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!
See other response:
http://www.jiskha.com/display.cgi?id=1252464742
abcdefghijklmnopqrstuvwxy and z
To determine whether the number .12122122212222122221222221... is rational or irrational, let's analyze the given information and your calculations.
Firstly, your understanding of rational and irrational numbers is correct. Rational numbers can be expressed as fractions, while irrational numbers cannot.
You attempted to convert the repeating decimal .1212212221222212222 to a fraction. The fraction you obtained, 1212212221222212222/10000000000000000000, is indeed an accurate representation of the decimal.
However, in order to determine whether this fraction is in its simplest form (reduced to lowest terms), you attempted to divide both the numerator and denominator by -2.14748e+09. This seems to be an error in your calculations since dividing by a negative number would not yield a simplified fraction.
To further determine if this repeating decimal is rational or irrational, let's try to identify a pattern.
Looking closely at the decimal, we notice that the number of "1"s between each "2" increases by 1 each time. This pattern continues indefinitely.
Although we can obtain a fraction representation for a finite number of digits, it is not possible to express an infinite repeating pattern like this as a fraction (unless the pattern eventually terminates).
Therefore, based on the pattern and the fact that the decimal has an infinitely repeating pattern, we can conclude that the number .12122122212222122221222221... is an irrational number.
In summary, you were correct in your initial inclination that this number is likely irrational. Your observation of the repeating pattern and your attempt to convert it to a fraction were steps in the right direction. The fact that the pattern is infinite and does not eventually terminate confirms that this number is indeed irrational.
To determine whether the number .1212212221222212222... is rational or irrational, we need to understand the properties of rational and irrational numbers.
Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. This means that rational numbers can be written as terminating decimals (like 0.25) or repeating decimals (like 0.333...).
Irrational numbers, on the other hand, cannot be expressed as a fraction of two integers. They are numbers whose decimal representation goes on forever without repeating.
In the given number .1212212221222212222..., we can see that there is a repeating pattern of digits. This repeating pattern suggests that it might be a rational number.
You attempted to convert the number to a fraction by placing it over a power of 10. You correctly identified that the number has 19 digits after the decimal point, so you wrote it as 1212212221222212222/10000000000000000000.
However, when you reduced this fraction to lowest terms, you encountered an error caused by a large number appearing in the calculation. The error indicates that the fraction cannot be reduced further.
This is a significant clue that the given decimal is actually an irrational number. Rational numbers can always be expressed in lowest terms, meaning there should be no error in the reduction process.
To further confirm this, you can observe that the repeating pattern .1212212221222212222... does not repeat exactly. There is an increasing number of "2"s between each set of "1"s. This lack of exact repetition is another indicator that the number is irrational.
Based on the evidence, it can be concluded that the given number .1212212221222212222... is an irrational number.
In summary, your rationale is correct. The fact that the fraction could not be reduced further, along with the lack of exact repetition in the decimal pattern, supports the conclusion that the number is irrational.