numbering+bases

=NUMBERING BASES=

numbering base is defined as the set of simbols used for the representation of quantities, and the rules governing such representation. a numbering system can be spotted by it´s base, that is the number of symbols it use, and is characterized by the coefficient that determines the value of each symbol depending on its position. The numbering system we usually use is the decimal system, of base 10. The decimal system use ten digits or symbols: 0,1, 2, 3, 4, 5, 6, 7, 8, and 9. There are ten because we have ten fingers and in the origins of man they used fingers to count.

in a numbering system, depending on the position they occupy within a cypher,the same digit can represent units,tens, hundreds, thousands, etc. Because of this is said that the numbering systems are positional.For example, in the decimal system , the value of the number 6839 can be expressed as sums of powers of the base 10 :

(6 * 10^3) + (8 * 10^2) + (3 * 10^1) + (9 * 10^0) = 6000 + 800 + 30 + 9 = 6839

We can also define the base of numeration as a set of rules and digits that allow represent numerical data. The main rule is __the same digit have different value depending the position__. That way, for example, if we have the number 555, the digit 5 have different value dependig the position. every position have an associated "weight" and in that way we can represent the units: 5, the tens: 50, and the cents: 500. the rightmost digit will be worth 0, the next 1, the following 2, and so on. We can represent this number as the sums of powers of the base 10 elevated to the position they occupy:

5 * 10^2 + 5 * 10^1 + 5 * 10^0 = Fundamental theorem of numbering =

This theorem relates a quantity expressed in any numbering system with the same amount in the decimal system ; i.e., the decimal value of an amount expressed in another number system that uses another base. Is given by the formula:

**N i =** ** E (digit)* (base) i **

** Where: ** E= sum of the result i= position relative to the comma. for the digits on the right, the i is negative, starting in -1; for the ones on the left, is positive, starting in 0. digit= every one that compose the number base= base of the numbering system. The decimal number will be the sum of multiplying each digit by the base elevated into his position. = = = intermediate codes = Early computers systems only used the binary code,so the programming tasks were quite tedious. It then turns to the use of intermediate codes, that allow an easy translation "fron and to" binary code. The intermediate codes are based on the ease of transforming a base 2 number to another base that isa power of 2 (2^2=4, 2^3=8, 2^4=16, etc) and vice versa. Usually used as intermediate codes are numbering systems base 8 ( octal) and base 16 ( hexadecimal).