Question

Prove that 5route 2 is irrational number

Answer 1

Answer:

Proved below

Step-by-step explanation:

$let5\sqrt2\;be \;rational\\i.e. \;5\sqrt2=\frac{p}{q} (where \;p\;and\;q\;are\;integars\;and\;co-prime)\\\\squaring\;both\;sides\\\\50=\frac{p^{2}}{q^{2}}...............(i)\\\=>50q^{2}=p^{2}\\=>50|p \;\;\;(50\;divides\;p)................(ii)\\=>50x=p(\;for\;some\;integar\;x).....(iii)\\Now, \;using\;(iii)\;in\;(i)\\50={(50x)^{2}}/{q^{2}} \\=>q^{2}=50x^{2}\\=>50|q..............(iv)\\=>50y=q...............(v)\\\\\\From\;(ii)\;and\;(iv),50|:is\;the\;common\;divisor\;of\;p\;and\;q$

$which\;is\;a\;contradiction\;to\;our\;supposition\;that\;p\;and\;q\;are\;co-priime.\\hence,\;5\sqrt2\;is\;not\;rationall\\i.e.\;It\;is\;irrational$

$Learning\;\;together\;:)$