Wednesday, April 20, 2011
octomatics revisited. a 16 base hexadecimal one
D - I am sorry for the quality of this image. My Win7 is disabled, so I ended up taking a webcam shot.
the octomatics project is about a new number system
which has a lot of advantages over our old decimal system.
the name comes from the mixture of 'octal' and 'mathematics'.
what do you think: why do we have the decimal system
in our western world? because of our 10 fingers? why
do we have 7 days a week? why are 60 seconds 1 minute
and 60 minutes 1 hour? why do we have 24 hours a day?
and 31 or 30 days a month? do you think thats a really
good solution? well, here is another one:
...welcome to octomatics !
the new numbers
how many numbers are the optimum? 8? 10? 12? 16?
i think it's 8 or 12. make it 8 and you will be able to read
and work with binary code without any transformation.
i think the numbers should look more technically than
letters. maybe they could look like the following:
D - I agree with him - though chose 12 instead.
CVN will have English-style names for 11 and 12, instead of using the usual 10 and 1/ 10 and 2 convention. If I name stuff after the #s, then this makes for neatness and brevity.
D - Anyway, Octomatics is based on 8. My proposed # system above is based on 16.
Well, really, I just wanted to reach 12 in some sensible fashion.
But now a little review of binary and hexadecimal.
The Indian scholar Pingala (circa 5th–2nd centuries BC) developed mathematical concepts for describing prosody, and in so doing presented the first known description of a binary numeral system. He used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), making it similar to Morse code.
Any number can be represented by any sequence of bits (binary digits), which in turn may be represented by any mechanism capable of being in two mutually exclusive states.
Counting in binary is similar to counting in any other number system. Beginning with a single digit, counting proceeds through each symbol, in increasing order. Decimal counting uses the symbols 0 through 9, while binary only uses the symbols 0 and 1.
Since binary is a base-2 system, each digit represents an increasing power of 2, with the rightmost digit representing 20, the next representing 21, then 22, and so on. To determine the decimal representation of a binary number simply take the sum of the products of the binary digits and the powers of 2 which they represent.
D - fractions are a real bugger though...
Binary may be converted to and from hexadecimal somewhat more easily. This is because the radix of the hexadecimal system (16) is a power of the radix of the binary system (2). More specifically, 16 = 24, so it takes four digits of binary to represent one digit of hexadecimal, as shown in the table to the right.
To convert a hexadecimal number into its binary equivalent, simply substitute the corresponding binary digits:
In mathematics and computer science, hexadecimal (also base 16, or hex) is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F (or alternatively a–f) to represent values ten to fifteen. For example, the hexadecimal number 2AF3 is equal, in decimal, to (2 × 163) + (10 × 162) + (15 × 161) + (3 × 160) , or 10,995.
Each hexadecimal digit represents four binary digits (bits) (also called a "nibble"), and the primary use of hexadecimal notation is as a human-friendly representation of binary coded values in computing and digital electronics. For example, byte values can range from 0 to 255 (decimal) but may be more conveniently represented as two hexadecimal digits in the range 00 through FF. Hexadecimal is also commonly used to represent computer memory addresses.
Most computers manipulate binary data, but it is difficult for humans to work with the large number of digits for even a relatively small binary number. Although most humans are familiar with the base 10 system, it is much easier to map binary to hexadecimal than to decimal because each hexadecimal digit maps to a whole number of bits (410). This example converts 11112 to base ten. Since each position in a binary numeral can contain either a 1 or 0, its value may be easily determined by its position from the right:
With surprisingly little practice, mapping 11112 to F16 in one step becomes easy: see table in Representing hexadecimal. The advantage of using hexadecimal rather than decimal increases rapidly with the size of the number. When the number becomes large, conversion to decimal is very tedious. However, when mapping to hexadecimal, it is trivial to regard the binary string as 4-digit groups and map each to a single hexadecimal digit.
D - check out purplemath.com for all sorts of goodies related to alternative #-base systems.
D - why use my 16-base # system?
1) it works with a 7-segment alphanumeric display - so any calculator.
2) the figures are the most minimal possible without unconnected stray "floating"
3) it has all the benefits of Octomatics
4) it works very well with computers due to hexadecimal.
Hope you like it.
I'll try to make a nicer diagram at some point.
Posted by Dino Snider at 12:58 AM