# ?, exponentiation, multiplication, addition, ?

1. May 10, 2007

### Loren Booda

What simple operations, if any, precede or succeed the series ". . . exponentiation, multiplication, addition. . ."?

2. May 10, 2007

### Moo Of Doom

..., hexation, pentation, tetration, exponentiation, multiplication, addition, ...

Maybe succession is a good candidate for the next operation in the sequence. I don't think there's much after that, though.

3. May 11, 2007

### chroot

Staff Emeritus
Are you trying to remember PEMDAS?

- Warren

4. May 11, 2007

### Loren Booda

chroot,

More like: multiplication represents repeated additions, and exponentiation represents repeated multiplications, etc. (Thanks for the PEMDAS blast to my algebraic past, though.)

5. May 11, 2007

### Moo Of Doom

Yep, that's the list I gave you.

Tetration ($\uparrow\uparrow$) is repeated exponentiation, right associated:

$$a\uparrow\uparrow b = a^{a^{...^{a^a}}}$$

where there are $b$ $a$'s on the left side.

$$2\uparrow\uparrow 2 = 2^2 = 4$$
$$2\uparrow\uparrow 3 = 2^{2^2} = 2^4 = 16$$
$$2\uparrow\uparrow 4 = 2^{2^{2^2}} = 2^{16} = 65536$$

Pentation ($\uparrow\uparrow\uparrow$) is repeated tetration, also right associated, etc.

Succession is adding 1, so you might say that adding a and b is like adding 1 to a, b times.

EDIT: I use Knuth uparrow notation for the higher operators. This is not strictly speaking a universally agreed upon thing, so you might have to explain it to pretty much anyone you show it to.

Last edited: May 11, 2007
6. May 11, 2007

### Loren Booda

Moo,

That's what I sought. "Succession" as you define it seems reasonable to me. My guess is that the next step involves fractals.