Das Kalenderblatt 110401

31/03/2011 - 09:21 von WM | Report spam
Where does Euclid's Elements leave us with respect to numbers?
Basically numbers were 1, 2, 3, ... and ratios of numbers were used
which (although not considered to be numbers) basically allowed
manipulation with what we call rationals. Also magnitudes were
considered and these were essentially lengths constructible by ruler
and compass from a line of unit length. No other magnitudes were
considered. Hence mathematicians studied magnitudes which had lengths
which, in modern terms, could be formed from positive integers by
addition, subtraction, multiplication, division and taking square
The Arabic mathematicians went further with constructible
magnitudes for they used geometric methods to solve cubic equations
which meant that they could construct magnitudes whose ratio to a unit
length involved cube roots. For example Omar Khayyam showed how to
solve all cubic equations by geometric methods. Fibonacci, using
skills learnt from the Arabs, solved a cubic equation showing that its
root was not formed from rationals and square roots of rationals as
Euclid's magnitudes were. He then went on to compute an approximate
solution. Although no conceptual advances were taking place, by the
end of the fifteenth century mathematicians were considering
expressions build from positive integers by addition, subtraction,
multiplication, division and taking nth roots. These are called
radical expressions.
By the sixteenth century rational numbers and roots of numbers were
becoming accepted as numbers although there was still a sharp
distinction between these different types of numbers. Stifel {{
}}, in his Arithmetica Integra (1544) argues that irrationals must be
considered valid: "It is rightly disputed whether irrational numbers
are true numbers or false. Because in studying geometrical figures,
where rational numbers desert us, irrationals take their place, and
show precisely what rational numbers are unable to show ... we are
moved and compelled to admit that they are correct "
However, he goes on to argue that, as they are not proportional to
rational numbers, they cannot be true numbers even if they are
correct. He ends up arguing that all irrational numbers result from
radical expressions. Well the obvious question the reader might feel
they want to ask Stifel is: what about the length of the circumference
of a circle with radius of unit length? In fact Stifel gives an answer
to this in an appendix to the book. First he makes a distinction
between physical circles and mathematical circles. One can measure the
properties of physical circles, he claims, but one cannot measure a
mathematical circle with physical instruments. He then goes on to
consider the circle as the limit of a sequence of polygons of more and
more sides. He writes: "Therefore the mathematical circle is rightly
described as the polygon of infinitely many sides. And thus the
circumference of the mathematical circle receives no number, neither
rational nor irrational."
Not too good an argument, but nevertheless a remarkable insight
that there were lengths which did not correspond to radical
expressions but which could be approximated as closely as one wished.
[J.J. O'Connor and E.F. Robertson: "The real numbers: Pythagoras to

Gruß, WM

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#1 WM
31/03/2011 - 09:47 | Warnen spam
Anmerkung: Es war nicht geplant und hat sich rein zufàllig ergeben,
dass Michael Stifel (Leo X = 666) im KB Nr. 666 vorkommt.

Gruß, WM

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