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In my excercise book of math , I have found one problem . In that problem I have been asked to detect whether the number $2.1234....$ is rational or irrational?

My concept is : "$2.1234....$ is irrational." But the answer of book says that the number is rational. I want to argue the answer. Is this answer right?

Can you guys help me to clarify my misconcepts?

Way to infinity
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    On one hand you write $2.1234$… and on the other you write $2.1234$, so which is it? The latter, as mixed said, is rational. The former could be either, because it's not really clear what comes after the ellipsis (the three dots). In case you didn't know, an ellipsis in math writing with nothing following it usually means that infinity is involved somehow; here it implies that there are an infinite number of digits after the decimal place. – Eric Stucky Jan 04 '13 at 05:15
  • The number is " 2.1234..... " – Way to infinity Jan 04 '13 at 05:19
  • Without further defining the number, we have no idea what the answer is. Adam W's answer gives you two answers, not one. – Joe Z. Jan 04 '13 at 05:23
  • If you are dealing with the number$$\alpha = 2.123456789101121314151617181920212223\ldots$$ then the number is irrational. To see why this is so, assume it is rational i.e. $$\alpha = 2.123456789101121314151617181920212223\ldots = \dfrac{p}q$$ A rational number either has a truncated decimal expansion or recurring expansion. Clearly, the number we are dealing with doesn't have a truncated decimal expansion. A number of the form $p/q$ with recurring decimal expansion will be of the form $$a.a_1 a_2 \ldots a_n \overline{b_1 b_2 b_3 \ldots b_m}$$ –  Jan 04 '13 at 05:30
  • However, in the decimal expansion of $\alpha$, at some stage the number $\underbrace{7777\ldots777}_{m+n+1 \text{ times}}$ has to appear. Argue that this cannot happen for the number of the form $p/q$. –  Jan 04 '13 at 05:31
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    Since nobody's said it, "2.1234..." is not a number, or even a numeral. It is a form of shorthand, and one of the meanings of that shorthand is that you're supposed to replace it with the decimal it is obviously suggesting. However, "2.1234..." doesn't make an obvious suggestion, therefore the question itself is ill-defined, and it doesn't really make sense to ask for its answer. –  Jan 04 '13 at 06:10

1 Answers1

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I am supposing that $2.1234\dots$ means that the digits continue: $2.12345678901234\dots$. Note also that a continuation could be $2.123456789101112\dots$, it is unclear exactly whether single digits continue, or if it counts ten, eleven, etc.

Note that in the first case, that means that the portion $1234567890$ is repeating.

adam W
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