REAL NUMBERS | Irrational Numbers | Part -4

Exploring Rational and Irrational Numbers

i) Prove that \( \frac{2}{3}\sqrt{5} \) is not a rational number, given that \( \sqrt{5} \) is irrational.

It is known that \( \sqrt{5} \) is an irrational number. is irrational. A rational number is defined as a number that can be expressed as the ratio of two integers.

Consider \( \frac{2}{3}\sqrt{5} \). Because \( \frac{2}{3} \) is a rational number and \( \sqrt{5} \) is irrational, their product—the multiplication of a rational and an irrational number—must be irrational.

Hence, \( \frac{2}{3}\sqrt{5} \) is not a rational number.

ii) Examples

• a) Sum of two irrational numbers is a rational number:
  Let \( a = \sqrt{2} \), \( b = -\sqrt{2} \)
  Then, \( a + b = \sqrt{2} + (-\sqrt{2}) = 0 \), which is rational.
• b) Sum of two irrational numbers is an irrational number:
  Let \( a = \sqrt{2} \), \( b = \sqrt{3} \)
  Then, \( a + b = \sqrt{2} + \sqrt{3} \), which is irrational.
• c) Difference of two irrational numbers is a rational number:
  Let \( a = \sqrt{7} \), \( b = \sqrt{7} \)
  Then, \( a - b = \sqrt{7} - \sqrt{7} = 0 \), which is rational.
• d) The difference between two irrational numbers is an irrational number:
  Let \( a = \sqrt{5} \), \( b = \sqrt{2} \)
  Then, \( a - b = \sqrt{5} - \sqrt{2} \), which is irrational.

iii) Product and Quotient of Irrational Numbers

• a) Product of two irrational numbers is a rational number:
  Let \( a = \sqrt{2} \), \( b = \sqrt{2} \)
  Then, \( a \cdot b = \sqrt{2} \cdot \sqrt{2} = 2 \), which is rational.
• b) Product of two irrational numbers is an irrational number:
  Let \( a = \sqrt{2} \), \( b = \sqrt{3} \)
  Then, \( a \cdot b = \sqrt{6} \), which is irrational.
• c) Quotient of two irrational numbers is a rational number:
  Let \( a = \sqrt{2} \), \( b = \sqrt{2} \)
  Then, \( \frac{a}{b} = \frac{\sqrt{2}}{\sqrt{2}} = 1 \), which is rational.
• d) Quotient of two irrational numbers is an irrational number:
  Let \( a = \sqrt{5} \), \( b = \sqrt{2} \)
  Then, \( \frac{a}{b} = \frac{\sqrt{5}}{\sqrt{2}} \), which is irrational.