> According to calculations that take into account the Cosmological Constant, the energy should be around 10^−9 joules per cubic meter.
> If on the other hand the calculation is based around Quantum Electrodynamic theory and the Planck Constant, the figure is 10^113 joules per cubic meter.
> One (or both) of the calculations must be wrong - with an error margin of more than 120 orders of magnitude - making it the most spectacular mathematical disagreement in all of physics.
"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.
Re:Wikenigma – an encyclopedia of scientific questions with ...
by blep_ on Friday December 30, 2022 @03:52PM
(#34189940)
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.
Re:Wikenigma – an encyclopedia of scientific questions with ...
by ComplexSystems on Sunday January 08, 2023 @11:21PM
(#34306710)
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), 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.
I particularly like the 120 orders of magnitude difference in vacuum energy calculations: https://wikenigma.org.uk/content/physics/quantum_physics/vac...
> According to calculations that take into account the Cosmological Constant, the energy should be around 10^−9 joules per cubic meter.
> If on the other hand the calculation is based around Quantum Electrodynamic theory and the Planck Constant, the figure is 10^113 joules per cubic meter.
> One (or both) of the calculations must be wrong - with an error margin of more than 120 orders of magnitude - making it the most spectacular mathematical disagreement in all of physics.
- gravity
- male pattern baldness
- Heller's syndrome
https://wikenigma.org.uk/content/physics/cosmology/cmb_exces...
Redshift quantization:
https://wikenigma.org.uk/content/physics/cosmology/redshift_...
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.
sqrt(2) is known (you can calculate it to arbitrary precision) but cannot be written as a decimal number, even with repeating-digits shorthand.
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), 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.