Rational number or irrational? And why?
I have been challenged in my college algebra class to determine whether or not .12122122212222122221222221... (continuing forever)is a rational number or not. I need to prove one way or the other. I am lost! :-( I know that rational numbers are number that 1. are whole numbers; 2. are fractions; and 3. are decimals. I also know that I can find a fraction for this number, or at least some of the number, but I'm not sure how to get it to repeat without end. The fraction that I came up with so far, is 12122122212222122221222221/99999999999999999999999999 and so on for repeating the pattern. However, I'm unable to prove what will happen if the number keeps repeating the pattern, by adding an additional "2" between each "1." At first I was inclined to believe that this was a rational number, because of the pattern and being able to come up with somewhat of a fraction for it, however, the more I look at it, and try to figure it out, I'm starting to think that this number is irrational, simply because it is not a "true repetition" of numbers.
After much research and study, I've come up with this so far:
Here's how to convert .1212212221222212222 to a fraction...
There is not much that can be done to figure out how to write .1212212221222212222 as a fraction, except to literally use what the decimal portion of your number, the .1212212221222212222, means.
Since there are 19 digits in 1212212221222212222, the very last digit is the "10000000000000000000th" decimal place.
So we can just say that .1212212221222212222 is the same as 1212212221222212222/10000000000000000000.
The fraction fmtterm: Can't handle [-2147483648/-2147483648] ? is not reduced to lowest terms. We can reduce this fraction to lowest
terms by dividing both the numerator and denominator by -2.14748e+09.
Why divide by -2.14748e+09? -2.14748e+09 is the Greatest Common Divisor (GCD)
or Greatest Common Factor (GCF) of the numbers 1.21221e+18 and 1e+19.
So, this fraction reduced to lowest terms is
So your final answer is: .1212212221222212222 can be written as the fraction
Which I think proves it as an irrational number, since the initial fraction can be reduced to a simpler form?!?!?!
Please help!!! I am going to continue to try to solve this question, but any help or advice you may have would be appreciated. If you could please explain to me if my rationale is correct, it'd be appreciated. If it is not, then please help me to understand...
Thank you so much! I hope you can help!