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.

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.