6 ms·

great point. My thought on (1/3 = 0.3333...) * 3 = 1 = 0.999... was that it is intuitively obvious that the "problem" is that we use base-10 for decimals. Ther

great point. My thought on (1/3 = 0.3333...) * 3 = 1 = 0.999... was that it is intuitively obvious that the "problem" is that we use base-10 for decimals. There is nothing magic or unknowable about the quantity 1/3. I've often wondered if there is some alternate base or mathematical system entirely that would be "better" in these respects. The thought usually comes up thinking about why pi is such an "ugly" number in base-10 decimal.

scrollaway 6y ago
Base 12 is a better base overall. It is divisible by more numbers. That is why it is used in various monetary, time and measurement systems. It's the one issue I have with the metric system... But that ship has sailed :) look up the dozenal society if you're curious how fervent some supporters might be.
- jacobparker 6y ago
  Any other argument for a different base aside, in base 12 you have the analogous “problem” that 0.BBB... = 1. None of the usual bases, equipped with this “...” power, avoid the “problem” (non-unique representation).
- Koshkin 6y ago
  The Babylonians though it was 60. (But imagine having to remember additional 50 numeric characters.)
  - bmn__ 6y ago
    > remember additional 50 numeric characters The numerals are not distinctly varied like our Arabic numerals. Quite the opposite, they are repetitive and completely systematic and require 80% less effort to remember. https://commons.wikimedia.org/wiki/File:Babylonian_numerals.svg https://commons.wikimedia.org/wiki/File:Babylonian_numerals....
throwaway894345 6y ago
I think the problem is that people are often only taught base-10 so they confuse numerals with numbers. If you learn base 2 and then base 16 and then base pi, you start to realize that numbers are something more abstract than whichever numeral system we use to represent them. Rightly or wrongly, the way I imagine integers now is an infinite set of different numerals (base infinity?) such that there is only ever 1 digit (I don't actually have concrete pictorials assigned to those numerals).
- bluquark 6y ago
  Interesting. I find it funny to imagine what if you did have pictorial representations. They'd have more and more complex strokes and knots, and you would also need an infinitely large paper or infinitely precise pen in order to not end up repeating a number sooner or later as you enumerate them. Come to think of it, this is one way of thinking about the relationship between symbols and geometry.
  - throwaway894345 6y ago
    Who says pictorials have to be 2-dimensional? ;)
pbhjpbhj 6y ago
I think the confusion comes when mathematicians say "a sheet of paper can be 0.333... (0.3_, ie 0.3 recurring) units long" or something because in experienced reality we can always choose a measure that's rational (in the maths sense). That 1/3 of a meter can be measured in one-third-meter units and be precise and easily written. Now sure, make a square using those measures and measure the diagonal, like an awkward mathematician - "see, see, we need irrationals!" - but then we can just cut another measure that's exactly that length ... stupid mathematicians! Yeah representation is not reality. Now, says the ratio of those two measuring sticks ...
brlewis 6y ago
Whatever base you use, it will only be useful for rationals. You can't use any base to significantly improve the representation of pi. In base 7, 3.1 would be a nice approximation, but you can't go beyond approximations.
- hansvm 6y ago
  It doesn't have a terrible representation in base pi ;)
  - brlewis 6y ago
    ;) As someone fond of integers, I can't advocate base pi.
lmkg 6y ago
You can't fix irrationals by changing base. But, there are other approaches. Continued fractions give very approachable representations for common irrational numbers like e and sqrt(2). While π doesn't have a good continued fraction, it has some very well-behaved generalized continued fractions. https://en.wikipedia.org/wiki/Continued_fraction https://en.wikipedia.org/wiki/Continued_fraction

scrollaway 6y ago

Base 12 is a better base overall. It is divisible by more numbers. That is why it is used in various monetary, time and measurement systems. It's the one issue I have with the metric system... But that ship has sailed :) look up the dozenal society if you're curious how fervent some supporters might be.

jacobparker 6y ago
Any other argument for a different base aside, in base 12 you have the analogous “problem” that 0.BBB... = 1. None of the usual bases, equipped with this “...” power, avoid the “problem” (non-unique representation).
Koshkin 6y ago
The Babylonians though it was 60. (But imagine having to remember additional 50 numeric characters.)
- bmn__ 6y ago
  > remember additional 50 numeric characters The numerals are not distinctly varied like our Arabic numerals. Quite the opposite, they are repetitive and completely systematic and require 80% less effort to remember. https://commons.wikimedia.org/wiki/File:Babylonian_numerals.svg https://commons.wikimedia.org/wiki/File:Babylonian_numerals....

throwaway894345 6y ago

I think the problem is that people are often only taught base-10 so they confuse numerals with numbers. If you learn base 2 and then base 16 and then base pi, you start to realize that numbers are something more abstract than whichever numeral system we use to represent them. Rightly or wrongly, the way I imagine integers now is an infinite set of different numerals (base infinity?) such that there is only ever 1 digit (I don't actually have concrete pictorials assigned to those numerals).

bluquark 6y ago
Interesting. I find it funny to imagine what if you did have pictorial representations. They'd have more and more complex strokes and knots, and you would also need an infinitely large paper or infinitely precise pen in order to not end up repeating a number sooner or later as you enumerate them. Come to think of it, this is one way of thinking about the relationship between symbols and geometry.
- throwaway894345 6y ago
  Who says pictorials have to be 2-dimensional? ;)

pbhjpbhj 6y ago

I think the confusion comes when mathematicians say "a sheet of paper can be 0.333... (0.3_, ie 0.3 recurring) units long" or something because in experienced reality we can always choose a measure that's rational (in the maths sense). That 1/3 of a meter can be measured in one-third-meter units and be precise and easily written. Now sure, make a square using those measures and measure the diagonal, like an awkward mathematician - "see, see, we need irrationals!" - but then we can just cut another measure that's exactly that length ... stupid mathematicians! Yeah representation is not reality. Now, says the ratio of those two measuring sticks ...

brlewis 6y ago

Whatever base you use, it will only be useful for rationals. You can't use any base to significantly improve the representation of pi. In base 7, 3.1 would be a nice approximation, but you can't go beyond approximations.

hansvm 6y ago
It doesn't have a terrible representation in base pi ;)
- brlewis 6y ago
  ;) As someone fond of integers, I can't advocate base pi.

lmkg 6y ago

You can't fix irrationals by changing base. But, there are other approaches. Continued fractions give very approachable representations for common irrational numbers like e and sqrt(2). While π doesn't have a good continued fraction, it has some very well-behaved generalized continued fractions. https://en.wikipedia.org/wiki/Continued_fraction https://en.wikipedia.org/wiki/Continued_fraction