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0-Based Counting

Date: 07/05/2003 at 15:37:40
From: Anonymous
Subject: 0-based counting: Why should we abolish 1-based?

I am trying to convince people, especially my friends, that you 
should count from 0. I.e. "Page 0, page 1, page 2."

For example, when I numbered a score (musical composition) from 0, a 
person complained and argued that 1-based counting is the standard [in 
this case]. So I need a little more information, and especially very 
convincing proof or statements.

One person suggests that I am going against the way everyone else 
does it, and "everyone is taught to count from 1."

I have many reasons why you should count from 0.

- Modulo based systems (like hour:minute:second, century:year, 
- Counting from 1 and 0 is inconsistent.
- 0 is the first digit in the number system, so why not start from it?
- Most programming languages have 0-based arrays.
- A lot of math stuff counts from 0, like the Taylor series.

Date: 07/06/2003 at 23:12:59
From: Doctor Peterson
Subject: Re: 0-based counting: Why should we abolish 1-based?


I certainly wouldn't propose that we abolish 1-based counting, because 
it is very useful. When you count objects, you are making a 
correspondence between the natural numbers and the objects: 1, 2, 3, 
... . The last number you name is the number of objects you counted. 
That doesn't work if you start at zero.

What you are talking about is not really counting, but naming items in 
a sequence, such as pages. Even there, it is very natural to start at 
1, so that the numbers you assign to the items correspond to their 
ordinals: first, second, third. It would be very confusing to call the 
first page zero, when "page 1" and "first page" are synonymous. So 
your friends are right when they say that this is the standard way to 
label page numbers, etc.; and trying to change that amounts to trying 
to change the English language.

But you are right that there are many situations in which zero-based 
counting is appropriate, particularly in mathematical settings. Most 
sequences and series are easier to express in a zero-based form, since 
then the index is the number of steps away from the first term, "term 
0," and we can add (for an arithmetic sequence) or multiply (for a 
geometric sequence) by the same thing each time, or use x^n in a 
series expansion of a function. In programming, similarly, if you use 
0-based indexing, the index is the offset from the storage location of 
the first element.

Yet when you do this, you introduce problems, since the number of 
terms or array elements is not the last index, but one more than that, 
which can be confusing. (Basic has a particularly poor way of handling 
this; C is just confusing to beginners.) So it would be hard to say 
that 0-based counting is the ideal way to assign indexes.

In general, I would say that zero-based counting is appropriate in 
technical contexts (computers and math), but one-based counting is 
standard in other settings, and to try to change that would be 

On the other hand, it is interesting that in many countries, such as 
England, a sort of zero-based counting is used for floors in a 
building: the bottom floor is called the ground floor, and the term 
"first floor" is used for what Americans call the second floor. That 
is, "floor" means a floor above the ground. They don't use the term 
"zeroth floor." Any other situation where the first of a sequence is 
distinct from those that follow might naturally follow the same 
scheme, such as "cover page" and "page 1, 2, ...". But if all pages 
are similar, then it makes most sense to follow tradition.

I'm not sure what you are saying about "modulo-based systems," by 
which you seem to mean systems of units of time. Are you saying we DO 
use zero-based counting there, or that it would solve some perceived 
problem? I do see that using times like 0:30 rather than 12:30 would 
work better, but that is not counting; and if you think that calling 
the first century the "zeroth century" would help, what would you call 
the first century BC? See this page for some thoughts on that:

   The Second Millennium 

- Doctor Peterson, The Math Forum 

Date: 07/07/2003 at 10:52:45
From: Anonymous
Subject: Thank you (0-based counting: Why should we abolish 1-based?)

Thank you so much for your answer. I've tried searching the Web for 
this topic, but the closest thing that I found was the awkwardness in 
handling 1-based years (i.e. "2001 is new millennium, not 2000). Once 
again, thanks a lot!
Associated Topics:
Middle School Number Sense/About Numbers

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