Numbers in a Fraction

Date: 05/20/2001 at 11:55:09
From: M. Lawrence
Subject: Is a fraction 1 number or is it 2 numbers?

Hi,

I'm a bit stuck on some schoolwork to do with fractions. Is a fraction 
one number or is it two numbers?

What I've figured out is that a fraction is made up of two numbers, 
the denominator and the numerator. 

For example 1/7. The denominator is 7 and the numerator is 1 - 
representing a fraction. So would this 'fraction' be considered one 
number representing one quantity but made up of two quantities?

On the other hand, it is representing the process of 1 divided by 7. 
So it would then be two numbers wouldn't it? Can it be both 1 and 2 
numbers? Very confused.

Thank you for your time.
M.

Date: 05/20/2001 at 19:53:25
From: Doctor Ian
Subject: Re: Is a fraction 1 number or is it 2 numbers?

Hi M,

If you say that 'Jack is the first child of Bob and Sally', you're 
talking about one person, even though you have to mention two other 
people to talk about him, right?

When you divide 3 by 4, you get a particular number. It's the same 
number you get when you divide 6 by 8, or 9 by 12, or 75 by 100. Those 
are all different pairs of numbers, but the number '3/4' is a single 
number, which happens to have a lot of names.

Does this help? Write back if you'd like to talk about this some more, 
or if you have any other questions.

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

Date: 05/20/2001 at 22:52:43
From: Russell Lawrence
Subject: Re: Is a fraction 1 number or is it 2 numbers?

So the fraction 3/4 is one number representing one quantity but has 
many different names (3/4 is 6/8 is 9/12 etc.) So because 6/8 is 
another name for 3/4 it is a process of dividing 6 by 8 to equal 3/4, 
or a process of simplifying it to equal 3/4. So is 6/8 a process?

So 3/4 then, is one number representing one quantity, and there is no 
process to derive it because it already has a given quantity. So 3/4 
is a single number and not a process? Am I on the right track? Are 
there any fractions that have only one name?

Lastly, is there a fraction for every number on the number line?

Thanks,
M.

Date: 05/21/2001 at 00:18:16
From: Doctor Ian
Subject: Re: Is a fraction 1 number or is it 2 numbers?

Hi M,

Your last two questions are the easiest to answer. There are no 
fractions with only one name, since you can always multiply a fraction 
by n/n to get a new name, e.g.,

     3   2   6
     - * - = -
     4   2   8

     3   3    9
     - * - = --
     4   3   12

     3   4   12
     - * - = --
     4   4   16

And there are many numbers on the number line that can't be expressed 
as fractions. For example, pi can't be expressed as a fraction. Nor 
can e (the base of the natural logarithms). Nor can the square root of 
2, the square root of 3, or the square root of any prime number. These 
are all called 'irrational' numbers, because they aren't rational, 
i.e., they can't be expressed as fractions.

Numbers are very subtle things, as you'll discover if you continue to 
ask the kinds of questions you're asking. We tend to think of integers 
as the sizes of sets - i.e., the number '7' is a label we use to 
signify the size of every set that contains 7 elements. It sounds 
circular, but it's not, since we can construct the sets without naming 
them.

But if numbers are supposed to be set sizes, then what sets have 
'sizes' like 3/4, or the square root of 2? That doesn't make sense, so 
we start to think of rational numbers in terms of subsets. But that 
doesn't work for irrational numbers, so we visualize those as 
distances along an imaginary 'number line'. (Then we see that this 
will also work for rational numbers, so we combine the rationals and 
irrationals to get the 'real' numbers.) And what about negative 
numbers? Well, we add a sign, and visualize it as a direction along 
the line. Then what about complex numbers? Okay, we add another 
dimension.

You seem to be assuming that there is some distinction to be made 
between a 'number' and a 'process'. But is this really the case? We 
determine set sizes by counting, or more accurately, by matching 
corresponding elements in sets. Counting and matching are certainly 
processes, so is a set size a number, or a process? We define certain 
classes of numbers in terms of the kinds of equations that they 
satisfy. Solving an equation is certainly a process, so are those 
solutions numbers, or processes?

When Warren McCulloch, one of the great minds in the theory of 
computation, was asked upon entering Haverford College what he wanted 
to do, he replied, "I have no idea, but there is one question I would 
like to answer: What is a number, that a man may know it, and a man, 
that he may know a number?"  

No one has answered that question yet, but a lot of people have had a 
lot of fun trying! If you're going to join the party (and I hope you 
will), one thing you'll have to learn to do is distinguish between a 
thing and its representations. Whatever this thing '3/4' is, it's not 
any of its representations - '3/4', '3 divided by 4', '4 into 3', 
'0.75', 'xxx_', a point on a number line, and so on - any more than 
you're the same thing as your name or a photograph or description of 
you. And it certainly can't be the process that gives rise to it, 
since there are lots of processes that can give rise to any number, so 
if number is the same as process, then all of the processes that give 
rise to the same number must be the same, which is only possible if we 
dilute the meaning of 'same' until the word is meaningless.

People don't think much about what numbers really _are_, because you 
don't need to know what they are in order to use them - in much the 
same way that you don't need to understand what fire is in order to 
cook with it.

The short answer to the question 'is a fraction one number or two?' is 
really just: A fraction is a representation of a single number, which 
happens to be constructed from two other numbers. If you want to go 
deeper than that, take a look at

   Was Mathematics Invented or Discovered?
   http://mathforum.org/dr.math/problems/angela.1.1.01.html   

to see where that kind of discussion can lead.

Does this help? Write back if you'd like to talk about this some more, 
or if you have any other questions.

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

Date: 05/21/2001 at 01:00:31
From: Russell Lawrence
Subject: Re: Is a fraction 1 number or is it 2 numbers?

Thank you very much for your time and detailed responses, it has 
helped a great deal and is very much appreciated.

M.