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### Numbers in a Fraction

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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.

M.
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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.

or if you have any other questions.

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

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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
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.

or if you have any other questions.

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

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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.
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Associated Topics:
Elementary Fractions