Question

रूट 3एक अपरिमेय संख्या है सिद्ध करे

Answer 1

$\sqrt{3} = \frac{x}{y} \\ \sqrt{3} y = x \\ 3 {y}^{2} = x {}^{2} \\ 3 \: divides \: x {}^{2} \\ 3 \: divides \: x \\ let \: x = 3c \: for \: some \: integer \: c \\ x {}^{2} = 9c {}^{2} \\ 3y {}^{2} = 9c {}^{2} \\ y {}^{2} = 3c {}^{2} \\ 3 \: divides \: y {}^{2} \\ 3 \: divides \: y$

Thus x & y have 3 as their common factor and hence it is a irrational number