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Thread: numerical bases

  1. #1 numerical bases 
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    Is there in history, or any form of mathematics where decimals (10) is not the basis for numeration? Or perhaps one where a consistence such as phi replaces the concepts of digits? Perhaps this is a stupid question, in which case it will be a short thread, but I know with my limited ability in math I will hopefully learn something.


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    The ancient Babylonians (IIRC) used base 12 which is why our days/hours/minutes are divided into multiples of 12.

    Base phi works mathematically, but I don't think it's ever been used historically. (Fibonacci base is easier for humans to work with anyway.)


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  4. #3  
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    You can have pretty much any numerical base. Base two is called binary. Base 16 is called hexadecimal. These are very commonly used.
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    Quote Originally Posted by Mayflow View Post
    You can have pretty much any numerical base. Base two is called binary. Base 16 is called hexadecimal. These are very commonly used.
    16 kinda makes sense, because of it's divisibility.
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  6. #5  
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    12 maybe more, divisible by 2,3,&4
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    Modern digital electronics mostly uses base 2 and 16. Base 2 is just 0 and 1 (on or off), meaning about +5V Dc for on, or close to 0V for off.

    This may help explain. HowStuffWorks "The Base-2 System and the 8-bit Byte"
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  8. #7  
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    Computers only really use base 2 (though that is kind of a minor difference). We just use base 16 to communicate with computers because it's very, very easy to convert between the two, and writing numbers out in base 2 takes forever. (Numbers are about 3 times longer in base 2, on average.)

    And yeah, base 12 (and 60) are really good for divisibility, and I think that is one of the reasons it became so popular. (You can take 1/2, 1/3, 1/4, 1/5, 1/6 of a minute and still get a whole number of seconds.)
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  9. #8  
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    will have to smoke a phatty and read that again
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  10. #9  
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    Quote Originally Posted by MagiMaster View Post
    Computers only really use base 2 (though that is kind of a minor difference). We just use base 16 to communicate with computers because it's very, very easy to convert between the two, and writing numbers out in base 2 takes forever. (Numbers are about 3 times longer in base 2, on average.)
    Yes sir. All digital devices that I am aware of use binary (base 2). Hexadecimal is a convenient way to represent binary numbers. x'C6' is a much clearer and easier representation than b'11000110' (decimal 198).
    Last edited by Chucknorium; May 13th, 2014 at 07:48 PM.
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    There were a few attempts at using base-3 for computers (Ternary computer - Wikipedia, the free encyclopedia), but binary won out for simplicity and reliability. (Detecting three states on a wire is much more error-prone.)
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  12. #11  
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    Quote Originally Posted by MagiMaster View Post
    There were a few attempts at using base-3 for computers (Ternary computer - Wikipedia, the free encyclopedia), but binary won out for simplicity and reliability. (Detecting three states on a wire is much more error-prone.)
    Interesting. Thanks for the reference.
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  13. #12  
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    Quote Originally Posted by MagiMaster View Post
    The ancient Babylonians (IIRC) used base 12 which is why our days/hours/minutes are divided into multiples of 12.
    Base 60, actually. I suppose having 60 digits isn't really a problem when you have an ideographic writing system with thousands of characters!

    (Actually, I can't remember now if they had 60 distinct digit characters - I'll try and find time to dig my textbook out and look it up...)
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  14. #13  
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    Quote Originally Posted by keeseguy View Post
    Is there in history, or any form of mathematics where decimals (10) is not the basis for numeration? Or perhaps one where a consistence such as phi replaces the concepts of digits? Perhaps this is a stupid question, in which case it will be a short thread, but I know with my limited ability in math I will hopefully learn something.
    The Maya of Central America used a base 20 system: Maya numerals - Wikipedia, the free encyclopedia

    The reason for the popularity of base 10 is the number of digits on the human hands. Perhaps the Maya also used their toes.

    I agree base 12 has a lot to recommend it due to its number of factors, and there are traces of this in terms such as dozen (= douzaine).

    A nice question is what base, if any, Roman numbering used. Partly base 10 and partly base 5 perhaps.
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    Quote Originally Posted by Chucknorium View Post
    Quote Originally Posted by MagiMaster View Post
    Computers only really use base 2 (though that is kind of a minor difference). We just use base 16 to communicate with computers because it's very, very easy to convert between the two, and writing numbers out in base 2 takes forever. (Numbers are about 3 times longer in base 2, on average.)
    Yes sir. All digital devices that I am aware of use binary (base 2). Hexadecimal is a convenient way to represent binary numbers. x'C6' is a much clearer and easier representation than b'11000110' (decimal 198).
    Binary is the only way the current computers operate. These are called bits. In this respect, things like hex and such are not really number bases.

    This has an explanation in what I will post a link to, but they have one thing wrong there. About Integer Types

    What was incorrect is that "
    • 16 bit - ushort - range 0 to +32767." should read
    • 16 bit - ushort - range 0 to 65535

    What this means is that 16 bits means that there are 16 on off (binary) switches.
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    Quote Originally Posted by exchemist View Post
    The Maya of Central America used a base 20 system: Maya numerals - Wikipedia, the free encyclopedia

    The reason for the popularity of base 10 is the number of digits on the human hands. Perhaps the Maya also used their toes.

    I agree base 12 has a lot to recommend it due to its number of factors, and there are traces of this in terms such as dozen (= douzaine).

    A nice question is what base, if any, Roman numbering used. Partly base 10 and partly base 5 perhaps.
    Thanks for this info! Good question. Roman numbering is kind of strange. There is no zero and they only use the digits I, V, X, L, C, M. The rules were also not constant because they sometimes used IIII and VIIII instead of IV and IX.
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    Quote Originally Posted by Strange View Post
    Base 60, actually. I suppose having 60 digits isn't really a problem when you have an ideographic writing system with thousands of characters!

    (Actually, I can't remember now if they had 60 distinct digit characters - I'll try and find time to dig my textbook out and look it up...)
    Please do. Let us know what you find out.
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    @exchemist, I don't think you can put a base to the Roman numeral system as it's not a positional number system (Positional notation - Wikipedia, the free encyclopedia). If you reimagined things a bit, you might could called it a mixed base system though (alternating 5 and 2).

    @Mayflow, hex definitely is a numerical base (base-16). It's just that computers don't use it directly. They convert it to binary and use that. Humans can use hex directly though.
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  19. #18  
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    Quote Originally Posted by Chucknorium View Post
    Quote Originally Posted by exchemist View Post
    The Maya of Central America used a base 20 system: Maya numerals - Wikipedia, the free encyclopedia

    The reason for the popularity of base 10 is the number of digits on the human hands. Perhaps the Maya also used their toes.

    I agree base 12 has a lot to recommend it due to its number of factors, and there are traces of this in terms such as dozen (= douzaine).

    A nice question is what base, if any, Roman numbering used. Partly base 10 and partly base 5 perhaps.
    Thanks for this info! Good question. Roman numbering is kind of strange. There is no zero and they only use the digits I, V, X, L, C, M. The rules were also not constant because they sometimes used IIII and VIIII instead of IV and IX.
    Yes I think the zero - and hence the system of decades - was the big contribution from the Arabic world, wasn't it? (Zero is "sifr" in Arabic, from which we get "cypher".)
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  20. #19  
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    Quote Originally Posted by MagiMaster View Post
    @exchemist, I don't think you can put a base to the Roman numeral system as it's not a positional number system (Positional notation - Wikipedia, the free encyclopedia). If you reimagined things a bit, you might could called it a mixed base system though (alternating 5 and 2).

    @Mayflow, hex definitely is a numerical base (base-16). It's just that computers don't use it directly. They convert it to binary and use that. Humans can use hex directly though.
    Yes that makes sense, thanks.
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  21. #20  
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    Quote Originally Posted by exchemist View Post
    Yes I think the zero - and hence the system of decades - was the big contribution from the Arabic world, wasn't it? (Zero is "sifr" in Arabic, from which we get "cypher".)
    I think the Babylonians had a symbol for zero in their positional system. I'm not really sure why some of their mathematics got passed on but some useful things like that didn't.
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  22. #21  
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    Quote Originally Posted by Strange View Post
    Quote Originally Posted by exchemist View Post
    Yes I think the zero - and hence the system of decades - was the big contribution from the Arabic world, wasn't it? (Zero is "sifr" in Arabic, from which we get "cypher".)
    I think the Babylonians had a symbol for zero in their positional system. I'm not really sure why some of their mathematics got passed on but some useful things like that didn't.
    I looked it up and found this: The Origin of Zero - Scientific American

    So I am, as so often, being Eurocentric in thinking it came from the Arabs. Fibonacci certainly brought it to Mediaeval Europe from them, but as you say it originated in ancient Babylon - and apparently independently among the Maya.
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    Quote Originally Posted by exchemist View Post
    A nice question is what base, if any, Roman numbering used. Partly base 10 and partly base 5 perhaps.
    Interestingly enough, the 'essence" — if you will — of Roman Numerals is base 1. Aha!

    In base 1, the position values are, for example: 1ⁿ ... 1, 1, 1, and 1, otherwise known as 1 ... 1, 1, 1, and 1. It's simply tally counting, So ...

    1 = 1111
    2 = 1111
    3 = 1111
    4 = 1111

    Just like Roman Numerals! And so, Roman Numerals is a positional system of sorts, although it's a mixed system because it tallies I's (ones), X's (tens), C's (hundreds), M's (thousands), etc.

    Because it's a mixed system, it was a pain to use for any kind of real math (not just representing a number as we use it today, for example, the year of a book's publication, MMXIV = 2014) — imagine trying to multiply or divide — which is why it fell out of favor when straight positional systems became known.

    I wonder about the origins of Roman Numerals.
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