Are there any almost irrational numbers that have deceived mathematicians?

[guysensei1]

Does any known rational number look irrational at first glance but when calculated to 100s or 1000s of digits actually resolve into a repeating sequence? Have they deceived mathematicians?

 

[HallsofIvy]

What do you mean by "look irrational"? Obviously, there exist a number whose decimal expansion is identical to $\pi$ for the first, say, 10 million decimal places, then is just "5"s after that. That is a rational number. Would you say it "looks irrational"?

 

[MathJakob]

I think what he means, is there a number which appears irrational, but then after a couple hundred or thousand digits it repeats, meaning it isn't actually irrational.

For example if after 14trillion digits $\pi$ "resets" and started $14159...$ again. Which of course it doesn't but this is what the OP means I think.

 

[guysensei1]

What I meant to say was:

Are there any numbers that mathematicians thought were irrational for some time then was proven rational? And as an expansion, what about the converse?

 

[Nugatory]

Does any known rational number look irrational at first glance but when calculated to 100s or 1000s of digits actually resolve into a repeating sequence? Have they deceived mathematicians?

We don't look at the decimal representation of a number to decide if it is rational or not, we prove that there are no integers a and b such that a/b is equal to the number.

For example: http://www.math.utah.edu/~pa/math/q1.html

 

[Boorglar]

I think guysensei1 is thinking about numbers that aren't artificially constructed for the purpose of "looking" irrational (like the example with pi's first 10^6 digits followed by 5's), but rather for 'natural' examples, where a given number was conjectured to be irrational by most mathematicians until someone proved it was in fact rational.

The Euler-Mascheroni constant is believed to be irrational, but no one has managed to prove this yet. If it turns out to be rational, this would be a perfect example. In any case, I believe a whole bunch of digits have been computed without any periodic pattern revealed yet.

A trivial example of the converse would be the square root of 2, since before the proof that it was irrational, Greek mathematicians believed every length could be expressed in terms of integer ratios. You could also look up "almost-integer", which give a lot of examples for numbers that are very close to integers:
http://en.wikipedia.org/wiki/Almost_integer

and another relevant link: http://xkcd.com/1047/