6 ms·

What would happen if you made Pi = 1, then adjust all other numbers accordingly?

ctdonath 14y ago
Relativistic mathematics. I think my head just exploded.
philh 14y ago
μ -- To clarify, that's not a thing you can meaningfully do. It makes sense to define your units such that, e.g. c=1, because c has units (distance per time). You're really adjusting your units, not your numbers. But π is dimensionless. Diameter and circumference are both measures of distance, and they are never equal. If you say "this number times one equals that number", and the numbers aren't the same, then when you say "one" you don't actually mean "one" the way anyone else speaks about it. (By convention: whenever you can do multiplication, if there's something called "one", then "one" is the multiplicative identity.) There's nothing to stop you from, e.g. renaming all numbers x -> x/π. But you wouldn't gain anything; you'd just say things like "1/π squared is 1/π", and "1 squared is π", and "π times 1/π is π". It looks weird, but what's happening is perfictly ordinary, just with weird names.
- anonymouz 14y ago
  One could sort of satisfy naturalethic's wish by choosing a numeral system with pi as basis. Then pi would have digit representation '1' in this system. And yes, mathematicians have studied transcendental bases for numeral systems (Knuth in particular, see the short paragraph on http://mathworld.wolfram.com/Base.html http://mathworld.wolfram.com/Base.html which explicitly mentions pi as basis). Of course it's not a good idea to base your day-to-day numeral system on pi, but I know that at least for irrational bases there are actual applications.
  - mej10 14y ago
    This is a good point as well, and more in line with what the original commenter was asking. Using pi as a base is even briefly discussed on wikipedia. I had some interest in using e as the basis for a while, since (as is pointed out on wiki) "The base e is the most economical choice of radix β > 1 (Hayes 2001), where the radix economy is measured as the product of the radix and the length of the string of symbols needed to express a given range of values." http://en.wikipedia.org/wiki/Non-integer_representation#Examples http://en.wikipedia.org/wiki/Non-integer_representation#Exam...
  - jerf 14y ago
    "One could sort of satisfy naturalethic's wish by choosing a numeral system with pi as basis. Then pi would have digit representation '1' in this system." I'm pretty sure pi comes out to 10 base pi, not 1 base pi.
    - anonymouz 14y ago
      Ah, right, of course! Still, I think this is the closest possible to the wish the OP had.
    - blacksmythe 14y ago
      Numeral systems are not restricted to the set (base N). "a numeral system with pi as basis" could be (base pi)pi, where pi =-> '1' and pipi -> '10' and unity -> '0.1'.
      - jerf 14y ago
        I'd refer you back to philh's answer. You can do that and it isn't "wrong", but it is surprisingly uninteresting, as compared to a discussion of irrational or negative or imaginary bases.
nthitz 14y ago
Sounds a little irrational...
mej10 14y ago
Depends on what you mean by setting pi = 1. You can subtract (pi - 1) from every number, giving you a translation of the real numbers. You can divide everything by pi, giving you a slight compression that ultimately doesn't mean anything. You could use it to create a field extension of the rationals, giving you a http://en.wikipedia.org/wiki/Algebraic_number_field http://en.wikipedia.org/wiki/Algebraic_number_field where {1, pi} is the basis. But to answer your question: probably nothing.
- anonymouz 14y ago
  Careful, none of these actually "set pi=1". Also, since pi is transcendental, the field would not be an algebraic number field, but in fact isomorphic to the rational functions over Q, i.e., quotients of polynomials with coefficients in Q.
  - mej10 14y ago
    Yep, you are right. I can't believe I forgot that.

Relativistic mathematics. I think my head just exploded.

philh 14y ago

μ -- To clarify, that's not a thing you can meaningfully do. It makes sense to define your units such that, e.g. c=1, because c has units (distance per time). You're really adjusting your units, not your numbers. But π is dimensionless. Diameter and circumference are both measures of distance, and they are never equal. If you say "this number times one equals that number", and the numbers aren't the same, then when you say "one" you don't actually mean "one" the way anyone else speaks about it. (By convention: whenever you can do multiplication, if there's something called "one", then "one" is the multiplicative identity.) There's nothing to stop you from, e.g. renaming all numbers x -> x/π. But you wouldn't gain anything; you'd just say things like "1/π squared is 1/π", and "1 squared is π", and "π times 1/π is π". It looks weird, but what's happening is perfictly ordinary, just with weird names.

anonymouz 14y ago
One could sort of satisfy naturalethic's wish by choosing a numeral system with pi as basis. Then pi would have digit representation '1' in this system. And yes, mathematicians have studied transcendental bases for numeral systems (Knuth in particular, see the short paragraph on http://mathworld.wolfram.com/Base.html http://mathworld.wolfram.com/Base.html which explicitly mentions pi as basis). Of course it's not a good idea to base your day-to-day numeral system on pi, but I know that at least for irrational bases there are actual applications.
- mej10 14y ago
  This is a good point as well, and more in line with what the original commenter was asking. Using pi as a base is even briefly discussed on wikipedia. I had some interest in using e as the basis for a while, since (as is pointed out on wiki) "The base e is the most economical choice of radix β > 1 (Hayes 2001), where the radix economy is measured as the product of the radix and the length of the string of symbols needed to express a given range of values." http://en.wikipedia.org/wiki/Non-integer_representation#Examples http://en.wikipedia.org/wiki/Non-integer_representation#Exam...
- jerf 14y ago
  "One could sort of satisfy naturalethic's wish by choosing a numeral system with pi as basis. Then pi would have digit representation '1' in this system." I'm pretty sure pi comes out to 10 base pi, not 1 base pi.
  - anonymouz 14y ago
    Ah, right, of course! Still, I think this is the closest possible to the wish the OP had.
  - blacksmythe 14y ago
    Numeral systems are not restricted to the set (base N). "a numeral system with pi as basis" could be (base pi)pi, where pi =-> '1' and pipi -> '10' and unity -> '0.1'.
    - jerf 14y ago
      I'd refer you back to philh's answer. You can do that and it isn't "wrong", but it is surprisingly uninteresting, as compared to a discussion of irrational or negative or imaginary bases.

nthitz 14y ago

Sounds a little irrational...

mej10 14y ago

Depends on what you mean by setting pi = 1. You can subtract (pi - 1) from every number, giving you a translation of the real numbers. You can divide everything by pi, giving you a slight compression that ultimately doesn't mean anything. You could use it to create a field extension of the rationals, giving you a http://en.wikipedia.org/wiki/Algebraic_number_field http://en.wikipedia.org/wiki/Algebraic_number_field where {1, pi} is the basis. But to answer your question: probably nothing.

anonymouz 14y ago
Careful, none of these actually "set pi=1". Also, since pi is transcendental, the field would not be an algebraic number field, but in fact isomorphic to the rational functions over Q, i.e., quotients of polynomials with coefficients in Q.
- mej10 14y ago
  Yep, you are right. I can't believe I forgot that.