3 ms·

Problem is, if you make sqrt(2) exact - say, by an appropriate choice of the number system - then 1 will not be exact. This is because the ratio sqrt(2) / 1 is

Problem is, if you make sqrt(2) exact - say, by an appropriate choice of the number system - then 1 will not be exact. This is because the ratio sqrt(2) / 1 is irrational in any number system.

thaumasiotes 6y ago
But this isn't true. As gowld points out, you can easily represent an exact 1 and an exact sqrt(2) simultaneously; all you need is compass and straightedge. Interestingly, that's as far as it goes; if you start with a line of length 1, compass and straightedge will let you construct a line with length equal to the square root of any rational number. You can do addition, multiplication, subtraction, division... and square roots, and there things come to a stop.
- whatshisface 6y ago
  But what singles out a compass and straightedge as a "the" construction system? There are infinity other ways to construct numbers, each one with a potentially different set of constructable numbers.
  - thaumasiotes 6y ago
    You can use them to draw lines which you can then use to cut leather. They don't need to be "the" construction system; they are a construction system, and a very simple one.
    - whatshisface 6y ago
      A pencil tied to the end of a string which is itself tied to the outside of a circle is also a construction system, one that happens to be able to construct the goat-grass area exactly. :)
      - thaumasiotes 6y ago
        > one that happens to be able to construct the goat-grass area exactly. How? How did you get a string of the correct length? Note that the tool you described is a compass.
- Koshkin 6y ago
  > compass and straightedge But those are not symbols. We're talking symbols (digits).
  - thaumasiotes 6y ago
    Who are we? The problem is to create a leash of a certain length. If you can make a leash exactly one unit long, then you can also make a leash exactly √2 units long; the fact that the ratio of the two numbers is irrational doesn't impede this in any way.
    - Koshkin 6y ago
      To remind, the context is: > you can't write it down as a decimal
      - thaumasiotes 6y ago
        No, the context is whatshisface claiming that there is a (resolvable) tension between the concept of an exact representation and an infinite decimal expansion, and gowld responding that decimal expansions are irrelevant. That doesn't support the idea that we're talking about digital representation; it undermines it.

Problem is, if you make sqrt(2) exact - say, by an appropriate choice of the number system - then 1 will not be exact. This is because the ratio sqrt(2) / 1 is irrational in any number system.

thaumasiotes 6y ago

But this isn't true. As gowld points out, you can easily represent an exact 1 and an exact sqrt(2) simultaneously; all you need is compass and straightedge. Interestingly, that's as far as it goes; if you start with a line of length 1, compass and straightedge will let you construct a line with length equal to the square root of any rational number. You can do addition, multiplication, subtraction, division... and square roots, and there things come to a stop.

whatshisface 6y ago
But what singles out a compass and straightedge as a "the" construction system? There are infinity other ways to construct numbers, each one with a potentially different set of constructable numbers.
- thaumasiotes 6y ago
  You can use them to draw lines which you can then use to cut leather. They don't need to be "the" construction system; they are a construction system, and a very simple one.
  - whatshisface 6y ago
    A pencil tied to the end of a string which is itself tied to the outside of a circle is also a construction system, one that happens to be able to construct the goat-grass area exactly. :)
    - thaumasiotes 6y ago
      > one that happens to be able to construct the goat-grass area exactly. How? How did you get a string of the correct length? Note that the tool you described is a compass.
Koshkin 6y ago
> compass and straightedge But those are not symbols. We're talking symbols (digits).
- thaumasiotes 6y ago
  Who are we? The problem is to create a leash of a certain length. If you can make a leash exactly one unit long, then you can also make a leash exactly √2 units long; the fact that the ratio of the two numbers is irrational doesn't impede this in any way.
  - Koshkin 6y ago
    To remind, the context is: > you can't write it down as a decimal
    - thaumasiotes 6y ago
      No, the context is whatshisface claiming that there is a (resolvable) tension between the concept of an exact representation and an infinite decimal expansion, and gowld responding that decimal expansions are irrelevant. That doesn't support the idea that we're talking about digital representation; it undermines it.