Friday, December 10, 2010

Numerical Bases

Since we were kids, we have all used decimals to express quantities. This nomenclature that seems so logical to us may not seem so to an inhabitant of Classical Rome. For them, each symbol that they wrote to express a number always represented the same value:

I      1
II     2
III    3
IV     4
V      5

All the I signs always represents the value 1 (one) wherever they are placed, and the V sign always represents a value of 5 (five). Nevertheless that does not take place in our decimal system. When we write the decimal symbol 1 we are not always talking about a value of one (I in Roman numbers). For example: 

1    I
 10    X
100    C

In these cases, our symbol 1 does not have always a value of one (or I in Roman numbers). For example, in the second case, the symbol 1 represents a value of ten (or X in Roman) and in the third one, 1 represents a value of one hundred (or C).

For example:

275 is not equivalent to 2+7+5, it could rather be decomposed as 200+70+5:

+ 70

therefore, the first "2" sign is equivalent to 200 (2 x 100), the second "7" sign is equivalent to 70 (7 x 10) whereas the last sign corresponds to the value 5 (5 x 1).

This is because our system is a positional numeral system. Therefor the value of a given digit depends on its position within the entire number being represented. All the above can be mathematically represented in a very simple way. For example, to represent the value 182736 we can assume that each digit is the product of itself multiplied by 10 powered to its place as exponent, beginning from the right with 100, following with 101, 102, and so on: 

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