Number Base Convention, Consistency -- and Context
Date: 05/08/2012 at 12:27:17 From: John Subject: What is An Internally Consistent Notation for bases I am having trouble with the "common" manner of writing a base, since the value ascribed to the base doesn't exist in the set of characters used. Binary, for example, cannot be "base 2" since in binary there is no such thing as "2." Worse still, "base 10" could mean base (1 + 1) or (2 + 1) or (3 + 1) or .... Similarly, hexadecimal is usually referred to as being "base 16," but 16 in hex is 22 in denary. To assume denary without any specific mention of that ground rule is surely misleading at best. To write the base used for any number, is there any generally recognized method that is internally consistent? I tend to use the "highest character in the set + 1," as this only requires that we agree on the order of the characters and is internally consistent. With my system, binary uses base (1 + 1) and hexadecimal uses base (F + 1). Simple! Both are absolutely clear and cause no confusion. See what I mean? Why the illogicality and lack of internal consistency of the implied assumption of denary? My method of expressing the base of any number appears to me to be unambiguous, given only the order of the counting numbers. So what do you experts use, and why? Yours with thanks, John P.S. I also tend to use the word "denary" for the base (9 + 1) system rather than the more common "decimal" because of the confusion with the use of "decimal point" or "decimal comma" amongst those who are learning about bases for the first time. I do, of course, explain why I prefer this term, and point out that my choice is less common.
Date: 05/08/2012 at 17:04:00 From: Doctor Peterson Subject: Re: What is An Internally Consistent Notation for bases Hi, John. I'm not entirely sure precisely what "your method" is. Are you saying that you would represent a number in, say, base 6, in the form on the left, below, rather than the one on the right, where the 6 is universally understood to be in base ten? 142 142 5 + 1 6 Using 5 + 1 instead of 6, so that there is no ambiguity about the base, is an interesting idea; and certainly it's possible for beginners to get a little confused. But the latter is a well-understood convention, so there is nothing "misleading" about it, while your method seems unnecessarily confusing, complicated, error-prone -- and would, in your example, essentially get read as "base 6" anyway. Some books (perhaps at an elementary level) write out the base in English rather than as a numeral for the same reasons: 142 six We probably don't do this at higher levels just because it is awkward (and assumes that English is the standard language). The reason we can unambiguously write this ... 142 6 ... is that we all agree that the base is written in base ten. If, say, different countries used different bases, then your concern would be justified. For bases ten and higher, there is certainly an ambiguity, especially for "base 10," which would mean no more than "this is written in my preferred base"! In particular, your suggestion of F + 1 for hexadecimal does remove the ambiguity, as long as there is no variation in the choice of digits beyond 9. If, for instance, we were to adopt the notation used in some elementary books, of representing T for ten and E for eleven in base twelve, base "E + 1" could mean either base twelve or base fifteen. But for bases less than ten there is no real ambiguity, since the digit 6 has the same meaning in any base that uses it. The main purpose for what you suggest, I think, is not ambiguity but just "internal consistency," by which I think you mean that it must use only symbols that are used within that base. Again, it's an interesting thought; but you only need to identify the base when you are communicating with people who use a different base. When we all use base ten every day, we don't bother writing a base subscript, because we know what we mean. A culture that knew only base six would just write "142" for our number, and likewise wouldn't need to specify the base. But once you communicate in the context of various bases, internal consistency is no longer really needed. We just choose a "universal language" in which to talk about other bases, and base ten seems perfectly acceptable. On the preference for denary to decimal, I'm in agreement with you. The word "decimal" has been so thoroughly misused as to have lost most of its meaning in practice, so it makes sense to use "denary" for the base and something else for the "decimal point" and related usage ("fraction separator?"). I don't know if you or I will change common usage in this area; here is one place where I've commented on this, without offering a solution to the problem: Defining Decimal Numbers http://mathforum.org/library/drmath/view/65238.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 05/08/2012 at 18:28:16 From: John Subject: Thank you (What is An Internally Consistent Notation for bases) Dr. Peterson, Thanks so much for a very fast and very extensive response. You got "my method" correct -- with a better layout, to boot! And yes, the problem was originally "teaching" (on separate occasions) a couple of young people in a tropical jungle village who approached me and said, "I didn't understand 'Miss' today when she was on about 'binary.' Can you help, please?" (Often, "Miss" apparently didn't understand it herself and was teaching by rote the algorithmic conversion of denary to binary and reverse but not teaching for understanding.) I had not met T and E for the top two characters in dozenal/duodecimal/base_12/base_B + 1, so that "problem" had not even occurred to me. Plus there are a few languages (peoples) which, I understand, don't (or didn't) use denary, Mayan being the obvious example (vigesimal). Memory says that one or more use duodecimal, though I can't find the reference. I just think it is illogical to not specify a base in the characters of the set delimited -- yes, internally inconsistent! I got to thinking about this problem because one of "my pupils" recounted what happened when "Miss" wrote the old chestnut on the board: "There are 10 types of people in the world: those who understand binary and those who don't." She got members of the class to read it -- and they all did it wrong until at last she let "my pupil" have a go. Now, generally he is thought to be pretty slow, but on that day, apparently, he was enthusiastic to answer. And he was the only one who got it "right," reading "10" as "One Zero" rather than "Ten." This surprised her, confused the other students, and increased his self-confidence no end as he understood why and could explain his answer: "There is no such thing as ten in binary!" It is on a par with the SI system's insistence that the unit of Mass is the kG but refuses to allow me to use mkG or kkG -- a problem my physics teacher acknowledged nearly 60 years ago when he stated that the Standards bodies were proposing the word "Einstein" (= 1kG) as the unit of mass. Try teaching decimal multiples and submultiples to a class in a jungle village with this anomaly in the system! ;-) And I loved your answer to Michael -- brilliant work. Again, very many thanks. John Retired electrical (power) engineer
Date: 05/08/2012 at 23:09:59 From: Doctor Peterson Subject: Re: Thank you (What is An Internally Consistent Notation for bases) Hi, John. With teaching as the context, probably writing the base as a word is the standard solution, and easiest to work with when the base must be specified. I think it's best wherever possible just to say "everything on this page is written in base 6" and not bother with subscripts at all. One thing that does is to get the student into the mindset of someone who knows only that base. (I even like to start with base 5 and have a backstory explaining how the imaginary culture developed this base, by using one hand to count.) Conversion to other bases would come in only when this culture has to, say, change to American currency, at which time they can invent some way to indicate which system is being used. You're right that many cultures formerly used other bases than ten, and many still have leftover fragments of it. I ran across a list here: http://en.wikipedia.org/wiki/Positional_notation#Applications Those cultures, of course, didn't have to state what their base was! I like the joke you mention (and others like it). The key to it is, of course, that the base is not indicated. You have to "think outside the box" and look at it as if another base were your native system. By the way, speaking of anomalies, if you look at the list of non-denary cultures in the link above, it's interesting to note how many of them (including the famous Mayan, in their calendar system not mentioned there) have some quirk that makes them not really consistent examples of their base. Inconsistency is part of human nature! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 05/09/2012 at 02:50:33 From: John Subject: Thank you (What is An Internally Consistent Notation for bases) Bless you; another very complete response. Thanks so much. John
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.