The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

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,


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:


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 ...


... 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

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

  Defining Decimal Numbers 

- Doctor Peterson, The Math Forum 

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.

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: 

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 

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.

Associated Topics:
High School Number Theory

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.