1. ## Prove irrationality

Prove that 0.1 2 3 4 5 6 7 8 9 10 11 12 13 ...
is not rational.

2. Originally Posted by fobos3
Prove that 0.1 2 3 4 5 6 7 8 9 10 11 12 13 ...
is not rational.
Since we know that there are infinite natural numbers, the decimal is non-terminating.

Since the decimal is written such that the natural numbers are in increasing order, numbers will not repeat themselves.

Hence 0.1 2 3 4 5 6 7 8 9 10 11 12 13 ... is a non-terminating, non-repeating decimal, hence not a rational.

3. Originally Posted by fobos3
Prove that 0.1 2 3 4 5 6 7 8 9 10 11 12 13 ...
is not rational.
The harder question is to prove that this number is transcendental.

By the way, the number is called Champernowne's number.

4. Originally Posted by Isomorphism
Since we know that there are infinite natural numbers, the decimal is non-terminating.

Since the decimal is written such that the natural numbers are in increasing order, numbers will not repeat themselves.

Hence 0.1 2 3 4 5 6 7 8 9 10 11 12 13 ... is a non-terminating, non-repeating decimal, hence not a rational.
I'm sure the numbers won't repeat themslves but what about the digits. For example 1 2 3 and 123 the digits 1,2,3 repeat. Let me give you another example. Consider the number 0.1 23 123 1231 2312 3123 12312...=0.123 123 123 123...
Numbers don't repeat but digits do.

5. Originally Posted by fobos3
I'm sure the numbers won't repeat themslves but what about the digits. For example 1 2 3 and 123 the digits 1,2,3 repeat. Let me give you another example. Consider the number 0.1 23 123 1231 2312 3123 12312...=0.123 123 123 123...
Numbers don't repeat but digits do.
0. 123 123 123 123 123 ... (Usually expressed as $\displaystyle 0.\overline{123}$) is a rational number because it can be written in terms of $\displaystyle \frac{p}{q}$. How? Using infinite geometric series:

$\displaystyle \text{S} = \frac{t_{1}}{1-\text{r}}$

Where t_1 is the first term in the series and r is the common ratio between each consecutive term.

$\displaystyle 0.\overline{123} = \frac{123}{1000} + \frac{123}{1000000} + \ldots$

$\displaystyle r = \frac{\frac{123}{1000000}}{\frac{123}{1000}} = \frac{1}{1000}$

Now substitute in the equation above:
$\displaystyle \text{S} = \frac{t_{1}}{1-\text{r}} = \frac{\frac{123}{1000}}{1-\frac{1}{1000}} = \frac{\frac{123}{1000}}{\frac{999}{1000}} = \frac{123}{999}$

6. Originally Posted by fobos3
I'm sure the numbers won't repeat themslves but what about the digits. For example 1 2 3 and 123 the digits 1,2,3 repeat. Let me give you another example. Consider the number 0.1 23 123 1231 2312 3123 12312...=0.123 123 123 123...
Numbers don't repeat but digits do.
A rational number has a decimal expansion which is eventually periodic, that is from some point on there is a string of digits of some length that is just repeated continually.

That the given number does not have that property just consider how far to the right the digit strings: 10, 100, 1000, 10000, ... first appear. What this shows is that for any given N the expansion has not become periodic before reaching N digits (since one of these strings first appears to the right of the N'th digit. Hence it does not become periodic period.

RonL