12-20-2012, 05:55 AM
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Spun a twirly fruitcake,
 
Join Date: Feb 2009
Age: 31
Posts: 3,865
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Grafting Numbers
I don't know if anyone knows about this, but there are these numbers called Grafting numbers.
Now a number that appears within the first digits of its square root is all fine and dandy, but I went to find the numbers that appear within the first fifty digits of its square root (up to 10 million). The result quite surprised me, I was expecting a couple in the thousands, but even seemingly random numbers in the millions repeat in the early decimal places. So what I'm thinking, there might be a regularity in how square roots (which are mostly said to be irrational numbers without repetition) actually DO repeat in base 10. It could also be a single moment at which the number appears in its decimals, but that's regularity nontheless. Here are the numbers with square roots that contain the original number (Big list): "Digit" is the digit at which the number repeats in the root In case you're questioning my method of acquiring these numbers, here's the Java code I made I'm not a highly schooled mathematician, I actually failed high school (not because of math, but because of biology). Maybe someone more bright can figure out the regularity. P.S: Trying to make a square root method for BigDecimal, so I can get up to 100 decimals. ----- Edit v1: Changed the code, max digits got inaccurate, narrowed it down to 40. Changed the println into a printf to make the results in the spoiler more synoptic. Edit v2: Got to run the program again, apparently it's "sqrt.indexOf(x.toString())+1;"
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Last edited by SKG_Scintill; 12-20-2012 at 06:27 AM.. |
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12-20-2012, 11:08 PM
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#2 |
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behanjc & me are <3'ers
Join Date: Jul 2006
Posts: 2,051
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Re: Grafting Numbers
I don't think the "regularity" you're looking for is this. Personally, I feel like this concept is a little bit too arbitrary. First few digits in the decimal representation might have some sort of significance, but to extend to 50 digits means nothing to me.
To see this, consider the following: You have 32 4-digit numbers in which the number occurs in the decimal representation within in first 50 digits. The probability that a 4-digit number can be found in these first 50 digits is (47*10^46)/(10^50) = 47/10000. There are about 9000 4 digit numbers, so the expected number of numbers that satisfy what we are looking for is 9000*(47/10000) = (9/10)*47 which is about 42, which is actually LESS than what we found. Similarly, you have 27 5-digit numbers. The expected probability here is 46/100000, which comes out to an expected value of about 41, which is once again less than the actual turnout. Generally, for a relatively small number of digits, based on chance alone, there are about 40 expected numbers that satisfy the given criteria for all of the integers with a certain number of digits, and since actual result turns out to actually be less than this, I don't see this as anything special.
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Rhythm Simulation Guide Comments, criticism, suggestions, contributions, etc. are all welcome. Piano Etude Demon Fire sheet music |
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12-21-2012, 09:01 PM
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FFR Player
Join Date: May 2008
Age: 32
Posts: 1,117
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Re: Grafting Numbers
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the fact that the square root of a number which isn't a perfect square is irrational can be proven. not just hinted at or shown for a few cases, but proven in full generality (it's a fun exercise imo, if its too tough, you can try proving it for primes first, or even for a specific prime (2 being the popular choice)). And the fact that no irrational numbers are repeating decimals can be proven as well (by showing that any repeating decimal is rational). this, however, does not mean you won't find the number repeated in it's square root (in fact, you'd expect to find it an infinite number of times) assuming the digits are random they should appear infinitely many times with a predictable regularity, though since they're not really random some patterns will happen more/less than expected. all that said, don't be discouraged if you can't find what you're looking for, hunting for patterns like this will probably do more for you math muscles than most of what you would do in high school anyway. |
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01-7-2013, 08:56 AM
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#4 | |
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Spun a twirly fruitcake,
Join Date: Feb 2009
Age: 31
Posts: 3,865
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Re: Grafting Numbers
Finally got to calculating the amount of times a number occurs in its decimals out of sheer chance.
I used this code: Up to a million and up to 100 decimals, the OP's code gives 419 results. This code estimates 538 results out of sheer luck. So you were right, it's just chance. Disappointing, but true... P.S: Yes I know it's off a bit because of the 10-1, which should be 10-0, though the difference is minimal and the conclusion is unchanged.
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Last edited by SKG_Scintill; 01-7-2013 at 09:26 AM.. |
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