Mathematical modelling of Signals – Processes – Systems - Signals, Processes, and Systems - page 468

In Chapter 5, within the framework of the symmetry of the frequency range, we
introduced the GAUSSian plane of complex numbers (see Illustration 97, Illustration 319
- 101). Two values precisely (a, b) belong to every point on this plane (and any other
plane). In this way a vector results for every frequency, whereby

• the length of the vector equates to amplitude Ꮅ ,

• the angle between frequency vector and positive horizontal axis equates to
phase angle ɔǤ

By means of the GAUSSian plane any complex number of the formula a + ib can be
shown as a point on this plane, whereby there is an initially very strange definition for i :

i =− 1

Precisely this inconspicuous expansion of the number range from real numbers to com-
plex numbers boosted mathematics in the past 200 years. The terms “complex” and
“imaginary” are also confusing: they suggest something difficult, mysterious and unclear.
This is precisely not the case. In fact, they simplify and demonstrate countless extremely
difficult calculations. The term efficien t or expanded number range would have been
better.

For many physical phenomena of electromagnetism, the field of
oscillatory, wave and quantum physics, are unthinklable without
complex calculation. The connection to this number range is so
extreme that complex calculation can be seen as an essential part
of a” language of nature”.

For this reason, complex calculation also dominates electro-
technology, which is based on physical phenomena, especially the
technology surrounding “signals - processes - systems””.

For those who have so far not realized the importance of complex numbers, the
following problem may be thought- provoking:

Natural numbers, rational, irrational or real numbers are a matter of common
knowledge. Here, the natural numbers (0), 1,2,3,4,... appear to be a “harmless”
range, about which there is little to be said. But there are also the prime numbers -
all those numbers that can only be divided by 1 and by themselves:

1, 2, 3, 5, 7, 11, 13, 17, ... ,167, 173, 179, ... ,9967, 9973, ...

Mathematicians have been dealing with these numbers for many centuries, many of
them spending their whole lives often attempting to answer the following questions:

• Can I understand their distribution in the pool of real numbers?

• How long do I have to go on counting to come to the next primal number?

• Why does the next prime number appear as if by chance after only a few
steps, some on the other hand only after a big gap?