IIT JEE , AIEEE Forum - Mathematics , Algebra

titun

Let x = 0.101001000100001........ + 0.27272727.............

Then x is

A) IS IRRATIONAL

B) IS RATIONAL BUT x^0.5 is irrational

C) IS A ROOT OF x^2 + 0.27x + 1 = 0

edison

In the expression for x first term is irrational and second is rational and the sum of irrational and rational is irrational only. so is the x.

titun

My dear friend,

how did you know that the first term is irrational?

krish

I think that all repeating or recurring decimals are said to be irrational .... so edison must have told that the first term is irrational
And he is right that sum of irrational and rational is always irrational..

titun

Krish, carefully observe that the 2nd term is also a recurring decimal part, but it can be reduced to the sum of infinite G.P. & hence it is rational as it can be expressed in p/q form.

lelin07

all non-repeating or non-recurring decimals are said to be irrational .

titun

Lelin07, I disagree with you that all non-repeating, non-recurring decimals are irrational.

0.25 = 1/4 WHICH IS RATIONAL.

0.4 = 2/5 IS RATIONAL.

Again, repeating, recurring decimal nos can also be rational.

0.3333333.......... = 1/3

0.6666666.......... = 2/3

Rational nos can be expressed in p/q form where p & q are relatively co - prime integers whereas irrational nos can't be expressed in such form.

edison

Well i think the topic and my solution has made the forum hot and invited lot of arguments. But its good to see that such forum can be highly interactive to such an appreciable extent like a lively classroom type situation.

The first term for x is irrational and second is rational for it can be reduced to p/q form where p and q are integers and q is non zero as follows:

Second term is = 0.27272727...

let this be a rational no.

so p/q = 0.2727272727... ----(1)

or 100p/q = 27.27272727.... -----(2)

subtracting (1) from (2) we obtain

99p/q = 27

or p/q = 0.27272727.... = 27/99

thus the second term is rational. Had it not been so we would have arrived at contradiction which in turn would suggest that our assumption is wrong and thus we are forced to consider the term in other way.

I was unable to obtain the p/q form for first term so i considered it as irrational.

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