Chapter 3
Numbers

We first want to remind the reader of the different kinds of numbers, that are used in arithmetics and to visualize their relationships with the elementary arithmetic operations. Here the number is the operand on which a certain operation is applied (arithmetic operations such as +,-,*,/,ˆ, =, >, <, logical operations such as and, or, not, if-then, otherwise, ...).

The definitions of the numbers and arithmetic operations are up to the complex numbers synchronized in such a way. that for all number z the following fundamental rules of arithmetic operations apply. here () means, that the operation in brackets is executed first.

z1 + z2 = z2 + z1(Commutative law of addition) z1 z2 = z2 z1( Commutative law of multiplication) (z1 + z2) + z3 = z1 + (z2 + z3) (Associative law of Addition) (z1 z2) z3 = z1 (z2 z3) (Associative law of multiplication) (z1 + z2) z3 = z1z3 + z2z3 (Distributive Law of multiplication) Consequence: It does not matter in which sequence the operations are executed. Shorthand notation in the text :z1z2 z1 z2;z2 = zz;z3 = z2z(= z 3);.....

The requirement to apply certain operations, that had been introduced for a certain kind of number, without restrictions lead to successive extensions of the usual concepts of numbers.

 3.1 Natural Numbers
 3.2 Whole Numbers
 3.3 Rational Numbers
 3.4 Irrational Numbers
  3.4.1 Algebraic Numbers
  3.4.2 Transcendental Numbers
  3.4.3 π and the Quadrature of the Circle according to Archimedes
 3.5 Real Numbers
 3.6 Complex Numbers
  3.6.1 Representation as a Pair of Real Numbers
  3.6.2 Normal representation with the “imaginary unit i”
  3.6.3 Complex Plane
  3.6.4 Representation in Polar Coordinates
  3.6.5 Simulation of Complex Addition and Subtraction
  3.6.6 Simulation of Complex Multiplication and Division.
 3.7 Extension of Arithmetics