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between addition and multiplication
#1
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.
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Messages In This Thread
between addition and multiplication - by lloyd - 03/10/2011, 09:10 PM

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