When (and why) did the convention that exponents are evaluated from right to left arise?

Question

Earlier, I saw this question on Quora: X^Y^Z Which one do I do first? and the current most-upvoted answer is this:

The ^ operator is not associative, so that:

(X^Y)^Z is not the same value as X^(Y^Z)

For some reason, the convention for exponents is to apply from the top down (or right to left), so that

X^Y^Z = X^(Y^Z)

I am not sure why this is the case. It's just a convention.

That is, $3^{3^3}=3^{27}=7625597484987$, not $27^3=19683$. So, why is it convention to evaluate a power tower from right to left and when did it arise?

Answer 1

This isn't a definitive answer (unfortunately I haven't been able to find any history on the subject when I've looked it up before), but I once noticed that you can write something like $2^{x^2+1}$ and it is clear in which order one must compute the operations. But if you divide that quantity by two it becomes $2^{x^2+1-1}=2^{x^2}$. Why should one suddenly start to have to write parentheses? In fact, by choosing the notation so that $x^y\cdot x^z=x^{y+z}$, we have somehow forced the convention.

Another thing to notice is baseline height: $(x^y)^z$ and $x^{y^z}$ both represent which thing is being exponentiated by the size and height of the superscript (in common math notation). In a strict sense I would say that left-to-right isn't even the convention when laid out typographically like this.

This says nothing, however, about left associativity of exponentiation in computer languages. In fact, TeX itself only knows about superscripts, and those are neither left nor right associative (hence you have to put parentheses and braces to group exponentiation yourself anyway).

Edit: I found more discussion about this at http://www.walkingrandomly.com/?p=4154, which notes that different computational packages (like Matlab and Mathematica) choose differing conventions for exponentiation from each other.

Answer 2

$(a^b){}^{{}^c}$ is the same as $(a^b)^c=a^{bc}$. If it was evaluated left-to-right, we'd have $a^{b^c}=a^{bc}$, so power towers would do nothing.

The way it's usually done, from right to left, has $a^{b^c}\ne a^{bc}$. Ultimately, that's what it comes down to.

EDIT: As mentioned elsewhere, right-to-left is the only way for something like $e^{x^2+1}$ to make sense. That hadn't occurred to me. (EDIT: This stops being a relevant point if we write exponentiation as $a\text^b$ instead of $a^b$, as you've done in part of the question — you'd write it as $e\text^(x\text^2+1)$, and the parentheses tell you how to evaluate it.)