Question

1. What are rational and irrational numbers ?
2. Rationalize 1/√7
3. Convert $\sf 2.\overline{27}$ and $\sf 0.\overline{47}$ in the form p/q.

Answer 1

rational number or those number which can be returned in the form of p upon Q

Step-by-step explanation:

1. and irrational number are those number which cannot return in the form

Answer 2

Question 1 :-

Rational numbers :-

Rational numbers are those numbers which can be expressed in the form $\dfrac{p}{q}$ , where p and q are co - primes and q ≠ 0. Natural numbers, Whole numbers, Integers, Fractions and Decimals, all comes under this category.

Decimal representation of rational numbers :-

Decimals which come under rational numbers are of 2 types.

• Terminating decimal forms
• Non terminating repeating decimal forms

Terminating decimals :-

The decimals which comes to an end after a few or so digits. It has finite number of digits.

Example :- 12.4, 3.45679 etc.

Non terminating repeating decimals :-

Non terminating decimals are never ending and have a infinite number of digits. Non terminating repeating decimals are those in which the digits are repetitive.

Example :- $\sf 12.\overline{35}$, $\sf 0.0\overline{124}$ etc.

Question 2 :-

Rationalize the denominator in $\sf \dfrac{1}{\sqrt{7} }$

To rationalize the denominator, we have to multiply the number by $\sf \dfrac{\sqrt{7} }{\sqrt{7}}$

Multiplying,

$\implies \sf \dfrac{1}{\sqrt{7}} \times \dfrac{\sqrt{7} }{\sqrt{7} }$

$\implies \sf \dfrac{1 \times \sqrt{7} }{\sqrt{7} \times\sqrt{7} }$

$\implies\sf \dfrac{\sqrt{7} }{(\sqrt{7} )^{2} }$

$\implies \sf \dfrac{\sqrt{7}}{7}$

∴ The rationalized form of 1/√7 is √7/7

Question :-

To convert $\sf 2.\overline{27}$ and $\sf 0.\overline{47}$ in the form p/q.

$\sf 2.\overline{27}$ :-

Let x = $\sf 2.\overline{27}$

Multiplying by 100,

$\implies \sf 100x = 100 \times 2.\overline{27}$

$\implies \sf 100x = 227.\overline{27}$

Subtraction x from 100x,

$\implies \sf 100x - x = 227.\overline{27} - 2.\overline{27}$

$\implies \sf 99x = 225$

$\implies \sf x = \dfrac{225}{99}$

Reducing to the lowest terms,

$\sf 2.\overline{27} = \dfrac{25}{11}$

$\sf 0.\overline{47}$ :-

Let x = $\sf 0.\overline{47}$

Multiplying by 100,

$\implies \sf 100x = 100 \times 0.\overline{47}$

$\implies \sf 100x = 47.\overline{47}$

Subtracting x from 100x,

$\implies \sf 100x - x = 47.\overline{47} - 0.\overline{47}$

$\implies \sf 99x = 47$

$\implies \sf x = \dfrac{47}{99}$

$\sf 0.\overline{27} = \dfrac{47}{99}$