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I am often amazed by how accurate a decimal numeral system is to describe the mathematics world. A lot of things feel very logical like the result of any multiplication by 5 will always end with a 0 or a 5. Or this again with this rule: a number is divisible by 3 if the sum of its digits is divisible by 3.(*)

Bu I wonder if a binary or an octal or a hexadecimal wouldn't have more of these rules, that would make the understanding of maths even simpler. I guess we have the decimal numeral system in Occident because we have 10 fingers, but this doesn't mean it's the more efficient way of understanding all the maths logic. We use the Qwerty keyboard but it's not the most efficient keyboard around.

So in brief my questions are:

  • Is the decimal numeral system the most efficient one to understand the mathematics rules and patterns of the universe, geometrics or simple multiplications/divisions?
  • Is not, which one is the best?

*) I am only using this two rules as examples because I don't know a lot of others maths patterns like those one. But I would be interested in knowing if others patterns, (I am tempted to say, more important patterns), exists in other number systems, that don't exist in the decimal one.

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    $\begingroup$ No. The decimal numeral system has little to do with mathematic as a whole. For example it is only suitable to represent numbers, in fact at most real numbers (or perhaps also complex numbers). Note that "understanding the patterns of the universe" happens mostly by physical laws expressed by formulas that do not involve numberical constants at all, or perhaps only the very simplest ones. $\endgroup$ Aug 17 '15 at 10:18
  • $\begingroup$ Thanks for pointing out about physics. I have edited the question to include more maths related subjects. $\endgroup$
    – MagTun
    Aug 17 '15 at 10:22
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    $\begingroup$ Using base eight would let us look at the last digit (octit?) to see whether a number was even, divisible by $4$ or by $8$. The sum of digits trick would tell us whether a number is divisible by $7$, and by adding every other digit, then subtracting the sum of the rest of the digits of a number you can tell whether a number is divisible by $9$ (most likely also $3$, I haven't looked into that), instead of $11$. $\endgroup$
    – Arthur
    Aug 17 '15 at 10:35
  • $\begingroup$ With regards to your comment "result of any multiplication by 5 will always end with a 0 or a 5", consider multiples of 4 in base 8 as one example and you will probably notice a pattern if you try a few other numbers and other bases $\endgroup$
    – BadAtMaths
    Aug 17 '15 at 10:43
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    $\begingroup$ You can get arbitrarily many such rules by choosing bases $b$ for which $b-1,b,b+1$ have as many divisors as possible between them. For example, for $b=21$, you can check divisibility by $3,7$ by checking the last digit; divisibility by $2,4,5$ by checking the digit sum; divisibility by $11$ by checking the alternating digit sum. Of course, higher $b$ are less and less convenient (and intuitive?) to deal with. $\endgroup$ Aug 17 '15 at 12:10
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the result of any multiplication by $5$ will always end with a $0$ or a $5$

Similar properties apply to a multiple of d in base b, if d divides b.


a number is divisible by $3$ if the sum of its digits is divisible by $3$

Similar properties apply to a multiple of d in base b, if d divides $b-1$.

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  • $\begingroup$ Thanks for this Lucian, please see my edit at the end of my question. $\endgroup$
    – MagTun
    Aug 17 '15 at 11:00
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Not at all. The concept of using 10 as a baseline probably grew out of the practice of calculating using one's fingers

But all kinds of number systems have been used in place of the decimal system and with relative successes. For instance, our computers expressly use the binary system, and switch to hexadecimal for representation purposes.

Decimal system only remains that important to us mathematically, because we have had a history of using 10 as a base (for reasons explained on the wiki and elsewhere). So naturally, any computation we do using the decimal system seems like a breeze. I am sure similar rules like those for the digits 3 and 5 do exist for numbers in other base systems.

(please suggest edits before downvoting)

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  • $\begingroup$ Thanks goelakash. Do you have any rules in mind that apply to other number systems? And do you know if they provide a better way of grasping the math world? $\endgroup$
    – MagTun
    Aug 17 '15 at 11:01
  • $\begingroup$ Not for any particular base, but as Lucian mentioned below, many of the rules can be generalised for different base systems. About grasping the math world, I'll stick with the decimal system :D $\endgroup$
    – goelakash
    Aug 17 '15 at 11:35

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