1. ## Properties of 1.212212221...

According to wikipedia "A real number has an ultimately periodic decimal representation if and only if it is a rational number". I can see how a number represented as a repeating decimals is a rational number. However, is there a proof that shows that all rational numbers can be represented as repeating decimals (I consider terminating decimals as a type of repeating decimal where the repeating decimal is 0)?

Also I haven't seen an adequate explanation of what repeating means in the context of repeating decimal. Can repeating here be thought of more generally as some recursive pattern as we see with the number 1.212212221...? Which brings me to the question: is the number 1.212212221... rational or irrational. The answers to the first questions I would think would help answer the previous question. If those answers are not sufficient enough to determine if this number is rational or irrational, please let me know how one would go about proving the rationality/irrationality for 1.212212221... If it helps, I came up with an exact representation of this number using summation notation, which is shown in the attached JPEG.

If 1.212212221... is irrational (which is where I'm leaning towards) then is it considered to be a special type of irrational number with interesting properties? Notice that in the JPEG, summation in the exponents is used, a use of summation that I'm not very familiar with.

Thanks in advanced for considering my questions.

2. Originally Posted by Sparky1
According to wikipedia "A real number has an ultimately periodic decimal representation if and only if it is a rational number". I can see how a number represented as a repeating decimals is a rational number. However, is there a proof that shows that all rational numbers can be represented as repeating decimals (I consider terminating decimals as a type of repeating decimal where the repeating decimal is 0)?
Yes, there is. Imagine x= A.BCCCCCCC...
where A is the "integer" part of A, B is a string of n digits and C is a string of m digits which repeats. Multiply both sides of the equation by $\displaystyle 10^n$ so $\displaystyle 10^nx= AB.CCCCCCC...$ Now multiply both sides of that equation by $\displaystyle 10^m$ to get $\displaystyle 10^{n+m}ABC.CCCCC...$ Although we have shifted the number m places, because that repeating section continues without ending, the decimal parts of both $\displaystyle 10^nx$ and $\displaystyle 10^{n+m}x$ are identical- subtracting the two equations eliminates the decimal part:

$\displaystyle (10^{m+n}-10^n)x= ABC- AB$

Now divide both sides by $\displaystyle 10^{m+n}- 10^n$
$\displaystyle x= \frac{ABC- AB}{10^{m+n}- 10^{n}}$m
a fraction with integer numerator and denominator.

As an example of how that works, suppose x= 145.32153153153... where the "153" repeats without stopping. Then 100x= 14532.153153153... Multiplying by another 1000, 100000x= 14532153.153153153... Subtracting, (100000- 100)x= 14532153- 14532 or 99900x= 14517621 so x is the fraction $\displaystyle \frac{15417621}{99900}$ proving that x is a rational number.

Also I haven't seen an adequate explanation of what repeating means in the context of repeating decimal. Can repeating here be thought of more generally as some recursive pattern as we see with the number 1.212212221...?
No, it cannot. It must means exactly what I had above- That there is some integer part (which may be 0 or negative), some string of digits after the decimal point and then a block of digits that repeats exactly the same without stopping.

Which brings me to the question: is the number 1.212212221... rational or irrational. The answers to the first questions I would think would help answer the previous question. If those answers are not sufficient enough to determine if this number is rational or irrational, please let me know how one would go about proving the rationality/irrationality for 1.212212221... If it helps, I came up with an exact representation of this number using summation notation, which is shown in the attached JPEG.

If 1.212212221... is irrational (which is where I'm leaning towards) then is it considered to be a special type of irrational number with interesting properties? Notice that in the JPEG, summation in the exponents is used, a use of summation that I'm not very familiar with.

Thanks in advanced for considering my questions.
Yes, 1.212212221... where there is always additional 2 in the block between 1s is irrational.

3. Originally Posted by Sparky1
According to wikipedia "A real number has an ultimately periodic decimal representation if and only if it is a rational number". I can see how a number represented as a repeating decimals is a rational number. However, is there a proof that shows that all rational numbers can be represented as repeating decimals (I consider terminating decimals as a type of repeating decimal where the repeating decimal is 0)?

Also I haven't seen an adequate explanation of what repeating means in the context of repeating decimal. Can repeating here be thought of more generally as some recursive pattern as we see with the number 1.212212221...? Which brings me to the question: is the number 1.212212221... rational or irrational. The answers to the first questions I would think would help answer the previous question. If those answers are not sufficient enough to determine if this number is rational or irrational, please let me know how one would go about proving the rationality/irrationality for 1.212212221... If it helps, I came up with an exact representation of this number using summation notation, which is shown in the attached JPEG.

If 1.212212221... is irrational (which is where I'm leaning towards) then is it considered to be a special type of irrational number with interesting properties? Notice that in the JPEG, summation in the exponents is used, a use of summation that I'm not very familiar with.

Thanks in advanced for considering my questions.
Ultimately repeating means that there is a number $\displaystyle K$ with terminating decimal representation and an $\displaystyle n$-digit positive integer $\displaystyle a$ such that:

$\displaystyle x=K+a\sum_{\lambda=0}^{\infty}\frac{1}{10^{\lambda n}}$

and yes if your number is what I think it is it is constructed to be non repeating and so is irrational and is related to Liouville's numbers/constant

CB