davidbau.com

Supermultiplication

When learning multiplication in the third grade, you are taught that "times is just repeated plus." So it is very natural to imagine that sometime in the fourth grade you will learn some new thing called "supermultiplication" that is "repeated times."

Unfortunately, multiplying repeatedly does not turn out to be nearly as beautiful as adding repeatedly, and so "supermultiplication" is never covered in fourth grade. Exponentiation behaves differently from other operations, because it is neither commutative nor associative.

But that seems a shame. Isn't it a blow to the beauty of mathematics that you can't just do the same thing again? Can't we have some sort of "supermultiplication" that is just as magic as multiplication itself, with commutative, associative, and distributive properties?

Of course you can!

Definining Supermultiplication

Let us steal the symbol # for supermultiplication. Then we define the superproduct of two numbers as one number raised to the log of the other:

a # b = a^log(b)

The base we use for the logarithm can be arbitrarily fixed. Although it might be most "natural" to use natural logarithms, let's suppose we use log₂ just to make some integer values easy to play with.

Then we have, for example:

2 # 2 = 2
3 # 2 = 3

Two seems to behave like one. And you can verify that one behaves like zero.

But unlike exponentiation, it seems to be commutative, and associative, and distributive over multiplication!

3 # 8   =   27   =   8 # 3
(4 # 4) # 8   =   16 # 8   =   4096   =   4 # 64   =   4 # (4 # 8)
        (2 * 5) # 4   =   10 # 4   =   100   =   4 * 25   =   (2 # 4) * (5 # 4)

Passing to logs it is simple to treat multiplication like addition and prove all these properties.

Can you do more?

Working with numbers that are not powers of two ends up passing quickly to nonintegral numbers. Here is a mini supermultiplication table:

2	3	4	5	6	7	8	9	10	11	12
3	5.704522	9.000000	12.818619	17.113567	21.849862	27.000000	32.541577	38.455858	44.726854	51.340702
4	9.000000	16.000000	25.000000	36.000000	49.000000	64.000000	81.000000	100.000000	121.000000	144.000000
5	12.818619	25.000000	41.971848	64.093096	91.676259	125.000000	164.316998	209.859240	261.841773	320.465480
6	17.113567	36.000000	64.093096	102.681405	152.949036	216.000000	292.874192	384.558576	491.995397	616.088429
7	21.849862	49.000000	91.676259	152.949036	235.770868	343.000000	477.416479	641.733812	838.607767	1070.643249

Certainly if you stick with powers of 2, you can end up with all the analogous concepts as with multiplication, such as superfactors of powers of 2, superprime powers of 2, and so on.

But there are a couple interesting twists. There is a whole separate relationship between the "two levels down" operation of addition and that of supermultiplication - is there anything that can be done there? And there is the problem of understanding superproducts of non-powers of two. Can the gaps be filled somehow?

Is there a super-super multiplication that you can build on top of supermultiplication?

Is there such a thing as "submultiplication"? That is, is there some commutative and associative operation that grows more slowly than addition, over which addition is distributive?

What lies beyond multiplication?

Posted by David at August 3, 2007 04:01 PM