Question

If a number can be represented using infinite fractions is it a rational number?

Answer 1

Is the number \[\large\rm \frac{1}{2\frac{3}{4\frac{5}{6\frac{7}{8...}}}} \] a rational number? Why or why not?

Answer 2

@ganeshie8 @zzr0ck3r

Answer 3

If it has a limit then we can call the limit rational

Answer 4

But shouldn't the limit be infinite?

Answer 5

only if it does not converge

Answer 6

See the denominator of the fraction keeps decreasing so the fraction must increase

Answer 7

in that case no. I was not sure if you were asking for a general thing, or for this example.

Answer 8

Oh I was concerned for both

Answer 9

But dont we consider infinite number rational?

Answer 10

no

Answer 11

its not a number

Answer 12

So the best answer to these question is undefined right?

Answer 13

If the limit is rational then we are good, but rational sequences can converge to irrational numbers

Answer 14

jerks :)

Answer 15

lol

Answer 16

these type of numbers aren't officially recognized in mathematics right?

Answer 17

I am not sure what you mean, but everything you can think of is defined to be something :)

Answer 18

I think your original question is just asking if sequences that live in \(\mathbb{Q}\) can ever converge to something not in \(\mathbb{Q}\) and they can

Answer 19

I mean no theorems about these type of numbers have been passed (like the type of development we have done on defining and manipulating infinities)

Answer 20

Still a bit confused on which numbers you are talking about. There are an infinite amount of types of infinity and some say an uncountable infinite amount of types of infinity. Not sure if that helps.

Answer 21

These type of numbers refer the numbers which can be representated using infinite fractions

Answer 22

But when we say that, we are talking about size of sets. When we say something does not converge, or converges to infinity, we are really just saying, that we can not say, that given any small number \(\epsilon\), there is some term of the sequence where, for the rest of the terms, will stay within \(\epsilon\) of some limit (some number).

Answer 23

they either converge or dont. If they do then it is a real number and we know much about them, if it does not, then it is not a number. But we don't clasify exactly what that means because we could have many things that happen. A sequence could flip flop or it could grow without bounds.

Answer 24

@zzr0ck3r Thanks for answering my weird question

Answer 25

They are the best questions :)

Answer 26

\(\color{#0cbb34}{\text{Originally Posted by}}\) @zzr0ck3r
They are the best questions
\(\color{#0cbb34}{\text{End of Quote}}\)
Thank you