## August 03, 2007

### 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 = alog(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

An excellent piece of thinking to come up with supermultiplication. I'm looking forward to experimenting with it on my calculator when I've got time.

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Posted by: Robert at December 5, 2018 01:55 AM