Interpreting 2^3^2

Date: 04/26/2001 at 10:10:52
From: Douglas
Subject: Mysterious exponent definition

I am currently taking a computer science course at a four-year 
university. In my class my professor asked for the answer to this 
problem:

     2^3^2

There were no parentheses in the problem at all. He purposely wrote it 
without them.

I told him that the answer was 512 (2^9). He informed me that I was 
correct, but he wanted me to document my results. He wanted to know 
where I found the rule or definition that is needed in order to solve 
the problem (to back up my work). I searched and searched, I looked 
for it my old Calculus textbooks and many other books. Finally I 
looked on the Internet, and that's where I found your site. He later 
told the class that he also has never been able to find the 'rule' or 
'definition' that states that this is the correct way in which to work 
the problem.

When sorting through your vast archives I came across a similar 
problem called Hard Powers at:

   http://mathforum.org/dr.math/problems/pearce12.11.97.html   

In it, the problem was 9^9^9. Dr. Mark responded:

   "When you write two exponents, as you did above or as in a^b^c, 
   this is (by definition) equal to a taken to the power (b^c)."

My question to you is, what definition are you referring to in the 
above problem?

My professor says that a lot of mathematicians take it on faith, that 
this is just the way to do it, as I did when I solved it.

I really hope that you can help me with this problem.

Thank you,
Douglas

Date: 04/26/2001 at 12:15:15
From: Doctor Peterson
Subject: Re: Mysterious exponent definition

Hi, Douglas.

I wouldn't quite call it "faith"; just an unwritten (or seldom 
written) convention. It's not that we "believe" it's true; we just 
"know" from experience that this is the usual way to read exponents.

I've occasionally wondered where I could find an "official" authority 
on this, myself. I know that a^b^c is generally taken as a^(b^c), and 
that the reason is that (a^b)^c can be written as a^(bc), while the 
other form, which is more interesting, has no alternative. We've 
occasionally mentioned this in Dr. Math answers, but are we an 
authority? I recall learning it from some book when I was in school, 
quite possibly not a textbook. But how can I prove it to a skeptic? 
I've been looking to see if some math organization has adopted 
standards, but have never found one to which I can refer.

Actually, the order of operations rules, so far as I have found, are 
not decreed by any authority, but have gradually become a commonly 
accepted standard through informal consensus, perhaps with the help of 
textbook authors, who tend to lay down rules more than actual 
mathematicians. The best thing to do is to show what mathematicians 
actually do.

Here are a few Web pages I've run across where such expressions are 
used, which will at least show common practice.

First, this page from Texas Instruments shows that not all calculators 
follow this rule, but that they acknowledge it as common:

   Order of operations - regarding exponents (2^3^4). Why does 2^3^4
   evaluate as (2^3)^4 instead of 2^(3^4)? - Frequently Asked
   Questions - TI-83
   http://education.ti.com/product/tech/83/faqs/faq30178.html   

It says that one option they had in designing calculators was to: 

   "associate right to left, as we learn in many algebra texts."

Next, here's a page on from MathSoft that uses this "customary" rule, 
but defines it in case we don't know:

   Iterated Exponential Constants
   http://www.mathsoft.com/asolve/constant/itrexp/itrexp.html   

It says:
                                   c            c
                                  b           (b )
   (Henceforth by the expression a   we mean a    , as is customary.)

So I think it's clear that a^(b^c) is the usual interpretation of 
a^b^c, but not so much so that authors can comfortably assume that all 
readers will follow it without a reminder to make sure we agree. As 
far as I know, everyone who uses such expressions uses them in this 
way; I've never seen the other way given as the "right" way. The most 
one could say in the other direction would be that there is no firm 
rule and that one should state the rule before using it.

Since you are in Computer Science, there is another approach. If, 
rather than simply writing math, you are writing in a computer 
language, just check the manual; that's your authority. You'll find, 
I believe, that in Visual Basic the exponent operator ^ associates 
left-to-right (as TI calculators do), while in Fortran ** associates 
right-to-left (as mathematicians do). I'll leave it for you to decide 
whether Microsoft or the creators of Fortran are more likely to get it 
right.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/