# Is there a notation for the repetition of basic operations?

I mean...

• multiplication is the repetition of addition:
• $2*2 = 2+2$
• $3*3 = 3+3+3$
• exponential is the repetition of multiplication:
• $2^2 = 2 * 2$
• $3^3 = 3*3*3$

.. it is an obvious pattern.

I propose:

• Multiplication is rank 2
• Exponential is rank 3
• etc ...

This would mean

$H(r, x)$ is the $r$th rank operation on $x$.

For example: $H(3, 2) = 2^2$

and $H(4, 3) = {^{^33}}3$

It is already kind of wire to get this value since it is quite big. But the function of the rank is certainly one that rises fast.

• @Simply Beautiful Art Now I can fully appreciate your reasons. But those come from the way hyperoperations are defined, whereby what precedes the addition "degenerates" to a unary operation. But I was talking about tropical operations: with them the tropical operation (immediately) preceding the addition is still binary. Over this new lower operation the addition is distributive, as the multiplication is distributive over the addition. This is the abstract definition, which more than one concrete operation may satisfy. One of this is $x*y=\max{(x, y)}$. You can see that $(x*y)+z=(x+z)*(y+z)$ – trying Jul 27 '17 at 17:25
• Hm, nifty $~{}~$ – Simply Beautiful Art Jul 27 '17 at 17:26
There is also the notation $a^{n*}$ for $\underbrace {a*\cdots*a}_{\text {$n $factors} }$. Using this notation we have $$na=a^{n+}$$ $$a^n=a^{n\cdot}$$