Question

prove that is it rational number √6​

Answer 1

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YOUR QUESTION SHOULD WE TO PROVE ROOT 6 IS IRRATIONAL.

LET US ASSUME THAT

$\sqrt{6}$

IS A RATIONAL NUMBER.

THEREFORE ,

$\sqrt{6} =( p \div q)$

,WHERE P AND Q ARE COPRIMES.

$\sqrt{6 }\times q = p$

SQUARING BOTH SIDES,

$( \sqrt{6} \times q ){}^{2} = {p}^{2}$

$6 {q}^{2} = {p}^{2}$

___(1)

${q}^{2} = {p}^{2} \div 6$

~2 DIVIDES p^2

~2 DIVIDES p

(because if a prime number can divide square of a number hence it can divide the number too)

THEREFORE,

$p \div 6 = c$

,FOR SOME INTEGER c

$p = 6c$

____(2)

FROM (1) AND (2),

$6 {q}^{2} =( 3 {c})^{2}$

$6 {q}^{2} = 36c {}^{2}$

${q}^{2} = 36 {c}^{2} \div 6$

${q}^{2} = 6 {c}^{2}$

${c }^{2} = {q}^{2} \div 6$

~6 divides q^2

~6 divides q

(because if a prime number can divide square of a number then it can also divide the number)

THEREFORE,3 DIVIDES p AND q BOTH WHICH IS THE CONTRADICTION TO OUR PREASSUMPTION.

THEREFORE,ROOT3 IS IRRATIONAL.





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