Symbol for exponential base?

Question

There's $\sum$ for summing, $\prod$ for multiplication, $\bigcap$ for intersection, $\bigcup$ for union, $\bigvee$ and $\bigwedge$ for logic, $\coprod$ for coproduct, and even $\bigsqcup$ for disjoint union.

Why don't we have one for exponential base? for example $$\LARGE \Delta\normalsize_{k=1}^{6}k=6^{5^{4^{3^{2^{1}}}}}$$

Why? I know that an exponent symbol would be useless because $(a^b)^c=a^{bc}$ but I can't think of any reason that a symbol for that doesn't exist?

Answer 1

There's a very simple reason the symbol doesn't exist:

It's not that widely useful

---

During my studying of mathematics, I used

• The symbol $\sum$ on a daily basis. There are periods of several months when I probably wrote the symbol down tens of times per day. I probably wrote down the symbol $\sum$ at least hundreds, if not thousands of times.
• The symbol $\prod$ on a semi-weekly basis. I probably wrote down that symbol at least 100 times in my life.
• The symbols $\bigcup$ and $\bigcap$ were used on a daily basis during my logic classes and during topology and measure theory. I'd bet I have at least $1000$ uses of the symbols in my life
• $\bigwedge$ and $\bigvee$ I used maybe a couple of times each month, still easily several hunderd times in my life (most of it during the introductiory logic classes).
• I went through an introduction to category theory class, and the notebook from that class is basically one $\coprod$ sign next to the other - again, easily hundreds of uses of the symbol
• I don't remember using $\bigsqcup$ all that much, but then again, I also don't remember using $\sqcup$ all that much.

Compared to that, if the symbol you propose existed, I would use it maybe 10 times during my whole time as a mathematitian. Orders of magnitute less than the other symbols.

Answer 2

Notation is not invented for the sake of notation (well, ideally it is not, anyway), it is invented to make our lives easier. We need to express sums, products, intersections, unions, disjunctions, conjunctions, coproducts and disjoint unions of more than than a few elements (or a variable number of elements) frequently; the same is not true for exponential towers.

Of course, you are free to define your triangle the way that you have; however, it is not likely to catch on more generally, since few people will have a use for it.

Answer 3

There is a name for what you're describing - it's called an 'exponential factorial', as referenced by Wolfram MathWorld and Wikipedia.

On both of those pages, I couldn't find any accepted symbols for the exponential factorial. This may be because the sequence gets big very quickly. For example, the exponential factorial of $5$ is $5^{262144}$, which is approximately $6.2 * 10^{183230}$.

This may make it not useful because I don't think there is any 'urgent' need to describe numbers this large. How often do you even come across a situation you need to use this situation? @5xum's answer shows how much all of these other symbols are used compared to this one.