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I'm not sure any mathematician would agree with this page: https://wikenigma.org.uk/content/mathematics/irrational_numbers https://wikenigma.org.uk/content/math

I'm not sure any mathematician would agree with this page: https://wikenigma.org.uk/content/mathematics/irrational_numbers https://wikenigma.org.uk/content/mathematics/irrational_numb... It says "An irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Put another way, it can never be specified with absolute accuracy." I guess their thinking is that since the decimal expansion goes on forever, the irrational number can never be "known." But that's a really narrow view of what it means to "know" a number. Similarly we can't ever "know" 1/7, for instance. A better set of numbers to focus on in the spirit of this would be the uncomputable numbers.

blep_ 4y ago
You're changing 'specified' to 'known' there, and that's breaking it. 1/7 can be specified as a decimal number (as "0.142857..."). It's an infinitely long string, but that is a complete description of the contents of it. sqrt(2) is known (you can calculate it to arbitrary precision) but cannot be written as a decimal number, even with repeating-digits shorthand.
- ComplexSystems 4y ago
  It isn't that I'm changing "specified" to "known"; I'm pointing out that this is an article on a purported "encyclopedia of unknowns," and it's only being included there because the authors clearly view these as "unknowable" in some sense. The thing you are calling "repeating-digits shorthand" is a highly nontrivial thing. It essentially equates to an algorithm that gives you the N'th decimal digit of the expansion for any N you want. It does so happen that 1/7 has an extremely simple pattern - it's just the same six digits repeating, so you can take N mod 6 and immediately know what the digit is. But there are plenty of irrational numbers which have similarly simple patterns, such as the Champernowne constant(https://en.wikipedia.org/wiki/Champernowne_constant https://en.wikipedia.org/wiki/Champernowne_constant), whose decimal expansion is just the concatenation of the decimal expansion of every integer: 0.123456789101112131415... . This is an irrational number, but it's a trivial programming 101 exercise to write a basic algorithm that can give you the N'th digit in the decimal expansion. The same thing is true for e.g. sqrt(2) and pi. These numbers have decimal expansions that would seem to be random and chaotic at first glance, but are really not - there is again another pattern there. For instance, there exist "spigot" algorithms that can give us the N'th digit in the decimal expansion of pi without having to compute any of the previous digits, because it utilizes such a pattern that is simply a bit less readily seen and perceived by human beings, but is still there. All of these numbers can easily be "known" in the sense that you can know as many digits as you want, whenever you want. Uncomputable numbers are the numbers for which this is not true.
- cwillu 4y ago
  1/7 can't be written as a decimal without adding extra notation: the “…” is as much a symbol as “/” or “√”

blep_ 4y ago

You're changing 'specified' to 'known' there, and that's breaking it. 1/7 can be specified as a decimal number (as "0.142857..."). It's an infinitely long string, but that is a complete description of the contents of it. sqrt(2) is known (you can calculate it to arbitrary precision) but cannot be written as a decimal number, even with repeating-digits shorthand.

ComplexSystems 4y ago
It isn't that I'm changing "specified" to "known"; I'm pointing out that this is an article on a purported "encyclopedia of unknowns," and it's only being included there because the authors clearly view these as "unknowable" in some sense. The thing you are calling "repeating-digits shorthand" is a highly nontrivial thing. It essentially equates to an algorithm that gives you the N'th decimal digit of the expansion for any N you want. It does so happen that 1/7 has an extremely simple pattern - it's just the same six digits repeating, so you can take N mod 6 and immediately know what the digit is. But there are plenty of irrational numbers which have similarly simple patterns, such as the Champernowne constant(https://en.wikipedia.org/wiki/Champernowne_constant https://en.wikipedia.org/wiki/Champernowne_constant), whose decimal expansion is just the concatenation of the decimal expansion of every integer: 0.123456789101112131415... . This is an irrational number, but it's a trivial programming 101 exercise to write a basic algorithm that can give you the N'th digit in the decimal expansion. The same thing is true for e.g. sqrt(2) and pi. These numbers have decimal expansions that would seem to be random and chaotic at first glance, but are really not - there is again another pattern there. For instance, there exist "spigot" algorithms that can give us the N'th digit in the decimal expansion of pi without having to compute any of the previous digits, because it utilizes such a pattern that is simply a bit less readily seen and perceived by human beings, but is still there. All of these numbers can easily be "known" in the sense that you can know as many digits as you want, whenever you want. Uncomputable numbers are the numbers for which this is not true.
cwillu 4y ago
1/7 can't be written as a decimal without adding extra notation: the “…” is as much a symbol as “/” or “√”