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

roots.

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 {{

http://www-history.mcs.st-and.ac.uk...tifel.html
}}, 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

Stevin"]

http://www-history.mcs.st-and.ac.uk...ers_1.html
Gruß, WM

## Lesen sie die antworten