I worked it out with a pencil, like all good mathematicians.
Prove me wrong!
A Yahoo! engineer has calculated pi's two quadrillionth bit using Hadoop, the open source distributed number-crunching platform based on Google's proprietary MapReduce technology. When pi is in binary, the two quadrillionth bit is 0. Hadoop-happy Yahooligan Tsz Wo (Nicholas) Sze set this pi world record on a cluster spanning …
> I worked it out with a pencil, like all good mathematicians.
> It's 4.
> Prove me wrong!
Sure, it's 3. The convention (unless otherwise specified) is to round down for decimals up to .4, and round up at .5 and higher.
Since it's already been established that pi to one decimal place is 3.1 (look it up if you don't believe me), it rounds to 3.
Do I get a prize for this?
Bugger that, I'm getting a beer.
I remain convinced that at some point along the way, pi actually starts repeating. Pi is a cosmic joke created by an unknown supreme force just to cause mathematicians to melt.
"Yeah, so...Pi? Remember Pi? We thought it was mostly a bunch of random numbers? mathematicians tried for centuries to discern a pattern and failed? Yeah...at 4Quadrillion + 4 digits it loops. It's a 4 Quadrillion + 4 digit periodic number. We through some algorithmic fuzzing at it so see if we could figure out any patterns, now that we have the full pre-loop sequence, and we scored a hit! It's actually a wav file:
<Nelson Muntz> HA HA! </Nelson Muntz>.
The conclusion of all Pi research to date is thusly that god (or the gods) or whom/whatever created the laws of physics in such a way as to cause Pi to exist...
...is a dick."
"I remain convinced that at some point along the way, pi actually starts repeating."
It is not hard to prove that the bits/digits/whatever of pi will never repeat. If they did, pi would be a rational number, and it was long ago proven that it is not. It is not even an algebraic number, so even the repeated fraction representation of pi is not periodic.
But I agree with another poster that it is a rather pointless waste of computer time, even more so than finding large Mersenne primes, which has used even more computer time than this.
If you read the BBC article carefully, you will see that he has calculated the 2 quadrillionth *bit* of the binary representation of pi. Not the 2 quadrillionth decimal digit.
They appear to have used a variation of the BPP formula:
These newer formulas allow for calculation of any binary (or hex) digit by calculation of only a few terms with a few neighbouring bits. This cannot be used to efficiently calculate any decimal digit, base conversion requires a majority of the previous bits to be calculated too.
See also http://en.wikipedia.org/wiki/PiHex
The 1 quadrillionth bit is 0 too. 50/50 chance, so who's to say it wasn't a lucky guess?
Well, the fact that it's the 2 quadrillion place rather than "Pi to 2 Quadrillion places" probably makes you consider it even more pointless, but let me make two observations;
1. It's a mathematical challenge, perhaps there is no point to this challenge other than "because it's there", however analysis like this does set the scene for, applicable mathmatics (folding@home, SETI etc.) some of which may be useful.
2. You have gone out of your way to comment, seeming not to try and understand, the "That is all" makes it a retorical question, I wonder which is more pointless?
Hmmm... 1000 nodes, 23 days, final (useless) answer is 0.
Shades of the classic Deep Thought supercomputer, 7,500,000 years, final (useless) answer is 42.
And the answers ARE useless.. I mean, how does one now go about specifying pi? After the first 2.7 trillion digits, does one write 1.9973 quadrillion 'x's and then a (binary) 0? Or what??
It's been known since as long as we've had decimal notation that pi is an irrational number (one that can't be represented exactly as a fraction) and that therefore it's decimal representation can contain no infinitely repeating strings (sorry Trevor!).
At the beginning of the 20th century, the concept of 'normal' numbers was invented. In the decimal expansion of a 'normal' number, the digits 0-9 occur with the same frequency* - so will any random string of digits - so 123456 will occur one in a million times. The same is true whatever base (binary, octal, hex, ...) is used.
Now, it's known that almost all irrational numbers are normal, but very few individual numbers are known to be normal. In particular, it's not known if any random square root or pi or e is normal. If pi was proven to be normal (and all the evidence - such as this test - shows and most mathematicians believe it is), it might be useful in cryptography - but that isn't why number theorists investigate these problems!
* the number 0 does not occur in the first 31 decimal places of pi, which is significant at the 95% level, but this fluke evens out when more decimal places are calculated.
What's wrong with the number 3.101? or
Brits used to have a different standard in which the common trillion was actually their billion, but since it didn't fit well with scientific units (especially when you get to giga- and tera- and so on), most of the UK has since adopted the more common standard.
The initial progression, each 1,000 times greater than the previous, is:
- thousand (10^3)
- million (10^6)
- billion (bi, 10^9)
- trillion (tri, 10^12)
- quadrillion (quadra, 10^15)
- quintillion (quinta, 10^18)
- sextillion (sexta, 10^21)
- septillion (septa, 10^24)
- octillion (octa, 10^27)
- nonillion (nona, 10^30)
- decillion (deca, 10^33)
And of course, you have the googol (10^100) and the googolplex (10^(10^100)).
Bellard is interesting because his calculation was done with an efficient method on a sub-$3000 PC, not because it is [still] a record to know that many consecutive digits. This, from August, details computation to 5 trillion digits: http://www.numberworld.org/misc_runs/pi-5t/announce_en.html (and additional detail at: http://www.numberworld.org/misc_runs/pi-5t/details.html).
...thinking the research might just be a little confused and "double the record" applies to Yee/Kondo vs Ballard, rather than Sze?
Both calculations were doubled compared to the previous ones, it transpires.
Sze's own announcement is at http://people.apache.org/~szetszwo/ (which looks transient; is also at http://arxiv.org/abs/1008.3171 but they've not preserved the link to http://oldweb.cecm.sfu.ca/projects/pihex/announce1q.html).
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