Educational

Irrational Numbers Between 5/7 and 9/11 Find Three Irrational Numbers Between \( \frac{5}{7} \) and \( \frac{9}{11} \) Solution (Direct Method): \[ \frac{5}{7} = 0.714285… \quad \text{and} \quad \frac{9}{11} = 0.818181… \] So, \[ 0.714… < x < 0.818... \] Choose non-terminating, non-repeating decimals between them: \[ 0.721001000100001…, \quad 0.752002000200002…, \quad 0.801003000300003… \] All these

Rational and Irrational Numbers Between Given Decimals Find a Rational and an Irrational Number Between Given Decimals Question: Find a rational number and also an irrational number lying between \(0.3030030003…\) and \(0.3010010001…\). Solution: First arrange the numbers: \[ 0.3010010001… < 0.3030030003... \] Step 1: Rational Number Choose a terminating decimal between them: \[ 0.302 \]

Irrational Number Between Given Decimals Find an Irrational Number Between Given Decimals Question: Find one irrational number between \(0.2101\) and \(0.\overline{2}\) (i.e., \(0.222…\)). Concept Used: An irrational number is a non-terminating, non-repeating decimal. :contentReference[oaicite:0]{index=0} Solution: We are given: \[ 0.2101 < \ ? \ < 0.2222... \] We need a number between these which is

Rational Numbers Between Given Decimals Find Two Rational Numbers Between Given Decimals Question: Give two rational numbers lying between \(0.515115111511115…\) and \(0.5353353335…\). Concept Used: Between any two real numbers, there are infinitely many rational numbers, so we can always choose convenient terminating decimals between them. :contentReference[oaicite:0]{index=0} Solution: First compare the numbers: \[ 0.515115111511115… < 0.5353353335...

Rational Numbers Between Two Decimals Find Two Rational Numbers Between Given Decimals Question: Give two rational numbers lying between \(0.232332333233332…\) and \(0.212112111211112…\). Concept Used: Between any two real numbers, there are infinitely many rational numbers. :contentReference[oaicite:0]{index=0} Solution: First, compare the given numbers: \[ 0.212112111211112… < 0.232332333233332... \] Now choose simple terminating decimals between them. We

Rational or Irrational from Equations Find Whether Variables Represent Rational or Irrational Numbers Question: In the following equations, find whether the variables represent rational or irrational numbers: (i) \( x^2 = 5 \) (ii) \( y^2 = 9 \) (iii) \( z^2 = 0.04 \) (iv) \( u^2 = \frac{17}{4} \) (v) \( v^2 =

Rational and Irrational Numbers with Decimal Representation Identify Rational or Irrational Numbers and Give Decimal Representation Question: Identify whether the following numbers are rational or irrational. Give decimal representation of rational numbers: (i) \( \sqrt{4} \) (ii) \( 3\sqrt{18} \) (iii) \( \sqrt{1.44} \) (iv) \( \sqrt{\frac{9}{27}} \) (v) \( -\sqrt{64} \) (vi) \( \sqrt{100}

Rational or Irrational Numbers Examine Whether the Following Numbers are Rational or Irrational Question: Examine whether the following numbers are rational or irrational: (i) \( \sqrt{7} \) (ii) \( \sqrt{4} \) (iii) \( 2 + \sqrt{3} \) Solution: (i) \( \sqrt{7} \) 7 is not a perfect square. Hence, \( \sqrt{7} \) cannot be expressed

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