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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

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.
- crabbone 2y ago
  I think it's kind of obvious what my definition of existence could be from the answer above: if it's possible to count up to that number in finite time, that number exists. By counting I mean a physical process that requires discrete non-zero intervals between counts. And you don't have to count in integers, you can count in fractions, not necessarily equal at each step: the only requirement is that the element used for counting exists (in terms of this definition) and that you are able to accomplish counting in finite time. To me, this pretty much captures what people understand the numbers to be used for outside of college math (so no transfinite, cardinals etc.)
  - tim-kt 2y ago
    That sounds plausible. I think that definition is equivalent to a number being rational. > To me, this pretty much captures what people understand the numbers to be used for outside of college math (so no transfinite, cardinals etc.) I'm in my fourth year of mathematics right now. I guess I'm not in the target group of articles such as these :P
    - crabbone 2y ago
      Yeah... the article has a lot of "simplifications" that the reader has to just kind of trust the author on... and those are meant for people with not a lot of mathematical sophistication. All this talk about "shrink fast enough" and why it's important is just some intense handwaving w/o any actual explanation. To be fair, it's kind of upsetting, but maybe there's no way to help it... some mathematical proofs can be "dumbed down" to the point that people with very little background can understand them. The proof of sqrt(2) being irrational might be one of those. But, what's given in the article feels like either the author didn't really understand the subject, or she couldn't explain what she understood in simple terms. But, it's really rare that there's such an easy to understand proof or concept. So, I don't blame her.
  - sunshowers 2y ago
    Do you mean the computable numbers? (there's an algorithm to compute them to arbitrary precision) The irrational numbers used outside of college math, like pi or e or sqrt(2), are computable, though almost all are not. You can do a lot of productive math using just computable numbers since they form a real closed field [1]. I believe they're a little harder to work with though. [1] https://en.wikipedia.org/wiki/Real_closed_field https://en.wikipedia.org/wiki/Real_closed_field
tsimionescu 2y ago
A lot of this is very much wrong. To the extent that your square is close to perfect and its sides are actually 1m, it's diagonal is just as truly sqrt(2)m long. And a circle that you draw correctly enough with a radius of 1m will have its circumference actually equal to 2pi. There is nothing more special about sqrt(2) than about 1, nor about a perfectly 90° angle versus a perfect circle. All of our drawings and constructions are approximations, but that doesn't make them naturally be integers or rationals any more than they are irrational. In other words, it would be just as accurate to say that the sides of the physical square are not 1, but x*pi for a pretty small x (that is, they are ever so slightly curved) as it would be to say that the circle's circumference isn't really 2*pi, it's actually some rational number, because the square is actually some very very many sided polyhedron. Even if we look at this from a purely physical perspective, elementary particles travel in perfectly straight lines and radiate in perfect circles in our models. And the directions of movement after a collision are not quantized, they can be arbitrary angles (just as space is not quantized, and in fact not even quantizable, in QM). And if you tried to look at a physical object and count the atoms to determine its length, you'd quickly find that it doesn't even have a constant number of atoms or a constant length, so in fact the least real concept is "an object of x meters in length", regardless of whether x is natural, rational, or irrational.
- crabbone 2y ago
  Well, no, no physical square, no matter how precisely its sides are 1m long has an irrational-lenght diagonal. That is simply impossible in the physical universe because the universe is discrete at this level. Irrational numbers are impossible in discrete context. The whole point they were invented is to capture the idea of continuous functions or continuous number line. But this is a "slight of hand", a definition that's made for convenience of solving useful problems, but isn't based in the physical reality. In a very similar way to how sqrt(-1) is not a thing in a physical reality, but it's useful to work with problems that can be described using complex numbers. > There is nothing more special about sqrt(2) than about 1 That fact that you don't understand what's special about it doesn't mean there isn't. It's a very different thing though. > All of our drawings and constructions are approximations Drawings: yes. Constructions: no. > but that doesn't make them naturally be integers or rationals any more than they are irrational. Here you've ventured into the territory you have no idea about... I'm sorry. You sound more like some LLM-generated gibberish here than anything a human with any expertise on the subject would write. Of course some things are naturally integers. We've invented integers to capture those things (in the physical universe). Similarly, rationals. There's nothing in the physical universe that's irrational in the same way how it can be an integer or a rational. Irrational numbers don't describe quantities or passage of time or forces acting on physical objects or the speed etc. because all those things are made up of small indivisible parts, and there's always a finite computable answer to how big something is, how long a process would take, how strong is the force applied to an object etc. Irrational numbers are a mathematical device to deal with different kinds of problems. Similar to how generating functions use "+" to mean a completely different thing from how it's used in algebra, or how it's used in regular languages, so are irrational called "numbers". But they aren't the same kind of thing as integers or rationals. To be honest, it would've been better not to call them "numbers" at all, to avoid this kind of confusion, but mathematics has a lot of old and bad terminology that's used due to tradition. > elementary particles travel in perfectly straight lines and radiate in perfect circles in our models The root of your problem in understanding this is: our models. Particles don't radiate in perfect circles in reality. Physical reality is discrete and cannot create perfect circles. You can imagine, however, a perfect circle and use it to a great effect to estimate the result of some physical process. But, if you truly measure the effect, you will never have an irrational number. There's no physical process of measuring anything that will end up with an irrational answer. That's simply impossible.

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.

crabbone 2y ago
I think it's kind of obvious what my definition of existence could be from the answer above: if it's possible to count up to that number in finite time, that number exists. By counting I mean a physical process that requires discrete non-zero intervals between counts. And you don't have to count in integers, you can count in fractions, not necessarily equal at each step: the only requirement is that the element used for counting exists (in terms of this definition) and that you are able to accomplish counting in finite time. To me, this pretty much captures what people understand the numbers to be used for outside of college math (so no transfinite, cardinals etc.)
- tim-kt 2y ago
  That sounds plausible. I think that definition is equivalent to a number being rational. > To me, this pretty much captures what people understand the numbers to be used for outside of college math (so no transfinite, cardinals etc.) I'm in my fourth year of mathematics right now. I guess I'm not in the target group of articles such as these :P
  - crabbone 2y ago
    Yeah... the article has a lot of "simplifications" that the reader has to just kind of trust the author on... and those are meant for people with not a lot of mathematical sophistication. All this talk about "shrink fast enough" and why it's important is just some intense handwaving w/o any actual explanation. To be fair, it's kind of upsetting, but maybe there's no way to help it... some mathematical proofs can be "dumbed down" to the point that people with very little background can understand them. The proof of sqrt(2) being irrational might be one of those. But, what's given in the article feels like either the author didn't really understand the subject, or she couldn't explain what she understood in simple terms. But, it's really rare that there's such an easy to understand proof or concept. So, I don't blame her.
- sunshowers 2y ago
  Do you mean the computable numbers? (there's an algorithm to compute them to arbitrary precision) The irrational numbers used outside of college math, like pi or e or sqrt(2), are computable, though almost all are not. You can do a lot of productive math using just computable numbers since they form a real closed field [1]. I believe they're a little harder to work with though. [1] https://en.wikipedia.org/wiki/Real_closed_field https://en.wikipedia.org/wiki/Real_closed_field

tsimionescu 2y ago

A lot of this is very much wrong. To the extent that your square is close to perfect and its sides are actually 1m, it's diagonal is just as truly sqrt(2)m long. And a circle that you draw correctly enough with a radius of 1m will have its circumference actually equal to 2pi. There is nothing more special about sqrt(2) than about 1, nor about a perfectly 90° angle versus a perfect circle. All of our drawings and constructions are approximations, but that doesn't make them naturally be integers or rationals any more than they are irrational. In other words, it would be just as accurate to say that the sides of the physical square are not 1, but x*pi for a pretty small x (that is, they are ever so slightly curved) as it would be to say that the circle's circumference isn't really 2*pi, it's actually some rational number, because the square is actually some very very many sided polyhedron. Even if we look at this from a purely physical perspective, elementary particles travel in perfectly straight lines and radiate in perfect circles in our models. And the directions of movement after a collision are not quantized, they can be arbitrary angles (just as space is not quantized, and in fact not even quantizable, in QM). And if you tried to look at a physical object and count the atoms to determine its length, you'd quickly find that it doesn't even have a constant number of atoms or a constant length, so in fact the least real concept is "an object of x meters in length", regardless of whether x is natural, rational, or irrational.

crabbone 2y ago
Well, no, no physical square, no matter how precisely its sides are 1m long has an irrational-lenght diagonal. That is simply impossible in the physical universe because the universe is discrete at this level. Irrational numbers are impossible in discrete context. The whole point they were invented is to capture the idea of continuous functions or continuous number line. But this is a "slight of hand", a definition that's made for convenience of solving useful problems, but isn't based in the physical reality. In a very similar way to how sqrt(-1) is not a thing in a physical reality, but it's useful to work with problems that can be described using complex numbers. > There is nothing more special about sqrt(2) than about 1 That fact that you don't understand what's special about it doesn't mean there isn't. It's a very different thing though. > All of our drawings and constructions are approximations Drawings: yes. Constructions: no. > but that doesn't make them naturally be integers or rationals any more than they are irrational. Here you've ventured into the territory you have no idea about... I'm sorry. You sound more like some LLM-generated gibberish here than anything a human with any expertise on the subject would write. Of course some things are naturally integers. We've invented integers to capture those things (in the physical universe). Similarly, rationals. There's nothing in the physical universe that's irrational in the same way how it can be an integer or a rational. Irrational numbers don't describe quantities or passage of time or forces acting on physical objects or the speed etc. because all those things are made up of small indivisible parts, and there's always a finite computable answer to how big something is, how long a process would take, how strong is the force applied to an object etc. Irrational numbers are a mathematical device to deal with different kinds of problems. Similar to how generating functions use "+" to mean a completely different thing from how it's used in algebra, or how it's used in regular languages, so are irrational called "numbers". But they aren't the same kind of thing as integers or rationals. To be honest, it would've been better not to call them "numbers" at all, to avoid this kind of confusion, but mathematics has a lot of old and bad terminology that's used due to tradition. > elementary particles travel in perfectly straight lines and radiate in perfect circles in our models The root of your problem in understanding this is: our models. Particles don't radiate in perfect circles in reality. Physical reality is discrete and cannot create perfect circles. You can imagine, however, a perfect circle and use it to a great effect to estimate the result of some physical process. But, if you truly measure the effect, you will never have an irrational number. There's no physical process of measuring anything that will end up with an irrational answer. That's simply impossible.