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Topic: [HM] Does Euclid recognize a ratio between equal numbers or equal magnitudes?
Replies: 13   Last Post: Feb 8, 2006 5:53 PM

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 Robert (Bob) Eldon Taylor Posts: 101 Registered: 12/3/04
Re: [HM] Not a fraction?
Posted: Jan 26, 2006 1:24 PM

Sam Kutler wrote: <<I do NOT understand that as OUR fraction 10/71.
Instead I regard it as the whole number TEN of new units each of which
is smaller than our original unit, exactly in the ratio of 1 : 71, >>

This was long my understanding of their fractions, but against this view
they seem to be very reluctant to use a fraction equal to or greater
than the whole. If I say I have a number of quarters in my pocket, you
will not be surprised to know there are four or five or more, because a
quarter is thought of as a distinct unit and not just a fourth of a
dollar. So if Archimedes or others were thinking of third or fifth as
new units, as you suggest, it seems they would speak also of four of
those thirds or five of those fifths. But they never seem to do this.
What they seem to be thinking of is the "part" and "parts" defined in
Euclid which, as we have recently explored, are never equal or greater

What do you make of Newton's definition of number in Universal Arithmetick?

By Number we understand not so much a Multitude of Unities, as the
abstracted Ratio of any Quantity, to another Quantity of the same Kind,
which we take for Unity. And this is threefold; integer, fracted, and
surd: An Integer is what is measured by Unity, a Fraction, that which a
submultiple Part of Unity measures, and a Surd, to which Unity is
incommensurable.

If a number is a ratio and a ratio is a relationship, and we can't add
relationships, how could Newton add numbers?

Barney Hughes wrote: <<They are taught that fractions are rational
numbers. From this they jump to the conceptual equality of ratios and
fractions. >>

Unless you are making an historical point, I think I will find myself as
mystified as your student teachers must have been. What am I to
understand the distinction between fraction and rational numbers in
the modern context? Is 2/3 a rational or a fraction? I guess my
puzzlement is because I am not clear what definitions are used for these
things. But let me bring this to a question:

if a and b are numbers (whole, fractions, rationals) and a = b, are a
and b necessarily the same?

I had supposed that, formulated in an Euclidean context, this question
yields answers different from the modern context, but perhaps I am wrong.

Regards,
Bob
Robert Eldon Taylor