Ravi Kant Kumar

Difference Between Rational and Irrational Numbers Explain How Irrational Numbers Differ from Rational Numbers Question: Explain how irrational numbers differ from rational numbers. Explanation: The difference between rational and irrational numbers is mainly based on whether the number can be written in the form \[ \frac{p}{q} \] where \(p\) and \(q\) are integers and \(q […]

Definition of Irrational Number Define an Irrational Number Question: Define an irrational number. Definition: An irrational number is a number that cannot be expressed in the form \[ \frac{p}{q} \] where \(p\) and \(q\) are integers and \(q \ne 0\). Its decimal expansion is non-terminating and non-repeating. Examples: \(\sqrt{2}\) \(\sqrt{3}\) \(\pi\) These numbers cannot be

Convert 0.6 + 0.7̅ + 0.47̅ into Fraction (p/q) Express \(0.6 + 0.\overline{7} + 0.4\overline{7}\) in the Form \( \frac{p}{q} \) Question: Express \(0.6 + 0.\overline{7} + 0.4\overline{7}\) (bar on 7) in the form \( \frac{p}{q} \), where \(q \ne 0\). Solution: Step 1: Convert each decimal into fraction \[ 0.6 = \frac{6}{10} = \frac{3}{5}

Convert 0.12̅3 into Fraction (p/q) Express the Decimal \(0.12\overline{3}\) in the Form \( \frac{p}{q} \) Question: Express \(0.12\overline{3}\) (bar only on 3) in the form \( \frac{p}{q} \). Solution: Let \[ x = 0.12\overline{3} \] Multiply by 100 (to remove non-repeating part): \[ 100x = 12.\overline{3} \] Now multiply by 10 (since one digit repeats):

Convert 0.4̅7 into Fraction (p/q) Express the Decimal \(0.4\overline{7}\) in the Form \( \frac{p}{q} \) Question: Express \(0.4\overline{7}\) (bar only on 7) in the form \( \frac{p}{q} \). Solution: Let \[ x = 0.4\overline{7} \] Multiply by 10 (to move non-repeating part): \[ 10x = 4.\overline{7} \] Now multiply by 10 again (since one digit

Convert 4.7̅ into Fraction (p/q) Express the Decimal \(4.\overline{7}\) in the Form \( \frac{p}{q} \) Question: Express \(4.\overline{7}\) in the form \( \frac{p}{q} \). Solution: Let \[ x = 4.\overline{7} \] Multiply both sides by 10 (since one digit repeats): \[ 10x = 47.\overline{7} \] Subtract the first equation from the second: \[ 10x –

Convert 125.3̅ into Fraction (p/q) Express the Decimal \(125.\overline{3}\) in the Form \( \frac{p}{q} \) Question: Express \(125.\overline{3}\) in the form \( \frac{p}{q} \). Solution: Let \[ x = 125.\overline{3} \] Multiply both sides by 10 (since one digit repeats): \[ 10x = 1253.\overline{3} \] Subtract the first equation from the second: \[ 10x –

Convert 0.621̅ into Fraction (p/q) Express the Decimal \(0.\overline{621}\) in the Form \( \frac{p}{q} \) Question: Express \(0.\overline{621}\) in the form \( \frac{p}{q} \). Solution: Let \[ x = 0.\overline{621} \] Multiply both sides by 1000 (since three digits repeat): \[ 1000x = 621.\overline{621} \] Subtract the first equation from the second: \[ 1000x –

Convert 0.54̅ into Fraction (p/q) Express the Decimal \(0.\overline{54}\) in the Form \( \frac{p}{q} \) Question: Express \(0.\overline{54}\) in the form \( \frac{p}{q} \). Solution: Let \[ x = 0.\overline{54} \] Multiply both sides by 100 (since two digits repeat): \[ 100x = 54.\overline{54} \] Subtract the first equation from the second: \[ 100x –

Convert 0.37̅ into Fraction (p/q) Express the Decimal \(0.\overline{37}\) in the Form \( \frac{p}{q} \) Question: Express \(0.\overline{37}\) in the form \( \frac{p}{q} \). Solution: Let \[ x = 0.\overline{37} \] Multiply both sides by 100 (since two digits are repeating): \[ 100x = 37.\overline{37} \] Subtract the first equation from the second: \[ 100x