Prove that V6 is not a rational number.

1.	Prove that V6 is not a rational number.
Answer» Suppose we consider ,√6 is a rational number . Then we can express it in the form of a/b ∴√6=a/b, where a and b are positive integer and they are co-prime ,i.e.HCF(a,b)=1 ∴√6=a/b =>b√6=a =>(b√6)²=a² [squaring both sides] =>6b²=a²………..(1) here,a² is divided by 6 ∴a is also divided by 6. [we know that if p divides a²,then p divides a] ∴6\|a =>a=6c [c∈ℤ] =>a²=(6c)² =>6b²=36c² [from (1)] =>b²=6c² here,b² is divided by 6, ∴b is also divided by 6. ∴6\|a and 6\|b we observe that a and b have at least 6 as a common factor .But this contradicts that “a and b are co-prime .” It means that our consideration of “√6 is a rational number” is not true. Hence,√6 is a irrational number. listen mister a rational number is of a fixed value I.e √6 gives a value which goes on. Thus √6 cannot be called as rational number

Answer»

Suppose we consider ,√6 is a rational number .

Then we can express it in the form of a/b

∴√6=a/b, where a and b are positive integer and they are co-prime ,i.e.HCF(a,b)=1

∴√6=a/b

=>b√6=a

=>(b√6)²=a² [squaring both sides]

=>6b²=a²………..(1)

here,a² is divided by 6

∴a is also divided by 6. [we know that if p divides

a²,then p divides a]

∴6|a

=>a=6c [c∈ℤ]

=>a²=(6c)²

=>6b²=36c² [from (1)]

=>b²=6c²

here,b² is divided by 6,

∴b is also divided by 6.

∴6|a and 6|b

we observe that a and b have at least 6 as a common factor .But this contradicts that “a and b are co-prime .”

It means that our consideration of “√6 is a rational number” is not true.

Hence,√6 is a irrational number.

listen mister a rational number is of a fixed value I.e √6 gives a value which goes on. Thus √6 cannot be called as rational number

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