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Re: sqrt2 and 2 equally mysterious, or not mysterious at all, and proof that 1 = 0.999...
Posted:
May 29, 2017 1:52 PM


Here is an rudimentary algorithm that produces all the digits of SQRT(2)/2 in base 2, one by one at each iteration:
http://www.analyticbridge.com/forum/topics/challengeoftheweeksquarerootoftwo
On Friday, April 21, 2017 at 12:02:41 AM UTC7, Franz Gnaedinger wrote: > Irrationals not being numbers, and 1 not being 0.999..., are everlasting > topics in sci.math, popping up all the time. Very simple yet forgotten > methods of early mathematics can dissolve the problem. > > Approximating the square root of 2 by a number column which I reconstructed > in 1979 > > 1 1 2 > 2 3 4 > 5 7 10 > 12 17 24 > 29 41 58 > 70 99 140 > and so on > > 99/70 is already a fine value for the square root of 2. (Analogous number > columns approximate the square roots of 3 and 5 and the cube root of 2.) > Now let us look at the square root of 4 by drawing up the analogous number > column > > 1 1 4 > 2 5 8 > 7 13 28 > 20 41 80 > 61 121 244 > 182 365 728 > and so on > > The ratios, for example 41/20 or 121/61, get ever closer to 2. This means > that 2 as the square root of 4 is equally mysterious as the square root of 2. > Or that the square root of 2 is no less mysterious than 2. > > As for 1 = 0.999... let me begin with the Horus eye series of Ancient Egypt, > 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 or simply '2 '4 '8 '16 '32 '64 > > 1 = '1 > 1 = '2 '2 > 1 = '2 '4 '4 > 1 = '2 '4 '8 '8 > 1 = '2 '4 '8 '16 '16 > 1 = '2 '4 '8 '16 '32 '32 > 1 = '2 '4 '8 '16 '32 '64 '64 > > From this you get > > '2 '4 '8 '16 '32 '64 ... = 1 > > (The Horus eye series had an astronomical function, since a month of 30 > days multiplied by '2 '4 '8 '16 '32 '64 yields 29 '2 '32 days or 29 days > 12 hours 45 minutes for one lunation of 29 days 12 hours 44 minutes 2.9 > seconds, modern average from 1989 AD  mistake of the old value less than > one minute per lunation, or half a day in a lifetime. One eye of the Horus > falcon was the sun, standing for a month of 30 days, his other eye was > the moon, standing for one lunation or synodic month.) > > Another infinite series > > 1 = '1 > 1 = '1x2 '2 > 1 = '1x2 '2x3 '3 > 1 = '1x2 '2x3 '3x4 '4 > 1 = '1x2 '2x3 '3x4 '4x5 '5 > 1 = '1x2 '2x3 '3x4 '4x5 '5x6 '6 > > 1 = '1x2 '2x3 '3x4 '4x5 '5x6 '6x7 '7x8 '8x9 '9x10 '10x11 '11x12 ... > > '1x2 '2x3 '5x6 '6x7 '9x10 '10x11 ... = pi/4 > > Now for 0.999... being 1 > > 1 = 10/10 > 1 = 9/10 + 10/100 > 1 = 9/10 + 9/100 + 10/1000 > 1 = 9/10 + 9/100 + 9/1000 + 10/10000 > and so on > 1 = 0.999... > > Mathematical education should begin with really simple methods, not on > the level of Euclid. Would prevent some people from wasting their life > on kooky notions.



