Defining Decimal Numbers

Date: 07/01/2004 at 19:45:01
From: Michael
Subject: examples of decimals

Can you tell me examples of decimals?  Is 6.5 a decimal?  Is 
3.141592.....a decimal?  I'm getting confused because I know that 6.5
has a whole number in it, so is it also a decimal?

Date: 07/01/2004 at 22:49:24
From: Doctor Peterson
Subject: Re: examples of decimals

Hi, Michael.

The word "decimal" is actually a very tricky one, and I'm not quite 
sure that there is a correct answer to your question!  The word is 
used so loosely that most uses of it are really wrong.  If you're 
confused, it's because the world is confused.

Properly speaking, since the "deci-" in the word means "ten", ANY 
number written in a base-ten system (that is, with each digit worth 
ten times as much as the one next to it) can be called a "decimal 
number".  When we write "123", the 3 is worth 3 ones, the 2 is worth 
2 tens, and the 1 is worth a ten of tens, or a hundred.  That is 
decimal.

But we also talk about "decimal fractions", which are non-whole 
numbers written in that way, using a decimal point.  The trouble 
comes when people shorten that to "decimals", dropping the part of 
the phrase that really says what they want to say (that it is not a 
whole number) and keeping the part that is true of any number we 
write today.  A decimal fraction, also called a decimal, is a number 
with a decimal point in it, like 1.23. 

Then people talk about the "decimal part" of a number, which really 
means the FRACTIONAL part, the non-whole part.  This is even worse, 
making it sound like the 1 in 1.23 is not decimal while the .23 is. 
The reality is that both parts are "decimal", but only the .23 is 
fractional.  But the common usage can make us think that 1.23 is not 
a decimal, but a mixed number, like 1 23/100, and that only numbers 
less than 1, like .23, can be called decimals.  The point that 
divides the two parts, of course, is called a "decimal point", when 
what it marks is not really the start of the decimal part, but of 
the fractional part.  I wish we called it something else, like 
"fraction point".  But of course, a fraction is written like 23/100, 
so that would be confusing too.

If you have heard about writing numbers in bases other than ten, such 
as base 5, we can get really confused when we want to talk about 
fractional numbers written in the place value system.  Are they "base 
5 decimals", when "decimal" means "base ten"?  I don't think so!  But 
we don't really have a good word for it.

I hope this hasn't just confused you more.  I just want to show why 
you may get different answers to your question.  In my opinion, any 
number with a decimal point in it would be commonly called a decimal, 
not just a number less than 1.

When I answer a question about words, I like to check with a 
dictionary, to see if the lexicographers, who define words for a 
living, and constantly watch how words are really used, agree with 
my impression.  Here is what Merriam-Webster (m-w.com) says:

  Main Entry: decimal
  Function: noun
  :any real number expressed in base 10; especially: DECIMAL FRACTION

  Main Entry: decimal fraction
  Function: noun
  :a fraction (as .25 = 25/100 or .025 = 25/1000) or mixed number
  (as 3.025 = 3 25/1000) in which the denominator is a power of 10
  usually expressed by use of the decimal point 

I think they agree exactly with me:  a "decimal" is any number in base 
10, but is usually used specifically for a number with a decimal 
point.  So the answer to your examples is "yes".

If you have any further questions, feel free to write back.

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