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Is the number $0.1234567891011121314\ldots$ a rational or irrational number?

The number has a very clear pattern but however in order for the number to be a rational number it would have to be written as a/b. The normal tricks of writing it as an equation and solving

$3.3333333... = x \\ 10*3.333333 = 10x\\ 33.333333 = 10x\\ 30 + x = 10x\\ 30=9x\\ x=30/9 \\= 10/3$

does not seem to work here

Bart Michels
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user1786726
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    If it were rational, then it would eventually repeat. Since it does not repeat, it is irrational. – Michael Burr Aug 17 '15 at 14:42
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    It's not hard to show that if $x$ is rational, its pattern of decimal digits must eventually hit a repeating pattern. Your number has a clear pattern. Does it ever start repeating? – Umberto P. Aug 17 '15 at 14:43
  • Some number theory books have dealt with this. Just search on the web – Shailesh Aug 17 '15 at 14:46
  • Clearly, this has the sequence $12$ somewhere in its decimal expansion. It also has $112$ somewhere, and it has $1112$ somewhere, and it has $11112$ somewhere… Can you see why no rational number can have this property? – Akiva Weinberger Aug 17 '15 at 15:14
  • In fact, can you prove that, if $\mathbf a$ is any sequence of digits (for example, $\mathbf a=13427$), then $\mathbb a$ appears somewhere in the decimal expansion of your number? – Akiva Weinberger Aug 17 '15 at 15:17
  • I voted to reopen this question by mistake. I will vote to close (as duplicate) if it gets reopened. – Surb Aug 17 '15 at 15:40

2 Answers2

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The Champernowne constant $$C = 0.12345678910111213141516\dots$$ is a transcendental real number, so it is also irrational.

user153012
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This is the Champernowne constant in base 10, which is in fact transcendental.

In order to be capable to write it as a fraction, you would need to have a repeating block of digits. Evidently, there does not exist such a repeating block, and thus it is irrational.

miradulo
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