Rational or Irrational???????

Recently, I was doing the question 'nested radicals' posted by Pranay Singh.The question that came to my mind is that how does the fraction give an irrational value i.e a fraction upon a fraction upon a fraction and so on should also be rational,but that is not true.Why????

#Algebra #RationalNumbers #InfiniteSequence

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Hey Aman Jain

Is this the question you are reffering to ?Nested radicals....

A Former Brilliant Member - 6 years, 3 months ago

Yes, but this can also be treated as ageneral question for all continued fractions that how do they give an irrational value despite being a fraction.

Aman Jain - 6 years, 3 months ago

Well , I also want to know answer of your question ......

A Former Brilliant Member - 6 years, 3 months ago

Hi, Aman it's Parv here - first of all you need to understand that a fraction with non zero rational numbers will always yield a rational value as it can be represented in the form p/q but in the question posted by pranay singh it is not a rational divided by a rational and the answer is also not equal to , but it is approaching that value, every time it is divided by one more (one +one/one.... ) it's value increases but by a very small extent, If you have not understood then you can think if it was not (.............infinity) but was (........20 terms ) then it would have been rational even at (..........9999999999999 times) it would have been rational but infinity is a strange number because "na to kabhi infinity ayega , na to kabhi yeh rational hoga, agar koi number equal to infinity hota toh yeh definitely rational hota"[sorry for members who are unfamiliar to this language - hindi , but just wanted to say that infinity is not equal to , but is approaching to , if it would have been equal to , then it would have been rational]

Parv Maurya - 6 years, 3 months ago

In the 3rd line of your comment,how is it not a rational divided by a rational?? even after doing it an infinite times it will still be a rational divided by a rational;or am i wrong anywhere???

Aman Jain - 6 years, 3 months ago

REMEMBER THAT I AM NOT GOOD IN EXPLAINING, SO PROCEED WITH COURAGE, I'LL TRY MY BEST You are still missing that link that infinite means that never ending and the basic definition of irrational is also never terminating never repeating Now , recall that the question was 1+1/(1+.....infinite times)

Now let us take the case if it was not infinite but the following numbers, but i am also confused in explaining the cases =>If it was equal to 0 time, then expression would have been equal to 0 or 1( I Got Confused! but i think it should be 0 as we added 1 none times , i'll take first term as 0 in latter cases)

=>If it was equal to 3 time, then expression would have been equal to (5/3)

=>If it was equal to 4 time, then expression would have been equal to 8/5)

=>If it was equal to infinite times, then expression would have been equal to an expression whose last digits are not finite.

I think , but not at all sure or confident that there is a pattern in these given cases, but you will not be able to define an nth term as a rational when n tends to infinity and the graph of it will form an asymptode but will only approach the value of the answer of pranay's question. I'll try to work on the point of sequence .

Parv Maurya - 6 years, 3 months ago

@Parv Maurya – sorry for replying so late but just wanted to put forward another argument that can i say that the answer to his question is the rational number which is just less than the irrational number the answer is???????

Aman Jain - 6 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$