Decimal - Is it rational or irrational?

I have been thinking about a continuing decimal, i.e. 0.707007000700007...

obviously, the decimals composite parts are 0.7+0.007+0.000007+...

My question is this, is this decimal rational or irrational? And a reason why?

I first started thinking of geometrical progressions and summing to infinity, but I found that my common ratio, r, had to change to create the decimal, so I personally think it's irrational as the repeating block is changing? Does anyone have a neat proof of this or even better, think that I am wrong?

Kind regards, tom

Re: Decimal - Is it rational or irrational?

Very sorry i posted this in the wrong place!

Re: Decimal - Is it rational or irrational?

Hello, Tom!

Quote:

I have been thinking about a continuing decimal, i.e. .$\displaystyle X \:=\:0.707007000700007\hdots$

Obviously, the decimal's composite parts are: .$\displaystyle 0.7+0.007+0.000007+ \hdots$

My question: is this decimal rational or irrational?

I first started thinking of geometrical progressions and summing to infinity,

but I found that my common ratio, r, had to change to create the decimal,

so I personally think it's irrational as the repeating block is changing?

Does anyone have a neat proof of this or, even better, think that I am wrong?

Kind regards, Tom

I too suspect that it's irrational, but I don't have a proof yet.

I've noted the following:

. . $\displaystyle \begin{array}{ccc} 0.7 &=& 7\cdot10^{-1} \\ 0.007 &=& 7\cdot 10^{-3} \\ 0.000007 &=& 7\cdot10^{-6} \\ 0.0000000007 &=& 7\cdot 10^{-10} \\ \vdots && \vdots \end{array}$

The exponents are *Triangular Numbers.*

. . Hence: .$\displaystyle X \;=\;7\sum^{\infty}_{n=1} 10^{-\frac{n(n+1)}{2}} $

And that is as far as I got . . .

Re: Decimal - Is it rational or irrational?

I love the insight into trianglular numbers!

I think what i've came up with is that it cant be rational for this reason:

conjecture: a rational number must have a repeating block of digits

if we consider a number with a repeating block of three, for example, then, in it's rawest form, we know our fraction is of the form:

n/999, where n is some integer

However, we have a block of increasing magnitude each time, and as the fraction continues forever with the same pattern, you will have m lots of different blocks in the decimal, hence it's irrational, as it will never repeat?

Please correct me if I'm wrong, I'm not the best at maths in the world (foundation year maths student)