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Why the interest in whether a number is irrational or not? Is it just a researcher's fun pastime or does it tell us something useful? From the article: Even th

Why the interest in whether a number is irrational or not? Is it just a researcher's fun pastime or does it tell us something useful? From the article: Even though the numbers that feature in mathematics research are, by definition, not random, mathematicians believe most of them should be irrational too. So is there some kind of validation happening where we are meant to be suspicious of numbers that aren't irrational?

treyd 2y ago
The interesting thing about irrational numbers is that they can't be constructed from a finite number of symbols from basic algebra. This is especially interesting when they have relationships with other irrational numbers, like the unexpected relationship between pi and e (and i) demonstrated in Euler's formula.
- nh23423fefe 2y ago
  Constructed is the wrong word. sqrt(2) is constructible and irrational
  - jjtheblunt 2y ago
    what's the right word?
  - crabbone 2y ago
    OP said constructed from finite number of symbols from basic algebra. There's no finite construction of sqrt(2) using addition and multiplication.
    - aidenn0 2y ago
      Surely exponentiation is basic algebra? And sqrt(2) is trivially constructable from exponentiation.
      - crabbone 2y ago
        No. exponentiation and square root are not algebraic operations. Only addition and multiplication are. That's kind of the whole point of this theory / subfield of mathematics. The operations you mentioned can be found in many different subfields of mathematics, eg. real analysis, number theory, or even arithmetic (using a more broad, but a well-accepted definition). But not in algebra. It's the point of algebra to only have addition and multiplication. And it's why, for example, algebraic geometry exists (because algebraic geometers want to avoid transcendental functions like sqrt(), sin() etc.)
    - LPisGood 2y ago
      I feel like you mean basic arithmetic, not basic algebra.
      - crabbone 2y ago
        Algebra is a very well defined mathematical term: > an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear" from: https://en.wikipedia.org/wiki/Algebra_over_a_field https://en.wikipedia.org/wiki/Algebra_over_a_field Historically though, the word "algebra" was used more broadly, and the further into the past you go, the more vague this term becomes. But, today, if you ask a mathematician, the definition above is how they would immediately understand algebra, and other kinds of algebras would need a qualification, eg. "linear algebra" or "abstract algebra" etc. Another way to look at this is to say that various subfields of mathematics that are called "algebra" are studies of particular kinds of algebra (from the first definition). And so they will still have all the same elements: a set (with some restrictions on it), a multiplication and addition. It could be surprising that so few basic elements give rise to such a rich field, but that's how math is... In a way, the elements you work with act more as constraints rather than extra dimensions. So, theories with very few basic elements tend to capture more stuff and be richer in terms of theorems than theories with more basic elements.
        LPisGood 2y ago
        You’re confusing “algebra” with “an algebra.” You’re misunderstanding the terms here. For a simple example, group theory is absolutely a branch of algebra, and a group is not “an algebra”
        crabbone 2y ago
        I don't see how any of that matters to answering the question. Anything that's labeled "algebra" will have addition, multiplication and a field over which those operations are defined. This is the whole point of the term. Transcendental functions s.a. sqrt() are to algebra like the trolley problem is to physics: deliberately excluded from the domain of discourse.
- erehweb 2y ago
  I think you may be thinking of transcendental numbers, which are a subset of irrational numbers. https://en.wikipedia.org/wiki/Transcendental_number https://en.wikipedia.org/wiki/Transcendental_number
  - treyd 2y ago
    Ah I'm mixing up my terms, you're right.
CassianAI 2y ago
Practically there's uses in areas like cryptography and simulation where Pseudo-random number generators (PRNGs) are used. If the numbers aren't irrational then there may be flaws in the assumptions being used. Beyond direct application, knowing a number is irrational can be a form of validation for theoretical modelling. If a number arising in a model turns out to be rational, it could mean an unexpected simplicity or symmetry, which is worth exploring further. Conversely, irrationality is often expected in complex systems and may confirm the soundness of a mathematical construct or physical model. I guess a good example of that is the relationship of light spectra and Planks constant.
- wat10000 2y ago
  How could it be relevant to cryptography/simulation? Short of a symbolic algebra system, all numbers on a computer are rational. Pi is irrational but M_PI is rational. How can a PRNG be based on an assumption of irrationality when it has no access to irrational numbers?
  - AlotOfReading 2y ago
    You can make PRNGs based on approximations to (disjunctive) irrational numbers, and the irrationality of the number being approximated is important to its quality. I'm not aware of any widespread real-world PRNGs constructed this way because they're less efficient than traditional PRNGs. It's mostly a mathematical trick to be used in proofs and thought experiments. I suspect they're referring to the more common practice of taking the first N digits of a well known number like Pi or e that happens to be irrational as a magic constant of known provenance. 1245678 is another common one though, which obviously isn't irrational.
    - CassianAI 2y ago
      Agreed re approximating and use of constants. Admittedly, I haven’t looked into PRNGs much since my numerical analysis college days! On simulation I did once do some work with Monte Carlo using quasi-random sequences (e.g Sobol), which can provide better coverage than pure randomness for certain problems.
crabbone 2y ago
Rational numbers are a lot more useful than irrational. Eg. everything that happens in digital computers is rational. If you need a measuring tool, the scale is going to be rational. Irrational numbers, in practice, cause lack of precision. So, for example, if you draw a square 1m x 1m, its diagonal isn't sqrt(2)m. It's some rational number because that square is made of some discrete elements that you can count, and so is its diagonal. But, upfront, you won't be able to tell what exactly that number is going to be. Another way to look at what irrational numbers are is to say that they sort of don't really exist, they are like limits, or some ideals that cannot be reached because you'd need to spend infinity to reach that exact number when counting, measuring etc. So, again, from a practical point of view, and especially in fields that like to measure things or build precise things, you want numbers to be rational, and, preferably with "small" denominators. On the other hand, irrational numbers give rise to all sorts of bizarre properties because they aren't usually considered as a point on a number line, but more of a process that describes some interesting behavior, sequences, infinite sums, recurrences etc. So, in practical terms, you aren't interested in the number itself, but rather in the process through which it is obtained. * * * Also, worth noting that there's a larger group that includes rationals, the algebraic numbers, which also includes some irrational numbers (eg. sqrt(2) is algebraic, but not rational). Algebraic numbers are numbers that can be expressed as roots of quadratic or higher (but finite) power equations. These, perhaps, capture more of the "useful" numbers that we operate on in everyday life in terms of measuring or counting things. And the practical use of these numbers is that they can be "compactly" written / stored, so it's easy to operate on them and they have all kinds of desirable mathematical properties like all kinds of closures etc. Algebraic numbers are also useful because any computable function has a polynomial that coincides with it at every point. Which means that with these numbers you can, in principle, model every algorithm imaginable. That seems pretty valuable :)
- tim-kt 2y ago
  > Another way to look at what irrational numbers are is to say that they sort of don't really exist, they are like limits, or some ideals that cannot be reached because you'd need to spend infinity to reach that exact number when counting, measuring etc. Depending on your definition of "existence", rational numbers (or any numbers) don't exist either.
sunshowers 2y ago
These things are useless for centuries until they suddenly underpin all of modern society.
octachron 2y ago
Mathematicians are more interested in the gap in our proof framework. Like stated in the articles, many "interesting" constants appearing in mathematics feels like obviously irrational. However, proofs that they are irrational have been eluding mathematicians for centuries. This contrast is seen as a sign that we may be just missing the right mathematical insights. And if we find this insight, we might be able to adapt it to unlock other open problems in mathematics (or computer science?). This is one of these cases where the path (the new proof framework) is expected to be much more interesting than the initial destination (the fact that yes the Euler constant is irrational, of course).
alpple 2y ago
Imagine you're a character in a Lord of the Rings novel. And math is the imaginary landscape that you are going on an adventure through. When a number is irrational, it is not easy to work with compared to natural numbers. So they're marking the map as a hard pass in the mountain ridge that is on your journey, assuming you want to explore that world.
yen223 2y ago
The interesting thing to me is how poorly-understood irrationality is. Despite it being relatively common knowledge nowadays that pi is irrational, we've only proven that pi is irrational in the past 300 years or so. And the proof is not simple (at least to me) As the article states, we didn't have a general purpose "plug this number in and it'll spit out whether the number is rational or not" formula. The irrationality proofs that we do have tend to bespoke to the structure of the number itself. That's why this research is exciting.

treyd 2y ago

The interesting thing about irrational numbers is that they can't be constructed from a finite number of symbols from basic algebra. This is especially interesting when they have relationships with other irrational numbers, like the unexpected relationship between pi and e (and i) demonstrated in Euler's formula.

nh23423fefe 2y ago
Constructed is the wrong word. sqrt(2) is constructible and irrational
- jjtheblunt 2y ago
  what's the right word?
- crabbone 2y ago
  OP said constructed from finite number of symbols from basic algebra. There's no finite construction of sqrt(2) using addition and multiplication.
  - aidenn0 2y ago
    Surely exponentiation is basic algebra? And sqrt(2) is trivially constructable from exponentiation.
    - crabbone 2y ago
      No. exponentiation and square root are not algebraic operations. Only addition and multiplication are. That's kind of the whole point of this theory / subfield of mathematics. The operations you mentioned can be found in many different subfields of mathematics, eg. real analysis, number theory, or even arithmetic (using a more broad, but a well-accepted definition). But not in algebra. It's the point of algebra to only have addition and multiplication. And it's why, for example, algebraic geometry exists (because algebraic geometers want to avoid transcendental functions like sqrt(), sin() etc.)
  - LPisGood 2y ago
    I feel like you mean basic arithmetic, not basic algebra.
    - crabbone 2y ago
      Algebra is a very well defined mathematical term: > an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear" from: https://en.wikipedia.org/wiki/Algebra_over_a_field https://en.wikipedia.org/wiki/Algebra_over_a_field Historically though, the word "algebra" was used more broadly, and the further into the past you go, the more vague this term becomes. But, today, if you ask a mathematician, the definition above is how they would immediately understand algebra, and other kinds of algebras would need a qualification, eg. "linear algebra" or "abstract algebra" etc. Another way to look at this is to say that various subfields of mathematics that are called "algebra" are studies of particular kinds of algebra (from the first definition). And so they will still have all the same elements: a set (with some restrictions on it), a multiplication and addition. It could be surprising that so few basic elements give rise to such a rich field, but that's how math is... In a way, the elements you work with act more as constraints rather than extra dimensions. So, theories with very few basic elements tend to capture more stuff and be richer in terms of theorems than theories with more basic elements.
      - LPisGood 2y ago
        You’re confusing “algebra” with “an algebra.” You’re misunderstanding the terms here. For a simple example, group theory is absolutely a branch of algebra, and a group is not “an algebra”
        crabbone 2y ago
        I don't see how any of that matters to answering the question. Anything that's labeled "algebra" will have addition, multiplication and a field over which those operations are defined. This is the whole point of the term. Transcendental functions s.a. sqrt() are to algebra like the trolley problem is to physics: deliberately excluded from the domain of discourse.
erehweb 2y ago
I think you may be thinking of transcendental numbers, which are a subset of irrational numbers. https://en.wikipedia.org/wiki/Transcendental_number https://en.wikipedia.org/wiki/Transcendental_number
- treyd 2y ago
  Ah I'm mixing up my terms, you're right.

CassianAI 2y ago

Practically there's uses in areas like cryptography and simulation where Pseudo-random number generators (PRNGs) are used. If the numbers aren't irrational then there may be flaws in the assumptions being used. Beyond direct application, knowing a number is irrational can be a form of validation for theoretical modelling. If a number arising in a model turns out to be rational, it could mean an unexpected simplicity or symmetry, which is worth exploring further. Conversely, irrationality is often expected in complex systems and may confirm the soundness of a mathematical construct or physical model. I guess a good example of that is the relationship of light spectra and Planks constant.

wat10000 2y ago
How could it be relevant to cryptography/simulation? Short of a symbolic algebra system, all numbers on a computer are rational. Pi is irrational but M_PI is rational. How can a PRNG be based on an assumption of irrationality when it has no access to irrational numbers?
- AlotOfReading 2y ago
  You can make PRNGs based on approximations to (disjunctive) irrational numbers, and the irrationality of the number being approximated is important to its quality. I'm not aware of any widespread real-world PRNGs constructed this way because they're less efficient than traditional PRNGs. It's mostly a mathematical trick to be used in proofs and thought experiments. I suspect they're referring to the more common practice of taking the first N digits of a well known number like Pi or e that happens to be irrational as a magic constant of known provenance. 1245678 is another common one though, which obviously isn't irrational.
  - CassianAI 2y ago
    Agreed re approximating and use of constants. Admittedly, I haven’t looked into PRNGs much since my numerical analysis college days! On simulation I did once do some work with Monte Carlo using quasi-random sequences (e.g Sobol), which can provide better coverage than pure randomness for certain problems.

crabbone 2y ago

Rational numbers are a lot more useful than irrational. Eg. everything that happens in digital computers is rational. If you need a measuring tool, the scale is going to be rational. Irrational numbers, in practice, cause lack of precision. So, for example, if you draw a square 1m x 1m, its diagonal isn't sqrt(2)m. It's some rational number because that square is made of some discrete elements that you can count, and so is its diagonal. But, upfront, you won't be able to tell what exactly that number is going to be. Another way to look at what irrational numbers are is to say that they sort of don't really exist, they are like limits, or some ideals that cannot be reached because you'd need to spend infinity to reach that exact number when counting, measuring etc. So, again, from a practical point of view, and especially in fields that like to measure things or build precise things, you want numbers to be rational, and, preferably with "small" denominators. On the other hand, irrational numbers give rise to all sorts of bizarre properties because they aren't usually considered as a point on a number line, but more of a process that describes some interesting behavior, sequences, infinite sums, recurrences etc. So, in practical terms, you aren't interested in the number itself, but rather in the process through which it is obtained. * * * Also, worth noting that there's a larger group that includes rationals, the algebraic numbers, which also includes some irrational numbers (eg. sqrt(2) is algebraic, but not rational). Algebraic numbers are numbers that can be expressed as roots of quadratic or higher (but finite) power equations. These, perhaps, capture more of the "useful" numbers that we operate on in everyday life in terms of measuring or counting things. And the practical use of these numbers is that they can be "compactly" written / stored, so it's easy to operate on them and they have all kinds of desirable mathematical properties like all kinds of closures etc. Algebraic numbers are also useful because any computable function has a polynomial that coincides with it at every point. Which means that with these numbers you can, in principle, model every algorithm imaginable. That seems pretty valuable :)

tim-kt 2y ago
> Another way to look at what irrational numbers are is to say that they sort of don't really exist, they are like limits, or some ideals that cannot be reached because you'd need to spend infinity to reach that exact number when counting, measuring etc. Depending on your definition of "existence", rational numbers (or any numbers) don't exist either.

sunshowers 2y ago

These things are useless for centuries until they suddenly underpin all of modern society.

octachron 2y ago

Mathematicians are more interested in the gap in our proof framework. Like stated in the articles, many "interesting" constants appearing in mathematics feels like obviously irrational. However, proofs that they are irrational have been eluding mathematicians for centuries. This contrast is seen as a sign that we may be just missing the right mathematical insights. And if we find this insight, we might be able to adapt it to unlock other open problems in mathematics (or computer science?). This is one of these cases where the path (the new proof framework) is expected to be much more interesting than the initial destination (the fact that yes the Euler constant is irrational, of course).

alpple 2y ago

Imagine you're a character in a Lord of the Rings novel. And math is the imaginary landscape that you are going on an adventure through. When a number is irrational, it is not easy to work with compared to natural numbers. So they're marking the map as a hard pass in the mountain ridge that is on your journey, assuming you want to explore that world.

yen223 2y ago

The interesting thing to me is how poorly-understood irrationality is. Despite it being relatively common knowledge nowadays that pi is irrational, we've only proven that pi is irrational in the past 300 years or so. And the proof is not simple (at least to me) As the article states, we didn't have a general purpose "plug this number in and it'll spit out whether the number is rational or not" formula. The irrationality proofs that we do have tend to bespoke to the structure of the number itself. That's why this research is exciting.