- 3’ and less than ‘3’, so, 0 will lie in between – 3 and 3. So, 0 is the correct option for the given problem because it is a rational number and lying between – 3 and 3.

Option (b): - 4.3

- 4.3 is less than ‘- 3’, hence it will not lie in between - 3 and 3. So, ignore this option.

Option (c): - 3.4

- 3.4 is also less than ‘- 3’, hence it will not lie in between – 3 and 3. Hence, this option is incorrect.

Option (d): 1.101100110001…………

As the given number in this option is an irrational number as it is not terminating and non-recurring. Hence, ignore this option as well, as we need to find rational numbers between – 3 and 3.

Note: One may go wrong if he/she does not know the property of rational and irrational numbers with respect to the termination and recursion. And hence, one may tick the 4th option as well as it is lying between – 3 and 3. So, be careful and clear with the properties mentioned above.

Don’t confuse it with the option given as ‘0’. As one may think, how will someone represent in the form of$\left( \dfrac{p}{q} \right)$. So, p can be ‘0’ in the definition of rational number. Hence, we can represent ‘0’ as$\dfrac{0}{1}\left( \dfrac{\text{p}}{\text{q}}\text{ form} \right)$. So, it is a rational number. Don’t be confused with it.

Complete step by step answer:

In mathematics, rational numbers are the numbers which can be expressed in the form of$\dfrac{p}{q}$, where p and q are integers and q will never be 0. As any integer (5, 4, -3) can be written in the form of $\dfrac{p}{q}$ by putting the value of q as 1, It means integers are rational numbers as well.

Irrational numbers are the numbers which can not be represented in form of ‘$\dfrac{p}{q}$’ i.e. just opposite to rational numbers. Another properties of rational number can be given as

i) Terminating: It means the rational number has some end. Ex: 0.1, 0.00023, 5, 7.88996 etc.

ii) Recurring: Numbers which digits get repetitive are termed as recurring numbers and they also termed as rational numbers. Example: 5.55555……or$5.\overline{5}$ , 7.366666……or$7.3\overline{6}$, 8.45454545…….or $8.\overline{45}$ etc.

As these numbers are getting repeated after decimal. So, these are rational as well and we can represent them in $\dfrac{p}{q}$ form as well. $Ex:\dfrac{1}{6}=0.16666,0.1\overline{6}$. Hence, we get that rational numbers can be terminating or recurring but irrational numbers are non-terminating and non-recurring.

Now, we can observe the given options to get a rational number between - 3 and 3.

Option (a): 0

As ‘0’ can be represented in the form of $\dfrac{p}{q}$ as $\dfrac{0}{1},q\ne 0$. Hence 0 is a rational number.

And we can observe that ‘0’ is greater than ‘- 3’ and less than ‘3’, so, 0 will lie in between – 3 and 3. So, 0 is the correct option for the given problem because it is a rational number and lying between – 3 and 3.

Option (b): - 4.3

- 4.3 is less than ‘- 3’, hence it will not lie in between - 3 and 3. So, ignore this option.

Option (c): - 3.4

- 3.4 is also less than ‘- 3’, hence it will not lie in between – 3 and 3. Hence, this option is incorrect.

Option (d): 1.101100110001…………

As the given number in this option is an irrational number as it is not terminating and non-recurring. Hence, ignore this option as well, as we need to find rational numbers between – 3 and 3.

Note: One may go wrong if he/she does not know the property of rational and irrational numbers with respect to the termination and recursion. And hence, one may tick the 4th option as well as it is lying between – 3 and 3. So, be careful and clear with the properties mentioned above.

Don’t confuse it with the option given as ‘0’. As one may think, how will someone represent in the form of$\left( \dfrac{p}{q} \right)$. So, p can be ‘0’ in the definition of rational number. Hence, we can represent ‘0’ as$\dfrac{0}{1}\left( \dfrac{\text{p}}{\text{q}}\text{ form} \right)$. So, it is a rational number. Don’t be confused with it.