Fractional Representations of Irrational Numbers
- 8:11 am |
This all started with Pi and now I can’t stop. Let me summarize what I have done so far:
- Pi day in the US is March 14th. This doesn’t work for non-USAians since they write the date in a reasonable fashion.
- July 22 seems to be nice day for Pi day (22/7 is close to Pi). What are better fractional representations of Pi?
- 355/113 is an awesome fraction to represent Pi. In order to get one that is just a little bit better, you have to go to 52,163/16,604.
So, here is the question. Is this huge void in best fractional representations a common thing? Or is it odd? It looks odd at first – but I just don’t know. How about I use the same method of looking for fractional representations for other irrational numbers? Here are the numbers I chose (I used the first 150 digits of each of these.) Oh, here is a one of the sources I used for the digits of these irrational numbers.
- Pi – of course.
- square root of 2
- square root of 3
- square root of 5
- square root of 7
- a list of 150 random numbers turned into a fake irrational number (I used this website random number generator
As before, represent the irrational number as some nu over d fraction. I either increase nu or d for each “step” to get a better representation. If the fraction is closer to the irrational number than the previous “best” fraction, I record it.
What to plot? Here is a plot of the iteration number (n = nu + d) vs. the step difference between that best fraction and the previous best fraction. Let me just show you this plot for pi.
Here the red arrow points to the first several “best fractions” ending with 355/113. You see the big gap. It is also odd that following the big gap (or should we call it the fractional pi-void?) the difference in iteration numbers for following best fractions is quite small. So, just think about an ideal case. In an ideal case every iteration would make a better fractional representation. The difference between iterations would be like 1 or something and you would get a nice straight line.
Ok, what about all these other representations of irrational numbers? Here you go.
Here, I point out three lines. Pi, e and the fake irrational number (random). The other lines seems to be pretty steady (but maybe that is because they are all square roots). Let me go to a larger iteration number – I get this.
These are the best fractions for 1 million iterations. I like to point out three things. First, the arrow pointing to the black line. Sure looks a lot like pi, doesn’t it. I have a feeling that this ‘random’ number list somehow used pi to generate its numbers. I could be wrong. Next, the small spike on the blue pi-curve. That is the jump from the 355/113 fraction being so awesome. However, take a look at the green curve for e. There is the fraction 49171/18089 (which matches e up to the 9th decimal place). The next best fraction is 271801/99990 (which also matches to the 9th decimal). This is an iteration gap of over 300,000. Boom. That is a big fractional void.
I would assume that as you get to larger and larger iteration values, the gaps would get bigger. Notice that none of the other irrational numbers have something like this – a jump much greater than the ‘average’ except for e and pi. Also, after the void, both e and pi have a series of regularly spaced increases. Odd.
I guess I need to look at even more irrational numbers.