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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?

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4  
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 4 '13 at 5:15
    
The number is " 2.1234..... " –  Way to infinity Jan 4 '13 at 5:19
    
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 4 '13 at 5:23
    
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 $$a.a_1 a_2 \ldots a_n \overline{b_1 b_2 b_3 \ldots b_m}$$ –  user17762 Jan 4 '13 at 5:30
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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. –  Hurkyl Jan 4 '13 at 6:10

1 Answer 1

up vote 1 down vote accepted

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.

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Due to 2nd reason I suspect that the number "2.1234..." is irrational . –  Way to infinity Jan 4 '13 at 5:22
    
Yes, that would be a bit more interesting, and my thought also is for irrational in that second possibility. However, I believe that the number is meant to be taken as in the first case with the repetition. –  adam W Jan 4 '13 at 5:26
    
The second possibility is irrational. –  Jonathan Christensen Jan 4 '13 at 5:27

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