between addition and multiplication - Printable Version +- Tetration Forum ( https://math.eretrandre.org/tetrationforum)+-- Forum: Tetration and Related Topics ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1)+--- Forum: Mathematical and General Discussion ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3)+--- Thread: between addition and multiplication ( /showthread.php?tid=608) |

between addition and multiplication - lloyd - 03/10/2011
This has probably been thought of before, but here goes anyway. I was thinking about the "sesqui" operation intermediate between adding and multiplying; I'll write "@" here. Obviously a @ b should lie between a+b and ab. Maybe we should take the mean. But which one, arithmetic or geometric? Since one applies to addition and the other to multiplication, why not take both? Then we'll take the mean of these two. But which mean? Again, take both; the proposed value for the sesqui-operation is then the limit of this process when iterated many times. In fact the two values converge quite quickly and for 10-digit precision we usually have convergence within 3 or 4 iterations. Here are some values for a @ a: 1 @ 1 = 1.456791031 2 @ 2 = 4.000000000 3 @ 3 = 7.424041309 4 @ 4 = 11.654328248 5 @ 5 = 16.644985716 6 @ 6 = 22.363401399 7 @ 7 = 28.784583111 8 @ 8 = 35.888457285 9 @ 9 = 43.658368718 10 @ 10 = 52.080163811 11 @ 11 = 61.141591230 12 @ 12 = 70.831889817 13 @ 13 = 81.141493853 14 @ 14 = 92.061815491 15 @ 15 = 103.585079914 16 @ 16 = 115.704197683 17 @ 17 = 128.412664031 18 @ 18 = 141.704478131 19 @ 19 = 155.574077463 20 @ 20 = 170.016283797 21 @ 21 = 185.026258257 22 @ 22 = 200.599463552 23 @ 23 = 216.731631979 24 @ 24 = 233.418738077 25 @ 25 = 250.656975101 26 @ 26 = 268.442734648 27 @ 27 = 286.772588895 28 @ 28 = 305.643275047 29 @ 29 = 325.051681631 30 @ 30 = 344.994836377 31 @ 31 = 365.469895439 32 @ 32 = 386.474133787 I discovered this forum after asking a question recently on sci.math. It looks like people here have been thinking about the same thing: I asked if the next operation after exponentiation should require new numbers, the way that addition/subtraction, multiplication/division, exponentiation/root-taking/logarithms lead from the counting numbers to negative, real and complex numbers respectively. RE: between addition and multiplication - lloyd - 03/10/2011
D'oh, I looked at the FAQ and the mean I proposed is covered in detail as a possible basis for the sesqui operation--it is the agm, ArithmeticalGeometricMean in mathematica, also called the Gauss mean. Oh well if I had to invent something that already exists, at least it's something that Lagrange and Gauss also thought of. Back to lurking. Lloyd. RE: between addition and multiplication - tommy1729 - 03/10/2011
i think you just posted the ancient Arithmetic-Geometric Mean ? tommy1729 RE: between addition and multiplication - tommy1729 - 03/10/2011
lol , seems i was typing my reply while you were posting ... what a waste of time ... RE: between addition and multiplication - JmsNxn - 03/10/2011
What would happen if we created this: x {0} y = x + y x {0.5} y = x @ y x {1} y = x * y x {2} y = x ^ y And then {0.25} will be the same arithmetic-geometric algorithm of {0} and {0.5}; {0.75} will be the arith-geo-algo of {1} and {0.5}, so on and so forth. We could then solve for x {1.5} n, n E N, since: x {1.5} 2 = x {0.5} x Perhaps Taylor series will be derivable giving us complex arguments. It'd also be very interesting to see what happens with logs, i.e: log(x {1.5} 2) = ? since normal operators undergo a transformation I wonder if something happens for these. RE: between addition and multiplication - lloyd - 03/11/2011
Surely, though, {0.25} should be weighted 3/4s towards the arithmetic mean, and 1/4 towards the geometric mean. Ah but is the weighting carried out arithmetically or geometrically? Apply a 3/4 arithmetic : 1/4 geometric weighting there too! And take the limiting case again. In other words, for a {0.25} b, with a<b, m1 of a and b = a + (b-a)*0.25 (0.25 of the way between a and b, judged arithmetically) m2 of a and b = a * (b/a)^0.25 (0.25 of the way between a and b, judged geometrically) Now plug m1 and m2 into a and b, and iterate until you get something stable (i.e. m1 = m2 to whatever degree of precision you need) This seems like a good generalisation to real values between 0 and 1! (03/10/2011, 11:42 PM)JmsNxn Wrote: What would happen if we created this: RE: between addition and multiplication - bo198214 - 03/11/2011
These intermediate operations were topic already in this thread. For me it seems important that the curve t |-> a [t] b is smooth (or better analytic) for fixed a and b. I mean you define it on the interval [1,2], i.e. between addition and multiplication, and then you would continue it to the higher operations t>2 by a [t+1] (b+1) = a [t] ( a [t+1] b ) And then it would be interesting whether the curve is (infinitely) differentiable at t=2, t=3, etc. Of course it would between the endpoints, i.e. on (2,3) and (3,4), etc. RE: between addition and multiplication - JmsNxn - 03/11/2011
(03/11/2011, 12:35 AM)lloyd Wrote: Surely, though, {0.25} should be weighted 3/4s towards the arithmetic mean, and 1/4 towards the geometric mean. Ah but is the weighting carried out arithmetically or geometrically? Apply a 3/4 arithmetic : 1/4 geometric weighting there too! And take the limiting case again. Yes! I like that a lot. RE: between addition and multiplication - martin - 03/11/2011
(03/11/2011, 12:42 PM)bo198214 Wrote: For me it seems important that the curve t |-> a [t] b is smooth (or better analytic) for fixed a and b. IIRC, this is exactly what I tried to do, some three years ago or so... It never really worked out, though some sample graphs for 2 [4] x and 3 [4] x looked quite good. RE: between addition and multiplication - bo198214 - 03/11/2011
(03/11/2011, 10:07 PM)martin Wrote:(03/11/2011, 12:42 PM)bo198214 Wrote: For me it seems important that the curve t |-> a [t] b is smooth (or better analytic) for fixed a and b. Hey Martin, we are not talking about the function f(x) = a [4] x but about the function g(x) = a [x] b. But you are right that we expect the same smoothness also for f(x), which is how Andrew Robbins came to his tetration extension. |