History of Writing Decimals and Number Lines

Date: 10/28/98 at 19:51:21
From: Kim
Subject: Integers and decimals

A student asked me why on a number line the negative numbers are on 
the left, and when you deal with decimals, the whole numbers are on the 
left. I am new to teaching math, and I understand his need to know 
but have a difficult time explaining it. Help!

Date: 10/29/98 at 12:31:04
From: Doctor Peterson
Subject: Re: Integers and decimals

Hi, Kim,

This is one of those "why" questions that can be hard to answer. The 
basic answer is that there is no connection between the two. People 
just chose those directions. But it is interesting to think about why 
each is the way it is.

The number line probably was first drawn before there were negative 
numbers. It would be natural to start at zero (or one) and count to 
the right, in countries that write from left to right. Then of course 
the negative numbers go on the left.

The same sort of thing would have happened with decimal digits. Once 
people were used to writing integers with the least significant digit 
(the ones place) at the right, then when decimals were invented the 
decimal point would have to go to the right of that, and decimal places 
farther to the right.

So the real question is simply, why are integers written left to right 
starting with the most significant digits? 

As far as I know, this is an old tradition, used not only in our arabic 
numerals but also in Roman, Greek, Egyptian, Babylonian, and whatever 
else there was. It makes some sense to start with the most important 
digits, just as you might start a list of people with the king, then 
his advisors, and so on down to the peasants. That's different from the 
number line, because there is no highest number to put down first on 
the line! You can't start at the high end, so you start at the low end, 
where there is a definite beginning (at least until negative numbers 
were invented) and you usually count up, not down.

You and your student may be interested to know that computer designers 
have argued over the direction numbers should be "written" in the 
computer's memory. Many computers, such as the Motorola 68000 series 
microprocessors I work with (used in Macintosh), store the most 
significant digit first, the way we write them. This is called, in a 
humorous reference to a dispute in Gulliver's Travels over which end 
of an egg should be eaten first, the "big-endian" method. Others, such 
as the Intel series (used in PCs) store the least significant digit 
first, which makes a lot of sense for some purposes, such as when you 
don't know whether a small number is stored as 00000052 or as 52. On a 
"little-endian" computer, you just read the first two digits, 
regardless of whether it is stored as "25" or as "25000000". On a big-
endian computer, you have to be told how it is stored. On the other 
hand, the big-endian method allows you to sort numbers the same way 
you alphabetize words, looking first at the leftmost digit and working 
your way to the right.

So the choice of writing numbers in decreasing order of significance 
as we do is entirely arbitrary, but has some reasons behind it.

Thanks for the question! Hope it stimulates some good conversations.

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