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 between addition and multiplication lloyd Junior Fellow  Posts: 10 Threads: 1 Joined: Mar 2011 03/10/2011, 09:10 PM 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. « Next Oldest | Next Newest »