Question

prove that 1 by root 2 minus 1 is irrational ​

Answer 1

Hope this will help uu...

Answer 2

$\huge{\bigstar{\bigstar{\underline{\underline\blue{\mathcal{\red{solution!!...}}}}}}}$

$we \: have \: to \: prove \frac{1}{ \sqrt{2} - 1} as \: an \: irrational \: number$

take 1/√2-1 as a rational number...

so, it cam be written in the form of a/b where a & b are co prime numbers and integers...

since,

rationalize it...

$\frac{1}{ \sqrt{2} - 1 } \times \frac{ \sqrt{2} + 1}{ \sqrt{2} + 1}$

$\frac{ \sqrt{2} + 1}{( {2})^{2} - ({1})^{2} }$

$\frac{ \sqrt{2} + 1 }{4 - 1}$

$\frac{ \sqrt{2} + 1 }{3}$

so, it can be written as...

$\frac{ \sqrt{2} + 1 }{3} = \frac{a}{b}$

square both side...

$( \frac{ \sqrt{2} + 1 }{3} ) ^{2} = (\frac{a}{b} ) ^{2}$

$\frac{2 + 1}{3} = \frac{a^{2} }{ {b}^{2} }$

hence √2 divides a² and b²...

therefore, this contradicts our wrong assumption...

1/√2-1 is a rational number...

hope!!...it helps uhh...❣️