Rational or Irrational?

[cougalier]

Problem:
Is .121221222122221.... (the number pattern continues forever in this decimal) a rational number or irrational number?

I'm stuck! Please help!


This is an explanation of why I need this and the research I've already done:

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.

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


I know that it can be written as a fraction for a set value (if the number were to end), but can it be written for a number that goes forever in this pattern?

 

[mmm4444bot]

… I know that rational numbers are [numbers] that 1. are whole numbers; 2. are fractions; and 3. are decimals …

Careful!

Not all decimal numbers are Rational. (The decimal form of Pi is a good example of a non-Rational.)

Here's the definition of Rational numbers that I use:

A Rational number is any number than can be written as the ratio of an Integer number to a Natural number.



… can [such a ratio] be written for a [decimal] number that goes forever in [the given] pattern? Nope.

The decimal form of a Rational number either terminates (i.e., repeats nothing but zeros from some digit onward) or repeats the same string of digits. ALWAYS!

Here's an example of a Rational number whose decimal form terminates:

1/4 = 0.250000000000000000000000… = 0.25

Here's an example of a Rational number whose decimal form repeats the same string of digits forever:

2/3 = 0.666666666666666666666666…

Here's another example of a repeating string of digits:

17/7 = 2.428571428571428571428571…

Again, unless the SAME string of digits repeats, the number is not Rational.

BTW, the definition of an Irrational number is any Real number that is not Rational.

What do you think about this exercise now?