Educational

Classification of Real Numbers Classification of Real Numbers Fill in the Blank: Every real number is either rational or irrational number. Explanation: Real numbers include all numbers on the number line. These are classified into: Rational numbers (can be written as \( \frac{p}{q} \)) Irrational numbers (cannot be written as \( \frac{p}{q} \)) Final Answer:

Sum of Rational and Irrational Number Sum of a Rational and an Irrational Number Fill in the Blank: The sum of a rational number and an irrational number is an irrational number. Explanation: Adding a rational number to an irrational number always gives an irrational result. Example: \[ 2 + \sqrt{3} = 3.732\ldots \] Final

Sum of Recurring Decimals Sum of Recurring Decimals Fill in the Blank: \(0.\overline{3} + 0.\overline{4} = \) \(0.\overline{7}\) Explanation: \[ 0.\overline{3} = \frac{1}{3}, \quad 0.\overline{4} = \frac{4}{9} \] \[ \frac{1}{3} = \frac{3}{9} \] \[ \frac{3}{9} + \frac{4}{9} = \frac{7}{9} \] \[ \frac{7}{9} = 0.\overline{7} \] Final Answer: \(0.\overline{7}\) Next Question / Full Exercise

Simplest Form of 1.6̅ Simplest Form of \(1.\overline{6}\) Fill in the Blank: The simplest form of \(1.\overline{6}\) is \(\frac{5}{3}\). Explanation: Let \(x = 1.666\ldots\) \[ 10x = 16.666\ldots \] \[ 10x – x = 16.666\ldots – 1.666\ldots \] \[ 9x = 15 \Rightarrow x = \frac{15}{9} = \frac{5}{3} \] Final Answer: \(\frac{5}{3}\) Next Question /

Product of Rational and Irrational Number Product of a Rational and an Irrational Number Fill in the Blank: The product of a non-zero rational number with an irrational number is always an irrational number. Explanation: Multiplying any non-zero rational number with an irrational number always results in an irrational number. Example: \[ 2 \times \sqrt{3}

Pi as an Irrational Number \( \pi \) as an Irrational Number Fill in the Blank: \( \pi \) is an irrational number. Explanation: The value of \( \pi \) continues infinitely without repeating, so it cannot be expressed as a fraction. Final Answer: Irrational Next Question / Full Exercise

Recurring Decimal as Rational Number Recurring Decimal as a Rational Number Fill in the Blank: Every recurring decimal is a rational number. Explanation: Recurring decimals repeat a pattern of digits and can always be expressed as a fraction \( \frac{p}{q} \), hence they are rational numbers. Final Answer: Rational Next Question / Full Exercise

Value of 1.999… in p/q Form Value of \(1.999\ldots\) in Fraction Form Fill in the Blank: The value of \(1.999\ldots\) in the form of \( \frac{m}{n} \), where \(m\) and \(n\) are integers and \(n \ne 0\), is 2. Explanation: Let \(x = 1.999\ldots\) \[ 10x = 19.999\ldots \] Subtract: \[ 10x – x =

Decimal Expansion of √2 The Decimal Expansion of \( \sqrt{2} \) Fill in the Blank: The decimal expansion of \( \sqrt{2} \) is non-terminating and non-repeating. Explanation: Since \( \sqrt{2} \) is an irrational number, its decimal expansion continues infinitely and does not repeat. Final Answer: Non-terminating, Non-repeating Next Question / Full Exercise

Decimal Expansion of Irrational Numbers The Decimal Expansion of an Irrational Number Fill in the Blank: The decimal expansion of an irrational number is non-terminating and non-repeating. Explanation: Non-terminating → Decimal never ends Non-repeating → No repeating pattern of digits Final Answer: Non-repeating Next Question / Full Exercise