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Thread: What if our math system was not based on 10s?

  1. #1
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    What if our math system was not based on 10s?

    Now this is probably a stupid question but I can't wrap my head around the idea. I need someone with a bit more brain power than mine.
    Our number system is based on tens. 0 through 9 then next branch, 10 through 19 and so on. It was probably easier for us humans to work in group on tens since we have ten fingers.
    Now I am aware that in the end it really doesn't matter how we express numbers. In group of tens or in group of 12, 15, 16, whatever. Because numbers value remains the same no matter how they are expressed. Or does it?
    My question is this:
    Theoretically, if I was to use a different system based on group of 16s, where 0 through 16 would be a 1 digit number, would it work the same way? Could it confuse a mathematician or would it be really obvious and easy method to bypass?
    This is a weird question.
    But thank you in advance for your help.
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    Re: What if our math system was not based on 10s?

    mathematics is not affected in any substantial way by the choice of base for your numerals.

    All bases are equivalent to one another and can be transformed from one to another without any loss of information.

    You can sort of think of it as mathematics itself being a box that has a numeric interface to the outside world.

    You can make that interface whatever numeric base you like, it doesn't affect the core stuff going on in the mathematics box.
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  3. #3
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    Re: What if our math system was not based on 10s?

    Quote Originally Posted by CFMR View Post
    Theoretically, if I was to use a different system based on group of 16s, where 0 through 16 would be a 1 digit number, would it work the same way? Could it confuse a mathematician or would it be really obvious and easy method to bypass?
    Minor correction: for hexadecimal (base-16), it would be 0 through 15.

    As early as 1859, a civil engineer by the name of John W. Nystrom proposed a hexadecimal system of notating number, arithmetic, measurement, currency, and even time (a 16 month calendar and a hexadecimal clock!). He wrote a book about it, which you can find here.

    I'm also aware of a website that advocates the use of hexadecimal.

    One can certainly add, subtract, multiply, divide numbers in hexadecimal. Converting from fractions to decimal positional notation would look different. For example, 1/2 = 0.8 in hexadecimal. You may want to look at this Wikipedia article for more examples.

    Base-16 isn't the only positional system where there is advocacy. There is also the The Dozenal Society of America, which advocates the use of base-12.


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    Re: What if our math system was not based on 10s?

    Thank you so much for pointing me in the right direction.

    Can you tell me why would someone choose to work with a base-16 or base-12?
    If it comes down to the same thing, what is the main reason or advantage?

    Thank you again!
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    Re: What if our math system was not based on 10s?

    more math is done in binary (base 2) than in any other base ... the only drawback is, there are only 10 kinds of people who understand it; those that do and those that don't.
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    Re: What if our math system was not based on 10s?

    Quote Originally Posted by CFMR View Post
    Thank you so much for pointing me in the right direction.

    Can you tell me why would someone choose to work with a base-16 or base-12?
    If it comes down to the same thing, what is the main reason or advantage?

    Thank you again!
    16 is $2^4$ which makes it a group of 4 binary digits and thus ideal for computer representation

    12 is clock arithmetic
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    Re: What if our math system was not based on 10s?

    Hey CMFR.

    The thing that stays the same is the algebra and arithmetic (in terms of all the laws like distribution, association and so on) and the thing that changes is how you do it.

    How you do it is a function of how you represent your numbers but what you do - and what kinds of results you should get (in terms of the algebraic constraints) is the same regardless of how its represented.
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    Re: What if our math system was not based on 10s?

    Thanks a lot to all of you! This will greatly help.
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    Re: What if our math system was not based on 10s?

    Until recently time and angles were measured in non decimal units.

    A further twist occurred with the introduction of the radian and steradian in angular measure.

    Radian measure is essential for much of the last two century's mathematical developments since most formulae involving angles require the angles to be in radians.
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    Re: What if our math system was not based on 10s?

    You are confusing numbers and numerals.

    The hexadecimal numeral 11 and the decimal numeral 17 both mean the same number.
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    Re: What if our math system was not based on 10s?

    Well I'm confused now. Isn't hexadecimal 0-9 a-f?
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    Re: What if our math system was not based on 10s?

    Quote Originally Posted by CFMR View Post
    Well I'm confused now. Isn't hexadecimal 0-9 a-f?
    Those are the hexadecimal DIGITS just as the decimal DIGITS are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In the decimal system of notation, numerals are constructed out of those ten digits. In the hexadecimal system, numerals are constructed out of 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F.

    $So\ 17\ (decimal\ numeral\ or\ Arabic\ numeral) = 11\ (hexadecimal\ numeral) = XVII\ (Roman\ numeral).$

    Numerals are ways to identify or to represent numbers. But the truths about numbers are not determined by how they are represented.

    $XV + II = XVII$

    $15 + 2 = 17$ and

    $F + 2 = 11$ are just three different ways to say the same thing.

    I think the distinction between numerals and numbers needs to be pointed out at the very beginning of algebra. It makes it more intuitive to see how we can use letters (pronumerals) to represent numbers whose numerals are not yet known.

    EDIT: To show how numerals are constructed in systems of numerals based on place values, it might have been clearer to write above:

    $15 + 2 = (10 + 5) + 2 = 10 + (5 + 2) = 10 + 7 = 17\ in\ decimal.$

    $F + 2 = (10 - 1) + 2 = 10 + (2 - 1) = 10 + 1 = 11\ in\ hexadecimal.$
    Last edited by JeffM; Aug 25th 2016 at 08:51 AM.
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    Re: What if our math system was not based on 10s?

    Ok I'm getting somewhere here. The only thing that I can't figure out is how 11 (hexadecimal) = 17 (decimal)?
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    Re: What if our math system was not based on 10s?

    I count in hexadecimal numerals this way

    1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

    I have run out of digits so what follows is

    10, 11 If you count those you will see that you have got to seventeen.

    It works just like decimal in terms of place values but 10 means sixteen rather than ten; so A0 means one hundred sixty, and 1E means thirty.
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  15. #15
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    Re: What if our math system was not based on 10s?

    Quote Originally Posted by CFMR View Post
    Ok I'm getting somewhere here. The only thing that I can't figure out is how 11 (hexadecimal) = 17 (decimal)?
    It's all about the place value. (By the way, instead of writing "(hexadecimal)" or "(decimal)" after a number, you can write the base using subscripts: 1116 = 1710.)

    Take a decimal 124810. The "1" indicates how many thousands, the "2" indicates how many hundreds, the "4" indicates how many tens, and the "8" indicates how many units. Another way to express this is by using powers of ten:
    1248_{10} = (1 \times 10^3) + (2 \times 10^2) + (4 \times 10^1) + (8 \times 10^0)

    = (1 \times 1000) + (2 \times 100) + (4 \times 10) + (8 \times 1)

    = 1000 + 200 + 40 + 8 = 1248_{10}

    Now look at a hexadecimal 15AF16. Since this is in hexadecimal, we express this using powers of 16, not powers of 10:

    15AF_{16}= (1 \times 16^3) + (5 \times 16^2) + (10 \times 16^1) + (15 \times 16^0)
    (A = 10 and F = 15)

    = (1 \times 4096) + (5 \times 256) + (10 \times 16) + (15 \times 1)

    = 4096 + 1280 + 160 + 15 = 5551_{10}

    Can you now see why
    1116 = 1710?


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    Last edited by yeongil; Aug 25th 2016 at 08:21 PM.
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