tag:blogger.com,1999:blog-20067416.post6959698550661676893..comments2021-09-24T17:58:19.157-04:00Comments on Recursivity: Math Challenge #1Unknownnoreply@blogger.comBlogger6125tag:blogger.com,1999:blog-20067416.post-27979032894257398372014-01-26T09:32:55.312-05:002014-01-26T09:32:55.312-05:00Of course, there is a stupidity in what I wrote. ...Of course, there is a stupidity in what I wrote. We also need to take into account the parity of the denominators. In particular I should have said "Since r/r' is a convergent and r' is odd, r/r' is close to an odd multiple of Pi, so cos(r) is very close to -1; since q' is even, cos(q) is very close to 1."<br /><br />Duh!Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-62776833855612609812014-01-25T07:07:06.595-05:002014-01-25T07:07:06.595-05:00All answers so far are good, but incomplete. Pseu...All answers so far are good, but incomplete. Pseudonym got the farthest, but even his/her answer leaves a bit to be desired. <br /><br />Commenters noticed that the numbers 333, 355, and 22 are the numerators of good rational approximations to Pi. But how do we find such good approximations? They come from the <a href="http://en.wikipedia.org/wiki/Continued_fraction" rel="nofollow">simple continued fraction</a> [3,7,15,1,292, 1, ...] for Pi. The terms of the continued fraction are called partial quotients.<br /><br />The numerators are, consecutively, 3, 22, 333, 355, 103993, 104348, ... But while sin(333)+sin(355) is very close to sin(22), it's not the case that that sin(355)+sin(103993) is close to sin(333). So it's not simply that these are three consecutive convergents to Pi.<br /><br />The solution is to take the numerators of three consecutive convergents to Pi,<br />say p, q, r <b>where the partial quotient associated with r is 1</b> and occurs in an odd position. Then we have p + q = r and we also have p/p', q/q', and r/r' are all close to Pi.<br />Now use the addition law for sines: sin(p) = sin(r-q) = sin(r) cos(q) - sin(q) cos(r). Since r/r' is an odd convergent, r/r' is slightly greater than Pi, so cos(r) is very close to -1. Since q/q' is an even convergent, q/q' is slightly less than Pi, so cos(q) is very close to 1. Thus we get that sin(p) is very close to sin(r) + sin(q). So, for example, another pseudo-identity is sin(103993)+sin(104348) = sin(355).<br /><br />If the 1 appears in an even position in the continued fraction, then we get a similar pseudo-identity with the order scrambled slightly: e.g., <br />sin(104348) is roughly sin(103993)+sin(208341).Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-59956835566767672742014-01-24T01:00:29.860-05:002014-01-24T01:00:29.860-05:00There's always something more after I press Pu...There's always something more after I press Publish.<br /><br />I forgot to mention there's a second requirement:<br /><br />sin x + sin y ≈ sin z<br />AND<br />x + z = y (note the slightly different order)<br /><br />I should have written:<br /><br />sin 103993 + sin 208341 ≈ sin 104348<br />AND<br />103993 + 104348 = 208341 <br /><br />Nevertheless I lack a good way to produce these numbers other than looking them up. It may be that there are other good answers using the values in {0,1,2,...,359}.Randyhttps://www.blogger.com/profile/06294841118508802764noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-506750323330682732014-01-24T00:18:38.628-05:002014-01-24T00:18:38.628-05:00sin 355 = cos 22 sin 333 + cos 333 sin 22
Since a...sin 355 = cos 22 sin 333 + cos 333 sin 22<br /><br />Since and 333 is approximately 106π and 22 is approximately 7π (these are from the continued fraction expansion of π), the result more or less follows.<br /><br />The next one in the series is (and again, it's not a true equality):<br /><br />sin 103993 + sin 104348 = sin 355Pseudonymhttps://www.blogger.com/profile/04272326070593532463noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-23217205921859598562014-01-24T00:01:10.912-05:002014-01-24T00:01:10.912-05:00Interesting. First thing I note is that the angle...Interesting. First thing I note is that the angles must be in radians. It doesn't work in degrees, even though the numbers appear to be in the interval suitable for degrees.<br /><br />So, since π is important in radians, what do these look like in π?<br /><br />355 ≈ 113.00π<br />333 ≈ 106.00π<br /> 22 ≈ 7.00π<br /><br />sin θ ≈ 0 where θ ≈ kπ, k∈ℤ<br /><br />sin repeats itself every 2π<br />|sin| repeats itself every π<br /><br />Further, because sin is nearly vertical at these locations, we can note:<br /><br />sin ε ≈ ε where ε is "small"<br /><br />Let's keep 10 digits after the decimal (so we can round to 9) and separate out the whole π pieces.<br /><br />(I just used subtraction of the above approximated values to get the decimal digits)<br /><br />355 ≈ 113π + 0.000 030 144 4 <br />333 ≈ 106π - 0.008 821 280 5 <br /> 22 ≈ 7π + 0.008 851 424 9 <br /><br />sin 355 ≈ sin (113π + 0.000 030 144 4)<br />sin 355 ≈ -sin 0.000 030 144 4<br />sin 355 ≈ -0.000 030 144 4<br /><br />sin 333 ≈ sin (106π - 0.008 821 280 5)<br />sin 333 ≈ -sin 0.008 821 280 5<br />sin 333 ≈ -0.008 821 280 5<br /><br />sin 333 + sin 355 ≈ -0.008 851 424 9<br /><br />sin 22 ≈ sin (7π + 0.008 851 424 9) <br />sin 22 ≈ -sin 0.008 851 424 9<br />sin 22 ≈ -0.008 851 424 9<br /><br />What is "similar"? It seems we're required to find unique whole numbers x,y,z such that sin x + sin y ≈ sin z. My guess is there aren't enough in the set {0,1,2,...,359} to appear to be degrees. <br /><br />If we're looking for small error values, approximations of π seem to be a good place to start.<br /><br />22/7 has been used.<br />223/71 is available, but has an error large enough that I don't want to deal with it.<br />333/106 has been used.<br />355/133 has been used.<br />I'll use these approximations (from Wolfram MathWorld and oeis.org)<br />103993/33102<br />104348/33215<br />208341/66317<br /><br />There's probably a better way, but trial and error produces this almost-identity, good to 15 or so digits:<br /><br />sin 103993 + sin 208341 ≈ sin 104348<br />(I can't bring myself to use =)Randyhttps://www.blogger.com/profile/06294841118508802764noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-20783129820495383802014-01-23T20:49:05.433-05:002014-01-23T20:49:05.433-05:00Well, if you think about it as a Taylor series it ...Well, if you think about it as a Taylor series it isn't so hard. 22/2*pi is very close to 3.5, likewise 333/2pi is 52.999 and 355/2pi is 56.000...<br /><br />So each of these have second order Taylor approximations of: 0 +/- ~1*(n - (pi*N))<br /><br />Where n is a small displacement and N is an integer. Arlohttps://www.blogger.com/profile/01043166301709318088noreply@blogger.com