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Is it true that by the method of contradiction that is irrational.

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True

False

answer is A.

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Detailed Solution

To prove that

is irrational, we will assume that the given statement is false, i.e. we will assume that

is rational.
Now, we know that any rational number can be written in the form of

where

and

are co-prime integers.
Thus, we can say that

Now, squaring on both sides, we get

Now, since, as mentioned above,

and

are integers, we can say from equation(1) that

is divisible by

.
Now, since

is divisible by

is also divisible by

.Thus, we can say that

is a factor of

.
Now, since

is a factor of

, we can write

as a multiple of

and

where

is any integer.
Thus, we can say that:

Now, squaring both sides we get

Now, putting the value of

from the equation obtained above we get:

Now, since

is an integer,

will also be an integer. Hence,

is also an integer which implies that

is divisible by

.Now, since

is divisible by

is also divisible by

.
Hence, we can say that

is a factor of

.
Now, we mentioned above that

and

are co-prime numbers but here we have proved that

is a factor common to both

and

.
Hence, the assumption we made, i.e.

is rational is wrong.
Thus, we can say that

is irrational.
Hence, the given statement is true.

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Is it true that by the method of contradiction that is irrational.

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