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Re: Re:apples and oranges





-----Original Message-----
From: Richard Grandy <rgrandy@ruf.rice.edu>
To: phys-l@atlantis.uwf.edu <phys-l@atlantis.uwf.edu>
Date: Friday, October 30, 1998 10:21 PM
Subject: Re:apples and oranges


Perhaps you could give an example of what other choices we might have
made
in inventing the negative or irrational numbers.

Richard Grandy
Philosophy
Rice University


Hi Richard,

I certainly am not able to come up with an alternate specification of an
irrational number, but history can. I can only quote from a dusty volume
on my shelf (thanks for prodding me to open it!) - "Mathematics in
Western Culture", Morris Kline, 1953, Oxford U Press, pp. 36 - 38.

Begin quote:

"The irrational number is a much neglected topic in the history of
thought and a troublesome member of our number system. . . . Such
numbers must be used in order to represent lengths and they are,
moreover, explicitly and implicitly involved in almost all of
mathematics. Yet how can we add, subtract, multiply, or divide such
numbers? For example, how can we add 2 and sqr(2)? How do we divide
sqr(7) by sqr(2)?

The Babylonians had a makeshift, though practical, solution of these
difficulties. They approximated the value of sqr(2). For example, since
the square of 14/10 or 1.4 is 1.96, and since 1.96 is nearly equal to 2,
1.4 must be nearly equal to sqr(2). An even better approximation to
sqr(2) is 1.41 because the square of 1.41 is 1.988.

The Babylonian approximation to sqr(2) does not permit exact reasoning
with irrational numbers, for no matter how many decimal places we are
willing to use we cannot write a rational number whose square is exactly
2. Yet, if mathematics is to merit its claim to being an exact study, it
must evolve a method of working with sqr(2) itself and not an
approximation of it. To the Greek mind, this difficulty was as genuine
and as prepossessing as the problem of food to a castaway on a coral
reef.

Not content to use the less scrupulous method of the Babylonians, the
Greeks undertook to face the logical difficulty squarely. In order to
think about irrational numbers with exactness they conceived the idea of
working with all numbers geometrically. They started out this way. A
length was chosen to represent the number 1. Other numbers were then
represented in terms of this length. To represent sqr(2), for example,
they used a length equal to the hypotenuse of a right triangle whose
sides were one unit in length. The sum of 1 and sqr(2) was a length
formed by adjoining a unit segment to the length representing sqr(2). In
this geometrical form the sum of a whole number and an irrational one is
no more difficult to conceive than the sum of one and one.

Similarly the product of two numbers, 3 and 5 for example, was expressed
geometrically as the area of the rectangle with dimensions 3 and 5. In
the case of 3 and 5 the use of area as a way of thinking about the
product may be no great advantage. But one can also think of the product
of 3 and sqr(2) as an area. To think about this second rectangle is no
more difficult than to think of the first one; yet it provides an exact
way of working with the product of an integer and an irrational number
or, for that matter, two irrational numbers.

The Greeks not only operated with numbers in the geometric manner but
went so far as to solve equations involving unknowns by series of
geometrical constructions. The answers to these constructions were line
segments whose lengths were the unknown values. The thoroughness of their
conversion to geometry may be judged from the fact that the product of
four numbers was unthinkable in classical Greece because there was no
geometric figure to represent it in the manner that area and volume
represented the product of two and three numbers respectively.
Incidentally, we still speak of a number such as 25 as the square of 5
and of 27 as the cube of 3 in conformity with Greek thought. . . .

Because the Greeks converted arithmetical ideas into geometrical ones and
because they devoted themselves to the study of geometry, that subject
dominated mathematics until the nineteenth century, when the difficulties
in treating irrational numbers on an exact, purely arithmetical basis
were finally resolved. In view of the clumsiness and complexity of
arithmetical operations geometrically performed, this conversion was,
from a practical standpoint, a highly unfortunate one. The Greeks not
only failed to develop the number system and algebra which industry,
commerce, finance, and science must have, but they also hindered the
progress of later generations by influencing them to adopt the more
awkward geometrical approach. Europeans became so habituated to Greek
forms and fashions that Western civilization had to wait for the Arabs to
bring a number system from far-off India."

End Quote.


Bob Sciamanda
Physics, Edinboro Univ of PA (ret)
trebor@velocity.net
http://www.velocity.net/~trebor