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/
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