The principals of the positional systems.

In order to limit the number of necessary symbols in expressing quantity, people invented positional representation of numbers. The system can use any choosen number as its base. Counted elements are grouped. Every completed group contains the number of objects equal to the base. New objects are created by grouping identical smaller groups then they can be grouped again.

To represent a given quantity the numbers of different kind of completed groups have to be written down from the left to the right starting with the number of the biggest group . If some possible groups are not present, zero is written. Finally the expression is ended by the number of remaining units.

The number of single units in different completed groups is given by the following formula:
g=b^p where g is the number of units, b is the number equal to the given base and p is the current position in the expression counted from the left to the right and started with 0 for the position of units. The number of units in the completed groups at a given position is called the weight of the position.

Example for binary system system: 1101=   1 group containing two groups of two groups of two, 1 group of 2 two groups of two, no group of two elements, and 1 units.

1	1	0	1
oo     oo oo     oo	oo oo		o

To understand the notations of any positional number system we have to remember, or be able to resolve using the blue formula, the weight of every position in the notation. That also can be counted knowing that each next to the left position has its weight base time bigger than the previous one.