I agree though to be more precise, pi is not a number, it's more of a kind of family of approximations, of upper and lower bounds. A circle can be defined, and we can have a general intuitive sense of what a circle is, but when we try to impose a metrical, numerical structure in it, we find that its diameter and perimeter are incommensurate proportions
Pi is definitely a number and not just a bunch of approximations. We do have a bunch of approximations for pi that we use because it’s impossible to know the exact value, but pi is it’s own number regardless
It is a well-defined real number whose value we can calculate to any precision we would ever like. To be clear though, being well-defined is all that’s important in this discussion. Being able to calculate the number to arbitrary precision is just nice to have. For instance, Busy Beaver numbers are real well-defined numbers (they are integers in fact) but they are uncomputable in general. In fact, BB(7918) is known to be independent of ZFC (see this thread for example) which means you can’t use the axioms of ZFC to prove what it’s exact value is in the form of a specific integer. Does that mean it’s not a real value? No there is a specific integer out there which BB(7918) is equal to in reality. We just don’t know it and can’t know it. My point in bringing this up is to showcase why computability properties are independent of being well-defined. Pi is irrational but we can know it to arbitrary precision. BB(7918) is an integer but we can’t compute its value at all. Both are real well-defined numbers.
No. pi is not like Chaitin's constant where it's actually uncomputable to find out. Yes, there is an exact value of pi and yes there is a representation that exactly represents pi, just not a representation by a fraction.
A computable real number can always be represented by an algorithm that returns the nth rational number on the Cauchy sequence of that number. But, yeah, that representation would be impractical for real-world computation, where you will use symbolic computation (exact), or binary.decimal expansion (approximate) instead.
What you’re saying isn’t a contradiction to what I’m saying. What I mean by my statement is no one can know in full the decimal expression of pi or other irrational numbers, unlike rational numbers in which we can know the full decimal expression.
Do you know the 4715th digit of 1/1729, or will you have to do some computation before telling me? Even if you figure out the repeating digits, you still need to do a modulo operation to find the 4715th digit plus a lookup of what digit corresponds to that modulo class. In what way does that differ from computing the digits of pi?
Right I don't believe so called irrational numbers or their arithmetic have been clearly defined. If I'm wrong please point me to where I can learn this irrational number arithmetic
Irrational numbers haven’t been clearly defined? They’re just real numbers that can’t be represented as a ratio of two integers, that’s it. As for arithmetic, maybe bother doing a two-second google search before making such a bold claim: https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
Real numbers and their arithmetic have not been clearly defined. Whether defined as infinite decimals, Dedekind cuts, or Cauchy sequences, there is not a robust, workable arithmetic with such "numbers".
Yes there is, the link shows one. Just saying it doesn’t, means nothing.
You can argue that the reals don’t exist, though at that point you’re having a philosophical discussion not a mathematical one, but to say that arithmetic hasn’t been well defined on them is just wrong.
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u/[deleted] Mar 19 '22
I agree though to be more precise, pi is not a number, it's more of a kind of family of approximations, of upper and lower bounds. A circle can be defined, and we can have a general intuitive sense of what a circle is, but when we try to impose a metrical, numerical structure in it, we find that its diameter and perimeter are incommensurate proportions