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Each position in a positional notation system is a value represented by a unique symbol or character. For each position, the resultant value of each position is the value of that character multiplied by a power of the base number for that number system. The position of each character or symbol (usually called a digit) counting from the right determines the power of the base that is to be multiplied by that digit.
For our familiar base 10 number system, each position starting from the right is a successive power of 10. The first postion represents 10 raised to the power of 0, the second position 10 raised to the power of 1 or just 10, the third 10 to the power of 2 or 10x10 = 100, the forth 10 to the power of 3 = 10x10x10 = 1000 and so on. (NOTE 10 to the power of 0 = 1 )
Fractional values are indicated by a separator (in some countires a period or full stop, in others a comma. Digits to the right of this separator are multiplied by the base (10 in this example) raised to a negative power or exponent. The first position to the right of the separator has a power value of 10 raised to the power of -1 which is the same as 1 divided by 10. The second position to the right has a power of the base (here 10) raised to the power of -2 and is just 1 divided by 10x10 or 1 divided by 100 and so on for each successive postion.
Stepping back it must be explained that the rule of POWERS or INIDICES (the value representing the number of times the base is multiplied by itself) results in the counterintuitive result that 10 raised to the power of zero (0) = 1. Furthermore, any number to the power of zero = 1.
The total value of a number in a positional system is in fact the total of each individual multiplication of a digit and its associated base multiplied by itself the number times represented by its position less 1.
As an example, the number 2674 in a base 10 number system is :
( 2 x 10³ ) + ( 6 x 10² ) + ( 7 x 10¹ ) + ( 4 x 10°) orIt should be emphasized that number systems can also be based on numbers other than 10. For eaxample, computers utilize storage of data by means of a BINARY or base 2 number system and values can be represented by numbers using a base of 8 and 16 called OCTAL and HEXADECIMAL respectively.( 2 x 1000 ) + (6 x 100 ) + ( 7 x 10 ) + (4 x 1 )
Furthermore numbers can be represented by any arbitrary choice of number base. The only difference in each these systems is the nuber of unique number symbols needed which is always the same as the number base. Thus in a base 2 or BINARY system only two symbols or digits are needed, 1 and 0. In a base 10 system our familiar ten digit sysmbols from 0 to 9 are needed. In an OCTAL sytem only digits 0 to 7 are needed. In a number system with a base greater than 10 we need to add extra symbols to represent values up to one less than the value of the base. In the HEXADECIMAL system, the letters A to F are used to represent values 10 to 15. In a DUODECIMAL or base twelve system, only two extra digits are needed to represent the values 10 and 11 as we normally understand them. In this sytem various characters have been proposed such as X = 10 in base ten and E = 11 in base 10. It is also emphasized that in any other number system, the value represented by the two digits 10 is 1 x the base raised to the power of zero.
In HEXADECIMAL this is 16 in base 10 and in DUODECIMAL would be 12 in base 10. Likewise 100 HEXADECIMAL = 16 x 16 = 256 base 10 and 100 DUODECIMAL = 12 x 12 = 144 base 10.
To summerize, the value of a number is the total of each unique value times a POWER of the base, or RADIX. A POWER is the number of times a value is multiplied by itself. The power is written above and to the right of the base and is called an EXPONENT.
Contributed by R. E. Greaves footrule.com
