Comment by crabbone | Hacker News Reader

Comment by crabbone | Hacker News Reader

Parent Full thread

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 :)

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.

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.

To the best of our knowledge, physical reality has both discrete quantities and continuous quantities. Space and time are continuous, for example. Energy, charge, mass - all discrete. And this is not a question of just models - we've tried to create models where time or space are discrete, and they don't work. They fail to predict reality correctly.

And you're still wrong about the nature of irrational numbers. They are not a model for computational irreducibility. They basically separate different types of quantities that are not directly relatable as ratios of one another. The circumference of a circle (or any non-trivial ellipse, for that matter) is a fundamentally different quantity than the length of a straight line: this is what Pi being irrational tells us.

And yet, you can do arithmetic with circle lengths just as much as you can with sides of squares. You can add them up, divide them, everything. If for whatever reason we decided to define 1m as the circumference of a particular circle, and base our geometry around circles rather than straight lines, we'd consider circle lengths to be integer/rational numbers, and then line lengths would be irrational (the radius of a circle with circumference 1 would be 1/(2pi), an irrational number). Similarly, if we decided that 1m is defined as the diagonal length of a particular square, we'd say that the lengths of that square have irrational length, and then you'd claim that no true square has lengths of exactly sqrt(2)/2.

As I said, it's just as correct to think of any physical circle/ellipse as a many-sided polyhedron as it is to think of any physical straight line as a segment of the circumference of a circle/ellipse with a really huge radius (focal length).

In fact, it's more correct physically to think of apparently straight lines as curved than to think of curves as composed of polygons, as straight paths are very rare in physics, any kind of bias will tend to induce a slight curvature. And atoms are more circular in nature than they are "boxy".

Ultimately we can only really measure ratios of things, and we know some things are not an exact ratio of another thing when we measure them precisely enough. Which of the quantities you consider to be represented by a rational number and which you consider to be irrational is purely a choice of definitions.

> And yet, you can do arithmetic with circle lengths just as much as you can with sides of squares.

No, you can't. There's no way to add or to multiply two numbers that don't have a finite expansion in some basis beyond just writing it as a sum outside of a very few special cases where there's a round-about way of finding the answer. I.e. if you try to do pi + pi, well, you may get a 2pi, if you pray hard enough and your faith is strong enough, but really, there's no proof that even that is true. You just choose to believe that it will check out somehow. But, even if you get a 2pi, it's still not an answer you want because to figure out what 2pi is, you still need to add a pi to a pi, so, you are back to square one.

> As I said, it's just as correct to think of any physical circle/ellipse as a many-sided polyhedron

Because you prayed hard enough and it was revealed to you in a dream? Based on what do you believe this?

Take a circle with a radius of 1cm. Unroll its perimeter and declare that length is 1 of new unit called a squelk.

You can measure things and build things in squelks just fine, but if you try to take something that is 100 squelks long and measure it in centimeters you will get an irrational number of centimeters because there is no rational conversation from squelks to centimeters.

A given length can be irrational in one unit of measure but not in another.

Of course we do have limits of precision in the real world, so in reality nothing lines up quite right.

> I.e. if you try to do pi + pi, well, you may get a 2pi, if you pray hard enough and your faith is strong enough, but really, there's no proof that even that is true.

Sure there is. pi + pi = 2pi <=> (pi + pi) / pi = 2pi / pi <=> pi/pi + pi/pi = 2 <=> 1 + 1 = 2, which we know is true. QED.

This is in fact exactly what I'm saying about the circle and its radius. We can't get rid of the irrationality when calculating the ratio between the circumference and the radius of a circle. But it's arbitrary which one we call rational and which we call irrational: a circle with a rational circumference will have an irrational radius, and vice versa.

> Because you prayed hard enough and it was revealed to you in a dream? Based on what do you believe this?

I don't know what exactly you are responding to here.

It's your claim that in the real physical world all "circles" have a rational circumference (perimeter), which is equivalent to saying that the "circle" is really a very very many-sided polyhedron (since only a polyhedron can have a rational perimeter if the sides are of a rational length and all angles are constructible). I don't need to pray (?!?) to see this.

And if you were responding to my full quote, that this comparison is equivalent to saying that all physical "squares" are in fact rounded-corner ovoid shapes (and so their actual side lengths are some multiple of pi, or at least some other irrational number that we don't even have a name for) then that follows from the observation above, that you can arbitrarily decide to call the circumference of a circle "2 pi" or the radius "1/2pi".

It also follows from how trajectories work in physics - if a particle is moving in a straight line and then some force starts acting on it in some direction other than directly in front or behind, its trajectory will become circular, not go at a straight angle. So if an electron in a perfectly isolated environment would follow a perfectly straight line, an electron in a real environment where there are electrical fields everywhere will follow a line that's curvy all around. In contrast, it's in fact impossible to create a trajectory for an electron that has any kind of angles, even in an ideally isolated environment - it's impossible for a physical object to turn on the spot like an ideal angle.

So, again, curves (and their associated irrational numbers) are in fact closer to physical reality, we just chose to approximate them using straight lines because its easier.

And as a final thought, related to the reality of the continuum. In all of the models that we have of physics that actually work, if I fire two particles away from each other arbitrarily in space, the distance between them will cover every real number in some interval [minDist, maxDist]. And any model that requires a minimum unit of distance to exist (so that the distances would be minDist + n*FundamentalMinimum, with n = 1, 2, 3...) doesn't work with special relativity, that says that lengths contract in the direction of movement (because if two particles are at a distance of FundamentalMinimum as measured by one observer, they will be at a distance of gamma*FundamentalMinimum to another observer moving at some speed relative to the first one, with gamma < 1, thus breaking the assumption that all lengths are > FundamentalMinimum).

> 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.

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.)

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

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.

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

No. I mean computable functions: https://en.wikipedia.org/wiki/Computable_function

Computable numbers are those that are described by computable functions. Irrationals like pi can be described by computable functions that take a precision as input.

“God created the natural numbers, all else is the work of man.”

Reply on news.ycombinator.com