# Irrational numbers

1. Aug 3, 2013

### cragar

Is it possible to have an infinite string of the same number in the middle of an irrational number?
For example could I have 1.2232355555555..............3434343232211
Where their was an infinite block of 5's. Then I was trying to think of ways to prove or disprove this. It does seem like it might not be able to happen because if I had an infinite a 5's in a row then it seems like it would be a rational number.

2. Aug 3, 2013

### Millennial

The "number" you are speaking of is nonsensical. You can't have an infinite string of 5's, and then have numbers come after that. If there are infinitely many 5's, you can't add anything after that; because it would mean the string of 5's eventually terminates at some point, which violates the definition of "being infinite", or "not having an end". This makes all of your question nonsensical as a whole. Perhaps you meant something else, but couldn't phrase it properly?

3. Aug 3, 2013

### pwsnafu

If the 3434... part of the number exists, then the string of 5's before it terminates, hence not an infinite string.

4. Aug 3, 2013

### micromass

You must be careful about what a decimal expansion actually is. Given a number $x\geq 0$, a decimal expansion is a sequence

$$(x_0,~x_1,~x_2,~x_3,~...)$$

where $x_0\in \mathbb{N}$ and where $0\leq x_i\leq 9$ for $i>0$. It must satisfy

$$x = x_0 + x_1\frac{1}{10} + x_2\frac{1}{10^2} + ...+ x_n \frac{1}{10^n}+...$$

What is a sequence? Well, it is a map $f:\mathbb{N}\rightarrow \mathbb{R}$. We write $f(n) = x_n$.

When you say you add an infinite number of $5$ in the middle, then this is invalid as it would not produce a sequence anymore. It is impossible to write something like that as a map from the naturals.

5. Aug 3, 2013

ok thanks