APPENDIX D
NUMBERS AND SPACES
A mathematician is a machine that transforms coffee into theorems.
Paul Erdös (1913, Budapest–1996)
Mathematical concepts can all be constructed from ‘sets’ and ‘relations.’ The
ost important ones were presented in the first intermezzo. In the following a few
more advanced concepts are presented as simply and vividly as possible,∗ for all those who
want to smell the passion of mathematics.
Ref. 1006
In particular, we shall expand the range of algebraic and the range of topological struc-
tures. Mathematicians are not only concerned with the exploration of concepts, but always
also with their classification. Whenever a new mathematical concept is introduced, math-
ematicians try to classify all the possible cases and types. Most spectacularly this has been
achieved for the different types of numbers, of simple groups and for many types of spaces
and manifolds.
More numbers
A person that can solve x2−92y2 = 1 in less than a year is a mathematician. Challenge 1315 d
Brahmagupta, (598, Sindh–668) (implied: solve in integers)
The concept of ‘number’ is not limited to what was presented in the first intermezzo.∗∗
The simplest generalisation is achieved by extending them to manifolds of more than one
dimension.
Complex numbers
Complex numbers are defined by z = a + ib. The generators of the complex numbers, 1 and
i, obey the well known algebra
· 1
i
1 1
i
i
i −1
(670)
∗ The opposite approach is taken by the delightful text by C AR L E. LI NDER HOLM , Mathematics Made
Difficult, Wolfe Publishing, .
∗∗ An excellent introduction into number systems in mathematics is the book H.-D. E B B I NGHAUS & al.,
Zahlen, 3. Auflage, Springer Verlag . It is also available in English, under the title Numbers, Springer
Verlag, .
964
Appendix D Numbers and Spaces
965
often written as i = +
√
−1 .
The complex conjugate z∗, also written z̄, of a complex number z = a + ib is defined as
z∗ = a− ib. The absolute value |z| of a complex number is defined as |z|=
√
zz∗ =
√
z∗z =
√
a2 +b2 . It defines a