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___________________I Topic: I_____________________
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\ HTML, I Base Number Systems I Author: /
> Forbze I I <
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Base Number Systems are the whole entire system that our computers run on, Binary
is a Base System. In this case Base 2.
Before the days of languages, such as C++, C, Visual Basic and so on, there
was actually Intelligent people slaving away at computers typing up endless
pages of machine code. If you want to know more about the history of Machine
code and etc. check out some my other tutorials for Blacksun. Well the point
im trying to make is that the average computer programmer, back then had done
Degrees in mathematics. 80% of the time the programmers had degrees anyway,
This degree in mathematics helped them understand the way a computer works,
eg. the electronic pulses caused by the computers output of 1's and 0's. This
At bottom, Computers
understand only one language -- the binary code of ones and zero's that represent
on-off electronic pulses. Because this code is so difficult for humans, programmers
have built more concise ways of expressing the binary numbers that constitute,
for example the contents of a computer's memory or the address in memory of each
peice of data. T wo numbering systems that can serve as convenient short hand
for the binary (base 2) are octal (base 8) and Hexidecimal (base 16). Hexidecimal
is sometimes known as "Hex" by programmers. Becuase 8 is raised 2 to the third
power ( 8 = 2 * 2 * 2) , one octal digit is the equivilant of three binary digits,
similarly, one hexidecimal digit represents 4 binary digits ( 16 is raised to
the forth power). The tables below list the decimal dumbers 0 through 16 and there
binary, octal, hexidecimal equivalents. In each system, the value of a digit is
determined by the value of its place column. The letters A through F
in hexadecimal represents the 11th through 16th digits in that system.
Subtract the highest
possible power of 2 from the decimal number - here, 4 from 5 - and continue subtracting
the highest possible power from the remainder, marking a 1 in each binary place
column where subtraction occurs and a 0 where it doesn't . Here. one 4, no 2 and
one 1 gives binary 101.
Binary to Decimal
Add the values of
all the binary places occupied by 1s. Here, to convert the 12- digit binary number
100101101001, add the place values of 2048,256,64,32,8 and 1. The result is the
decimal number 2409.
Binary to Octal
Starting with the righmost digit, group the binary digits in threes, treating
each three as a seperate binary number with the place values of 4,2 and 1. The
sum of each of trio's place values equals one octal digit. Here, the sums of the
values of each of the four groups ar 4,5, 5 and 1, making octal 4551.
Binary to Hexidecimal
Again from the right, group the binary digits in fours, treating each four as
1 binary number with the place values 8, 4, 2 and 1. The sum of each group's
place values equals one hexidecimal digit. Here, the sums of the three groups
are 9, 6 and 9, making hexidecimal 969.
Using the same rules as in decimal addition, start by adding the figures in the
rightmost, or 1s column: 1 + 1. The result - 2 - is expressed in binary as 10
(one - zero). Write down the 0 and carry the 1. In the 2s column, 1 + 1 again
equals 2, or binary 10;write down the 0 and carry the 1 into the 4s column. The
result is 100, the binary equivalent of decimal 4.
Adding the figuress in the 1s column - 7 + 1 - gives 8, expressed in the octal
system as 10 (one - zero). As in binary addition, write down the 0 and carry the
1. Next, add the figures in the 8s column, the sum of 6 and 1 is 7. The result
is Octal 70 - the equivalant of binary 111000, or decimal 56.
Adding the figures in the 1s column - 7 + 9 - Gives 16, the base of the hexidecimal
system, expressed as 10. Write down the 0 and carry the 1. In the 16s coloumn,
add 1 to the D (13 in decimal). D plus 1 is E (14 in decimal). The result is E0
(E-Zero), Hexidecimal shorthand for binary 11100000, or decimal 224
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