I remember reading (might have been here) that math and art both get difficult at the same moment for the same reason. When you're a kid, you live in the rational world. By this, I mean numbers that can be expressed as the ratio of two integers. Human creations are rational. The volume of a square is a neat, tidy equation. So is the area of a square. You can draw them easily, too, using a ruler and clean nifty lines, and they look great. Squares are all over human creation. You can draw a car with straight lines and squares. Wheels and other things bring in this inconvenient number, pi, that is "irrational", but let's just go with three point blah blah and it'll be fine. At least the curvature is constant.
So, where are the squares in nature? Hell, where are the circles. Where is the constant curvature. How do you draw a leaf, a tree, a face? How do you calculate the surface volume of a leaf, or the volume of a tree?
All of a sudden, you can't measure it with the numbers you know. There is no neat ratio of integers that will calculate the volume of that tree trunk. Or even the volume under an easily expressed mathematical equation on a graph. In fact once you start measuring nature, rather than the things people make, rational numbers aren't anywhere. All of a sudden, you have to deal with limits, sequences, strange numbers that can be made arbitrarily close to zero as other numbers approach infinity. It turns out every number is "irrational", pretty much nothing is rational. So, instead of irrational, let's call it Real.
Where do math and art get hard? When you start to describe things as they are, rather than as we imagine the to be. You know, Real.
> So, where are the squares in nature? Hell, where are the circles. Where is the constant curvature.
They exist! The crystal structures of molecules are rather Platonic, for example. Nature is elegant when you get very small, requiring fewer and fewer core concepts as you work your way down to more fundamental levels of understanding. (At lower levels those fundamentals might be irrational ones, but still, those few primitives [like spirals in complex space] become the only tools you need.)
The inelegance, then, comes from modelling the interactions of mind-bogglingly huge collections of these fundamental things, at high levels of abstractions, and then expecting your abstraction (which is just that: a formula that allows you to make some useful prediction of these super-high-level interactions) to be as elegant as the fundamental forces operating at the lowest levels.
I could make the opposite argument. In nature/reality there are no irrational numbers.
Say you want to measure the circumference of the visible universe. This is all the space we have, the rest is outside our light cone and we can't interact with it to the point that we can only infer it might exist.
You need about 55 decimals of pi to get this circumference to the accuracy of a Planck length. Or about that many, give or take.
So that's measuring the very largest real thing that could possibly matter, down to the accuracy of the very smallest thing we can conceive of.
So that's it for pi. Any further decimals are strictly theoretical. We can prove they must be these decimals and not others, using math, but it's only theoretical knowledge, these additional decimals serve absolutely zero purpose in nature or reality. You cannot get to them by measuring reality, you can only theorize about what these numbers would be if you could measure to infinite precision, which we can't, because there are limits.
The "Real" numbers is really a misnomer. Even if you don't buy the above accuracy argument, and want to describe nature as something infinite (even though we're strictly limited to interacting with a finite subset of it), then at least agree that it's countable. The real numbers are way too stretchy and insane (see the Banach-Tarski paradox). In nature you can't stretch things infinitely far, nor can you cut up things to arbitrary precision.
And yes occasionally we discover new smaller particles, or sub-particles, but what we don't discover is a continuum. And it would be really weird if we did, because you can do crazy tricks to the Real numbers.
Very interesting response. The thing that keeps me from entirely agreeing (with an admission that I don't have a background in the science you mentioned[1], so I can't really understand it) is that the number is pi - the 55 decimals is just an approximation that can be expressed as a ratio of two integers. The number isn't the rational that can be made arbitrarily close to a limit, the number is the limit. That number is pi. pi is every bit as much a number as 1, 2, 3. So is sqrt(2). So is e.
My understanding is that almost all numbers are real, but not rational. They can be expressed as the limit of a sequence that can be made arbitrarily close to the limit. But the number is not the sequence, the number is the limit of the sequence.
[1] = I had to look up what a Planck is. Should have taken more physics along with the math.
Try to cut that square in half diagonally and things get irrational really fast!
FWIW I thought calculus made a lot of sense and helped make the world make more sense. Algebra is a fancy set of rules to manipulate rather abstract symbols, calculus actually explains how real things work!
Good example. It's amazing how quickly you leave the rationals, even with a square, you almost immediately need numbers that can't be expressed as the ratio of two integers... historically, was that the first encounter with an "irrational" number (Pythagorean theorem applied to a right isosceles triangle)? I do remember the proof from number theory about 20 years ago, though I could never recreate it now from memory.
And algebra itself is one of those real things that calculus helps to explain!
Take, for instance, the Fundamental Theorem of Algebra, where the proof in terms of elementary complex analysis and/or topology closely related to complex analysis make the truth of the theorem geometrically obvious, the (mostly) algebraic proofs I've seen leave me with little more than the desire to re-check the proof, because I'm not at all certain that something equivalent to the F.T.A. hasn't been implicitly assumed at some point in the proof.
Now it may be "just me" — I've always had an easier time following analytical proofs than abstract algebraic ones — but just thinking of the necessary prerequisites — homotopy between maps defined by complex polynomials vs. what? Galois theory? — I don't think it's just me.
Incidentally, complex numbers are another case where, as with the irrationals, a poorly-chosen name has made simple and quite generally useful ideas seem esoteric to those not already familiar with the subject.
"once you start measuring nature, rather than the things people make, rational numbers aren't anywhere"
I thought that because nature is made of quantized things, that the opposite is true - all numbers are really rational, and it's irrational numbers that don't exist except in human imagination.
One way to look at why an irrational number cannot describe a physical object-
Suppose you had an object with a variable position in one dimension that could be described with an irrational number. Then that single object can, in principle, store an infinite amount of information in the decimal expansion of that number simply by positioning it and measuring its position.
The information is not encoded in the object but rather in the arbitrary set of coordinates imposed on it. You can always impose a set of coordinates where the object is at the origin, or where its position in all axes is a small integer.
Systems of coordinates don't have to linear, they don't even have to be monotonic. The underlying object is unchanged by a human making a choice of a set of coordinates in which to describe it, nor by switching to a different system of coordinates.
Your keyboard's "Q" key does not know when you think of it in terms of you-centric Cartesian-ish or spherical-ish coordinates. It doesn't know when someone nearby thinks of your "Q" key in a different set of coordinates, or with a different coordinate-origin, and it doesn't matter to the "Q" key or its behaviour even if those coordinates aren't relatable by e.g. a Lorentz transformation. A physicist might care about that. The engineers who blueprinted the keyboard might care about that too, for electronics-timing reasons, for example. The person nearby you might not.
It is that you are using a real valued coordinate system -- chosen by you, rather than by some law of nature, and certainly not by the "Q" key or any of its molecules or subatomic components -- that lets you encode the coordinates of its depressed position as containing some decimal representation of your login password. You are free to choose coordinates in which that information vanishes, just like you can choose a system of coordinates with the origin on the depressed Q key in which the entire works of Shakespeare can be found in the base-36 expansion of real-valued coordinates at some small spot in your left fovea.
> an irrational number cannot describe a physical object
Use a system of coordinates in which the centre of mass of your pineal gland is at the coordinate origin and the tip of your nose is at (x, \pi, z) at all times.
1. Why does bisecting a square (I assume you mean diagonally) create an irrational number? If matter, and space-time itself are not infinitely divisible, then wouldn't that mean any diagonal line is kind of a zigzag when you look at it closely enough? It's not that the sides aren't integers, it's that the diagonal isn't strictly possible.
2. This is not my logic nor what I wrote about someone else's argument. I didn't say that all infinite decimals store infinite information. Obviously you can talk about how much information is stored in a given infinite string but I didn't go there and it's not relevant. It doesn't matter whether you think all real numbers store the same amount of information or not.
I described the claim that if you could physically realize an arbitrary real number you could store unlimited information in the tiniest piece of matter. Which goes against the intuitive idea based on experience that more matter is required to store more information.
An object having a position described by an irrational number does not store infinite information. The object isn't storing anything at all; it's just sitting at that location.
The person doing the decimal expansion has to store an infinite amount of information. What if the person just uses an infinite-series expression?
Another question - what if I just shift my reference frame to be one irrational unit offset from yours? Does the information encoded by object-positions in my reference frame now change?
I don't think it makes much sense to go smaller than the planck length (10^-35m) because probing smaller distances would create black holes. So in a sense position is quantized according to current understanding of gravity and QFT.
From what I remember, the book 'Quantum Mechanics in Simple Matrix Form' explicitly discusses which variables are quantized and which are not, according to standard quantum mechanics. It takes a few chapters to develop the argument.
Yes, an abacus stores information, but not an infinite quantity; the information is only a few bits, and it's interpreted that way because we take a very low-precision view of the state of the abacus by dividing the rungs in half, mapping each side to 0 or 1. This is the key; the units of position used for measurement.
The object itself can be at any 'true' position (including an irrational position) but no measurement device has enough precision to say for sure; so I think the answer is that objects can take any position, but the fact that measurement-devices are limited makes it a moot point.
Real number is often just another approximation on the way to the true value. You are still counting spherical cows in vacuum.
Area of a platonic circle is a real number, area of an actual circle is discrete, but varies depending on how you draw the border.
You can count atoms and get your volume of the tree in integers that way. Bonus points for being temperature and pressure independent (more so than a volume of the spherical tree in vacuum anyway).
I guess they call those numbers irrational because they are never represented physically and thus are a pure figment of imagination. :)
If you want to appreciate how strange real numbers are, check out Weierstrass function. It's a function that is continuous everywhere, but differentiable nowhere. In other words, it has a kink (i.e. a sharp corner) on ALL points.
That is an example of how strange real valued functions can be, and one that tests your understanding of continuity.
It's a standard example trotted out in elementary real analysis courses along with Cantor sets, etc. If you have a good lecturer, they might make you try and come up with an example like it as an exercise before you've seen it...
A good part of becoming competent at analysis is building up a grab bag of examples like this you can throw at new situations.
I don't believe this is correct. That function isn't defined at 0. While it does have a limit, continuity at a point c requires that the limit as x->c = f(c), which obviously isn't the case here.
Honestly I don't think that's much worse from a "reals are weird" point of view, although it is another wrinkle that there is a strict subset which is even harder to pin down.
You can't define definability, because otherwise you could talk about the "least natural number not definable in fewer than n symbols" which is a contradiction for sufficiently large n.
For any system we could use to define a real number, (and not just because we are mortal), the system could only define or uniquely specify countably many real numbers. Any countable set of real numbers has measure 0, and therefore, almost all real numbers cannot be defined/specified using that system.
This holds for whatever system you use which defines a (possibly partial) function from [ the set of finite (but unboundedly long) sequences of characters over a finite alphabet], to real numbers
Or equivalently, from the natural numbers to the real numbers.
Hence dartboards have the mathematical property that a dart thrown into one is infinitely more likely to land on an integer than on a computable real (or a natural number, but for different reasons).
Not in darts. You can prove this to your satisfaction, by attempting to reach the continuum between the integers.
The dart will bounce off the metal framey bit and stick into the forehead of local barfly, 'Very Angry Ron'. Then when you graph the result, violent discontinuities will appear in the function, across a wide range of scales.
These already have their own special set in category theory and are sometimes referred to, within the field, as; 'Assorted and Unreasonable Associates of Very Angry Ron'. That is, the other side of a reasonably big field, ideally with some cattle in the way.
Part of any good introduction to real analysis is going to be walking you through a bunch of this stuff until you first understand how much you had glossed over the details, then than the details are strange, and finally that you have a working understanding or at least comfort with techniques to reason about this.
Is that the same as making sense? Well to paraphrase a physicist, you don't so much understand it as get used to it.
One of the fun things about teaching this material is that usually students have been through years of calculus prior to it, and you get to watch the moment when they realize all of this stuff has been "hiding in plain sight".
Intuition often comes from relating a new thing to something known. So maths, as an abstraction of reality, initially has many sources of intuition.
Later maths never has exactly the same patterns as earlier maths (it's already abstracted; so same patterns would be the same thing, though it cam build-on). Eventually, it doesn't relate to anything known, and you have to create that familiarity from scratch.
That's hard... but if some stuff became known in the first place, why not this too? (One counter is that the other stuff was instinctively known, e.g. 3D space, or at least our minds are pre-shaped to know it, e.g. language).
Assuming math is open-ended, there'll always be new stuff that doesn't relate.
To me at least it was when I took a real analysis course. Very difficult material but once it all 'clicked' I felt like it gave a lot of insight into the fundamentals of math and as a bonus finally offered a rigorous non-hand-wavey explanation of the fundamental theorem of calculus.
Most people who say the real numbers make sense doesn't know what they're talking about. The real numbers are pretty bizarre no matter how much mathematics you've studied.
You can learn to prove things about them rigorously, and memorize a lot of properties, but very few people will ever develop an accurate intuition for them.
There’s a famous story of a seminar on p-adic numbers at Princeton where Harish-Chandra was in attendance and one of the audience brings up some intuition from ordinary primes and the speaker replies with a joke about the degenerate case of the real numbers.
Edit: A great way to unlearn bad intuition is through the study of counterexamples. A good starting point might be chapter 1 of Counterexamples in Analysis [1].
The reals click for some people, and fail to click for others. What can I say?
For me it was once I really understood what a Cauchy sequence of rational numbers is, and how that is a real number. Let's see if I can explain that.
In math we have the following kind of construction in lots of places. We take some simple system, we construct some way of representing things from a more complex relationship. And then define some sort of equivalence. The new
This is a mouthful but you've seen it. Take the construction of the rational numbers from the integers. A rational number is just a pair of integers (n, m) with the second one not zero. It represents n/m. However there is an equivalence, 1/2 is the same number is 2/4. The equivalence is that (n, m) = (n', m') if and only if n * m' = n' * m.
You finish by defining operations as (n, m) + (n', m') = (n * m' + n' * m, m * m') and (n, m) * (n', m') = (n * n', m * m'). This looks like a mouthful, but it is exactly the rule that you're used to.
So we've seen this kind of construction before. (You do the same when constructing the integers from the natural numbers.)
So constructing the reals from the rationals is done as follows. Intuitively a real is a sequence of rationals that is converging. And two sequences of rationals are equivalent if they should converge to the same thing.
Where "converging" means that you have a sequence of rationals (x_1, x_2, x_3, ...) such that if we pick n, m "big enough", then x_n - x_m will be as close to 0 as we want. Or in usual Calculus notation, for every epsilon > 0 there is an N such that for every n and m both bigger than N, abs(x_n - x_m) < epsilon.
And (x_1, x_2, x_3, ...) should "converge to the same thing" as (y_1, y_2, y_3, ...) if (x_1 - y_1, x_2 - y_2, x_3 - y_3, ...) converges to 0. Or in usual Calculus notation, for every epsilon > 0 there is an N such that for every n bigger than N, abs(x_n - y_n) < epsilon.
Here is a sanity check. If you have a decimal representation, that gives us a sequence of rationals converging to that real, (3, 3.1, 3.14, 3.141, ....). Switch from base 10 to base 2, and you get a different sequence, but it is the same real. And the old chestnut, 1 = 0.99999... repeating is easy to verify.
And now work your way through the following axioms:
The algebraic axioms are easy.
1. There is a well-defined binary operation + such that x+y is always defined.
2. + is commutative, so x+y = y+x.
3. + is associative, so (x+y)+z = x+(y+z).
4. There is an additive identity 0 such that x+0 = x for all x.
5. Every x has an additive inverse called -x such that x + (-x) = 0
6. There is another binary operation called .
7. is commutative, x * y = y * x
8. * is associative, (x * y) * z = x * (y * z).
9. The distributive property holds. x * (y + z) = (x * y) + (x * z).
10. There is a multiplicative identity 1 different from 0.
11. Every x other than 0 has a multiplicative inverse 1/x such that x * (1/x) = 1.
And now the order axiom.
12. Every number is exactly one of positive, negative or 0. Or, more formally, there is a set P closed under addition and multiplication such that for all x, exactly one of three things is true: x is 0, x is in P, or -x is in P.
And then the tricky one. Completeness.
13. If X is a non-empty set of reals with an upper bound, it has a least upper bound. (For example the set of x such that x^2 - 2 < 0 is non-empty, it has an upper bound, and therefore it has a least upper bound. Which happens to be sqrt(2).)
To see that the order axiom holds, let x_1 be a rational number below the value of something in X, and y_1 be a rational number above an upper bound. And now we construct two sequences as follows.
At each step if (x_n + y_n) / 2 is an upper bound, then x_(n+1) = x_n and y_(n+1) = (x_n + y_n) / 2. Else x_(n+1) = (x_n + y_n) / 2 and y_(n+1) = y_n.
We can prove three things.
1. (y_1, y_2, y_3, ...) converges to an upper bound.
2. No upper bound can be below what (x_1, x_2, x_3, ...) converges to.
3. Both sequences are equivalent, they represent the same real.
The conclusion is that that real has to be the least upper bound.
If you can really get that, then congratulations! You understand the reals!
However I personally found it very helpful in real analysis to be able to take any question about the reals back to how it relates to this construction. This greatly helped my intuition.
IMHO, though, "an equivalence class of Cauchy sequences" is not the most intuitive representation of a real number. (A point on a line is; even Dedekind's cuts well-described in older editions of Baby Rudin are more intuitive.)
Dedekind cuts have a special case at all of the rationals. Which is weird. And the whole construct the completion using equivalence classes of sequences construction is one you'll encounter a bunch of times in topology. So it is worth learning it properly.
Plus if you're into that kind of thing, you can also construct the p-adics this way. :-)
Exactly. Moreover, a construction can be seen merely as a proof of the consistency of the set of axioms (i.e. it is the axioms rather than any particular construction that define what the reals are).
Propp himself (the author) wrote a paper in 2012 called Real Analysis In Reverse, which he links to in the post but doesn't really call attention to. It's a lovely paper, and it talks about precisely what properties of the reals are enough to tell you that you really are dealing with the reals - like his example of the Constant Value Theorem from the post.
One correction. The root of rational and irrational is the word "ratio". Rational numbers are ratios of integers. Irrational numbers are not ratios of integers.
It is a linguistic coincidence that we wound up with words that has another reasonable meaning.
It's not a coincidence. The root of both ratio and rational come from latin ratio, gerund of reri, 'to calculate, to reckon, to think'.
To the ancients, just like us, computation, thinking, and reasoning are linked. Irrational numbers were literally numbers that 'could not be reckoned' in the normal sense.
You can even go further with this analogy. The halting function is not computable because there is no finite procedure that tells whether any program halts. But this objection to infinity is also what the ancient Greeks had to continuity (Zeno's paradoxes) and irrational numbers, as there is no finite procedure that results in an exact answer. In fact, there is no bounded algorithm that can distinguish sqrt(2) on all possible numeric inputs.
Nonsense, the digits of sqrt(2) most definitely are computable and there are bounded algorithms that can check, given inputs n and k, whether or not n is the kth digit of sqrt(2).
That doesn't seem to be a fair argument, you can also do that with rational numbers: if I give you an infinite sequence that is exactly 1/3, can you in finite time say it actually is?
If you choose to represent numbers as infinite streams of digits, then obviously you can't compare them for equality in a finite amount of time.
The issue is: is it possible to use real numbers in a way that sidesteps this problem? In general, it's impossible -- most real numbers are uncomputable! -- but for some useful subsets (beyond the rational numbers) it's possible.
It may not be a fair argument, but it hits one of the key questions in the philosophy of math.
Classical math takes the attitude that absolute truth exists, and we can reason about reasoning fairly freely. In particular I can ask a question like, "Does this program halt?" and it will have a well-defined answer. Even if I don't know what it is. The set of programs that halt is a well-defined set, even if there is no procedure for that can always determine if a given program is in the set.
Constructivists do not accept this point of view. To a constructivist, a question has 3 possible answers. True, false, and unknown. Talking about whether a program "really" halts when nobody has verified it one way or another is nonsensical. A construction that requires knowing something we can't find out, even in principle, is not a valid construction.
Now in this point of view, we can carry out the construction of the real numbers as follows. A Cauchy sequence is a program that produces a sequence of numbers along with a proof that it converges to 0. Two programs define the same number if their sequences converge. Easy, peasy.
But consider the following. A program that conducts a search for a proof or disproof of the Riemann conjecture, at each step of the search giving (-0.5)^n. If it finds a proof or disproof, it will continue giving (-0.5)^N where N is the step where it found that answer. If it doesn't, it continues.
Now this is a Cauchy sequence. It converges to something. But to what? Is it positive or negative or 0? If there is a proof or disproof it will not be 0. It might be positive or negative. If neither proof nor disproof exist, it will be 0.
To a classical mathematician, there must be an answer, we just don't know what it is. To a Constructivist, this is a question whose answer is unknown and therefore undefined. This number therefore cannot be categorized as positive, negative, or 0. Exactly because of the problem that you state, we have no way in guaranteed finite time to figure out whether this sequence becomes constant or forever approaches 0.
I advocate learning constructivism. Not because it is useful - it is not. But because it shows that many things that mathematicians confidently claim do not actually follow by pure reasoning and cannot be proven. For example the existence of numbers that cannot be written down. To a constructivist, all numbers can be written down. We just cannot always tell them apart!
There's a fair point there, which is that some computations may look hard simply because we picked a difficult representation, but I'm not sure that applies here.
There's finite, exact representation for 1/3 in whole numbers, namely itself, but as far as I know (?) there isn't one for sqrt(2) unless, say, your choice of representation is the root of some polynomial. Is there a bounded procedure that shows whether any two polynomials represent the same set of roots?
Getting back to the original etymological question, my point is that irrational in the sense of 'can't be reckoned, computed' is very close to how we would see it. The Greeks simply had a different conception of computation, one grounded in finitism and constructing things geometrically.
Yes, and in fact the ancient Greek term, of those who invented the thing, was neither ("illogical" or "not a ratio").
It was "alogos" (later "arritos"), with the meaning "inexpressible", "which cannot be spelt out" - in the sense that you can never fully write out e.g. the square root of two the way you can a natural or a rational number like 2/3 or 1/4.
Unfortunately "alogos" also meant "illogical", so that's where the confusion stems...
Looks like the intention of the author was to play with that word game in order to get an appealing title, but even for people familiar with the topic it can have the "not logical" meaning first.
There's uncountably many more noncomputables than computables too, but the noncomputables hardly crop up, even though they come along with the reals also.
If you did want to salvage a "calculus without real numbers" the computables are probably more promising. All of the criticisms of the article still entirely apply to them, but there's a better chance you could come up with substitute definitions that may yield some useful concepts; they're closed on a more useful operations. I believe this is an actual studied area of math. But it's certainly more complicated than using real numbers. We may not teach, ahem, "real" real numbers to high school students, but you'll get to it fairly early in dedicated college math courses at the undergrad level.
I've wondered before if there could be a well defined set of numbers between the integers and reals (perhaps with its own class of infinite set) that nicely excludes the uncomputables. If there is, I suspect that it might be what the universe is actually using and if I was going to go looking for it, I'd follow the breadcrumbs from umbral moonshine and go sniffing around the monster group.
Other than the rationals? Those are all computable. Or the algebraics.
Whether there's an infinity between the naturals and the reals depends on the continuum hypothesis, which is independent of the usual axioms of set theory, so you can pick.
In short, it isn't just that we have more computables than non-computables; there are categories of numbers like "normal" that nominally contain "all the numbers" but we can barely even name a handful.
This YT link is to a recent (2019) video on the Numberphile channel with Matt Parker as presenter, just so people know what level of quality to expect (14m26s runtime).
And, yeah, that's one of my favorites in the STEM genre. I can't wait until my son (now nine) is ready to watch along with me.
Also, if you add any noncomputable to a computable, the result is noncomputable.
Basically, since the reals were created by René Descartes, in a misunderstood attempt at numerical sarcasm, the number line has become infested with conceptual prions, or 'priorns' as the tabloids call them nowadays, that are worryingly both everywhere and conversely, far too small to detect.
If these priorns get into the brain tissue of a working mathematician, the neural substrates responsible for higher order mathematical abstractions and preferences for knitwear, can start to come up with really stupid number definitions, such as binary reals where every even nth digit is the nth digit of pi and every odd nth digit is from the result of an idealised coin flip.
If this should unfortunately occur to a mathematician in your area, the kindest thing to do is to try and find them a role in quantative trading and tell them that they are going to be working on something really important. The banking sector subsidises daycare office environments providing these fictional job roles, as a social service for those infected by priorns, as a way of providing thanks to the wider mathematical community for inventing all the numbers that banking relies on.
It is unfortunate to have to resort to such deceit, however by that stage in the pathology, mathematicians become constantly irrational and fintek is therefore provided as a safe and enclosed habitat where they cannot damage anything particularly critical to wider society. They may even thank you for the opportunity, the poor deluded things.
"Thank you very much, but I will buy the Computable numbers, instead. They do not require abandoning my sanity (although I am aware I must still abandon rationality)."
In particular, the reals mean you accept measure theory, which in one of its key results says you can build two spheres of radius x from one sphere of radius x, and no holes. The computables don't go there.
There are infinitely many computable numbers, but only countably many -- no more than of natural or rational numbers. That turns out to be enough for everything sane you want to do.
Of course you need altered versions of the key theorems of calculus, because the computables are not continuous in the "real" sense. The numbers blur a bit, instead, to cover the gaps, much as water manages to behave like a fluid despite being made of nothing but discrete particles. The differences are a PITA but keep you on the straight and narrow. You get the same answers for everything that makes sense, and no answer for things that don't.
All the supposedly real numbers you will ever encounter are computable (too). Roots, pi, e, anything representable with a Taylor series. So you don't really give anything up.
Reals make a good enough approximation, which means you don't need to go hungry. Pretend you're using reals. Nobody needs to know. Everybody else is, too. The sane, anyway.
Measure theory also tells you that if you managed to make two solid balls of radius 1 from cutting up a single solid ball of radius 1, then some of the intermediate pieces you cut the original ball into must have been non-measurable sets. Non-measurable sets are known to behave counterintuitively, and are avoided when using measure theory (since they cannot be assigned a volume).
You have to check if a number is zero before you divide by it, you need to check if a number is positive before you take its log, you need to check it a set is measurable before you measure it or integrate over it. I don’t really see how any of these are significantly different.
> In particular, the reals mean you accept measure theory, which in one of its key results says you can build two spheres of radius x from one sphere of radius x, and no holes. The computables don't go there.
You cannot accept or reject measure theory. There are no axioms in measure theory; there are just the definition of a measure and theorems about measures.
You're actually talking about the Axiom of Choice. If AC is true, then you can construct subsets of the reals that aren't measurable, and if AC is not true, then you can have a model of set theory where all subsets of the reals actually are measurable.
The problem with using computable numbers in mathematics is that if you define two numbers x and y according to two different computations, then the question of whether x equals y is itself uncomputable. That doesn’t inherently make math with computable numbers impossible, but it does mean that even when restricting yourself to computable numbers, you will inevitably start working with some uncomputable functions, so the whole thing doesn’t quite work cleanly.
The problem is that under the hood calculus is all about limits (or methods with a certain equivalence, we won't get into that).
Take a derivative? that's a limit. Take an integral? That's a limit. And limits of seemingly rational expressions are sometimes irrational, case and point the limit of
(1 + 1/n)^n
as n approaches infinity is euler's constant, e.
So if you're taking limits with rationals you're going to find yourself with undefined limits (because irrational numbers are no longer defined) breaking things constantly.
Formally, the irrationals are dense but not complete.
where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number.”
So, ε is a gap just above zero, π-ε a gap just below π, etc.
(The difference is that, when stepping from real to real, you won’t accidentally step on a surreal in the way you’ll frequently hit irrationals when you step from rational to rational)
The surreals are actually a problem in this context. As far as I'm aware, nobody's yet come up with a satisfying integral calculus that works in the surreal numbers generally.
These seem like "indefinite" numbers, that don't have a value in the same way as rationals or reals. Is there any truth to that intuition? Can you do calculus with surreal numbers?
Is there an obvious concrete example? I've never spent proper time studying Real Analysis, and I confess I have the same intuition as OP: That you could use rationals to approximate irrationals to arbitrary precision.
What's special about real numbers is that, if you have a sequence of reals for which the distance between two consecutive elements approaches zero, then there exists a real number that's the limit of the starting sequence.
This isn't the case with rational numbers, eg. the sequence 1.4, 1.41, 1.414 ... (EDIT: these are increasing approximations of sqrt(2)) satisfies the hypothesis but there's no single rational number this sequence approaches.
This property is called completeness. The real numbers are a complete topological space, whereas the rationals aren't.
The operative words there are "sort of". You can in fact get arbitrarily close. Just as you can with other sets of numbers, like those the form m/10^n (decimal approximations), or the numbers tan(n) where n is an integer.
The problem with doing this is that you give up a lot of properties of functions that you might want, for example that continuous functions which cross the x-axis somewhere actually intercept the x-axis. This is not true if your x-axis is made of rational numbers: consider the function x^2 - 2.
Yeah, you would think so (and I thought the same for a while). All the epsilon delta stuff seems to work fine with rationals. But really most theorems of calculus (intermediate value thm, etc...) just are false, with very easy counter examples, as we see in the post.
The missing property is that every bounded set has a least upper bounds or that every cauchy sequence converges.
A consequence is that you can divide the rationals cleanly into disconnected parts. For example A = the set of rationals < sqrt(2) and B = the set of rationals > sqrt(2)
Continuity is a property of functions on topological spaces, not a property of sets of numbers. The property you are describing seems more like the Archimedean property or density of a set.
This has nothing to do with calculus and everything to do with the fact that there are an infinite number of irrational numbers between any two rational numbers. Indeed, if you randomly select a real number, you can be almost certain that you will select an irrational number. Of course, almost certain has a specific rigorous meaning: the probability of not selecting (in this case, an irrational number) is greater than zero, but less than all real numbers, i.e. Infinitesimal.
Edit: spelling
Edit: the infinitesimal belongs not in the real numbers, but in the hyperreal numbers, if you want to learn more
I am reminded of the ironic irrationality of ancient thinking insisting that rational numbers numbers must exist with things like Pythagoreans murdering over it - apocryphal admittedly. But the same pattern appears for other numbers and systems like the complex and imaginary and non-euclidian geometry which caused "rational" people to be filled with such rage.
It is an interesting pattern that anyone who insists upon their "rational" or "sane" systems in the face of reality tends to actually deeply unhinged.
How do you define something 'like' calculus? How do you define 'number' system? If by number system you mean arbitrary ring and by calculus you mean the derivative for example, then yes, but it's not super interesting. For example, the structure of algebraic data types forms a ring, and you can define an operation 'like' the derivative on it. However, given that only two operations (multiplication and addition) are commonly defined here, it's not particularly involved, although the result itself is quite useful (the derivative of the algebraic form of a data type is the type of one-holed versions of that same data type).
It's unclear what precisely you mean by 'weaker', but if you mean with regard to completeness - the reals are the so-called completion of (that is, the smallest complete metric space containing) the rationals; so you're kind of stuck.
You can weaken "completeness" though. In each of its many manifestations, completeness asks for the existence of something for all subsets (sequences, functions, ...) of reals. We can instead ask for the ones that are definable by finite means (aka computable, constructive).
To find (=construct) the extreme value, for example, you probably won't be able to find a general procedure; it has to be treated case by case. Now you'll appreciate how easy real numbers are. It gives the assurance that the extreme values exist.
This is really cool. My advisor is a strict constructivist and will tell anyone with ears about finitism after a few beers, and I've always found it a useful philosophical position.
I liked his comment that irrationals really aren’t that unreasonable in the common sense of the term. There are so many questions on Quora which demonstrate people are really uncomfortable with transcendental/irrational constants like pi and e. While you can approximate those as much as you like, the true monsters like Chaitin’s constant cannot be approximated at all.
My first inkling that I had no clue about the reals was when I first thought hard about the fact that you can have a bijection between intervals of different length, eg (0, 1) and (0, 10).
> To paraphrase Obi-Wan Kenobe: The completeness property of the reals is what gives calculus its power. It surrounds the set of real numbers and penetrates it. It binds the number line together.
Correct. Incommensurable - lacking a common measure. The side and diagonal of an isosceles right triangle are incommensurable - there is no length that goes into both an integer number of times. There is a proof of this in Euclid.
So, TL;DR:, one cannot build a meaningful calculus using the rational numbers alone. Not unexpected, judging by its history... A more interesting (to me, anyway) fact is that one can build (or "model") any number system - all integers, rationals, reals, etc. - using just the set of natural numbers (i.e. positive integers).
Well building the reals from rationals is not easy at all. It took a few decades to get a reasonable definiton. And in constructive mathematics, there are several competing (non equivalent) definitions of the reals. It's a very deep topic, yet we seem to understand (at least in calculus) what real numbers are.
I don't like clickbait... The article starts to suggest that calculus is hard to reason about and "irrational". In the end all it is really saying is that calculus requires irrational numbers, well doh! And guess what, you even need complex numbers. And the question is not about "shopping" numbers, there is a history to all of that. If you want something universal, use complex numbers. Try to teach that to a pre-schooler and you will find that you might wanna start with something simpler.
That's not true, and that is the point. To prove the main theorems of calculus, you don't need complex numbers, but you do need irrationals.
When you first learn calculus, you do one or two epsilon-delta proofs, and then your teacher gets a little hand wavy about limits and you move on to the real work of derivatives and integrals, cause the limits stuff intuitively makes sense. When you continue on in Real Analysis or Topology of the Real Line, you discover that your intuition lied to you, and concepts like open and closed sets and intersections and accumulation points are important and are in general non-obvious.
> ... In the end all it is really saying is that calculus requires irrational numbers, well doh!
That's really not the point, it really isn't. In the middle it says:
"The theorems of the calculus that work for the reals but fail for the rationals, despite the very different claims they make, are all equivalent to each other!"
There's something quite deep going on with the reals, and it may be that you don't realise it, that you do realise it and don't care, or whatever, but it's not as simple as you seem to be implying.
"I don't like clickbait... The article starts to suggest that calculus is hard to reason about and "irrational". In the end all it is really saying is that calculus requires irrational numbers, well doh!"
Agreed. I did not want to click it, because I knew the author could not prove that Calculus made no sense. But due to the high amount of upvotes, I wanted to see just what were his arguments.
The title turned out to be just a pun; which also happens to be clickbait-y, though I don't think that was an accident. As the author said, nothing in the post is particularly deep or surprising to anyone who has taken a course on real analysis, or even Calculus with some amount of rigor (which is unfortunately getting increasingly rare).
This kind of subtle clickbait is, IMO, even more harmful than the obvious kind: whereas one can simply ignore the latter, in this particular case I couldn't figure out whether the title was a clickbait or not, and I had to waste time reading it to find out it's not what I thought it was, and it's not what I was interested in reading.
Complex analysis was definitely interesting. I hope there was an award for whomever figured out the easiest way to integrate some real-to-real functions was by side-stepping into imaginary numbers
I'd say it is quite smart clickbait and the one I can live with. If I have to compare with "Celeb X did so outrageous thing!!!!" which turns out X parked on a spot for disabled or whatever.
I remember reading (might have been here) that math and art both get difficult at the same moment for the same reason. When you're a kid, you live in the rational world. By this, I mean numbers that can be expressed as the ratio of two integers. Human creations are rational. The volume of a square is a neat, tidy equation. So is the area of a square. You can draw them easily, too, using a ruler and clean nifty lines, and they look great. Squares are all over human creation. You can draw a car with straight lines and squares. Wheels and other things bring in this inconvenient number, pi, that is "irrational", but let's just go with three point blah blah and it'll be fine. At least the curvature is constant.
So, where are the squares in nature? Hell, where are the circles. Where is the constant curvature. How do you draw a leaf, a tree, a face? How do you calculate the surface volume of a leaf, or the volume of a tree?
All of a sudden, you can't measure it with the numbers you know. There is no neat ratio of integers that will calculate the volume of that tree trunk. Or even the volume under an easily expressed mathematical equation on a graph. In fact once you start measuring nature, rather than the things people make, rational numbers aren't anywhere. All of a sudden, you have to deal with limits, sequences, strange numbers that can be made arbitrarily close to zero as other numbers approach infinity. It turns out every number is "irrational", pretty much nothing is rational. So, instead of irrational, let's call it Real.
Where do math and art get hard? When you start to describe things as they are, rather than as we imagine the to be. You know, Real.
They exist! The crystal structures of molecules are rather Platonic, for example. Nature is elegant when you get very small, requiring fewer and fewer core concepts as you work your way down to more fundamental levels of understanding. (At lower levels those fundamentals might be irrational ones, but still, those few primitives [like spirals in complex space] become the only tools you need.)
The inelegance, then, comes from modelling the interactions of mind-bogglingly huge collections of these fundamental things, at high levels of abstractions, and then expecting your abstraction (which is just that: a formula that allows you to make some useful prediction of these super-high-level interactions) to be as elegant as the fundamental forces operating at the lowest levels.
Say you want to measure the circumference of the visible universe. This is all the space we have, the rest is outside our light cone and we can't interact with it to the point that we can only infer it might exist.
You need about 55 decimals of pi to get this circumference to the accuracy of a Planck length. Or about that many, give or take.
So that's measuring the very largest real thing that could possibly matter, down to the accuracy of the very smallest thing we can conceive of.
So that's it for pi. Any further decimals are strictly theoretical. We can prove they must be these decimals and not others, using math, but it's only theoretical knowledge, these additional decimals serve absolutely zero purpose in nature or reality. You cannot get to them by measuring reality, you can only theorize about what these numbers would be if you could measure to infinite precision, which we can't, because there are limits.
The "Real" numbers is really a misnomer. Even if you don't buy the above accuracy argument, and want to describe nature as something infinite (even though we're strictly limited to interacting with a finite subset of it), then at least agree that it's countable. The real numbers are way too stretchy and insane (see the Banach-Tarski paradox). In nature you can't stretch things infinitely far, nor can you cut up things to arbitrary precision.
And yes occasionally we discover new smaller particles, or sub-particles, but what we don't discover is a continuum. And it would be really weird if we did, because you can do crazy tricks to the Real numbers.
My understanding is that almost all numbers are real, but not rational. They can be expressed as the limit of a sequence that can be made arbitrarily close to the limit. But the number is not the sequence, the number is the limit of the sequence.
[1] = I had to look up what a Planck is. Should have taken more physics along with the math.
Try to cut that square in half diagonally and things get irrational really fast!
FWIW I thought calculus made a lot of sense and helped make the world make more sense. Algebra is a fancy set of rules to manipulate rather abstract symbols, calculus actually explains how real things work!
Take, for instance, the Fundamental Theorem of Algebra, where the proof in terms of elementary complex analysis and/or topology closely related to complex analysis make the truth of the theorem geometrically obvious, the (mostly) algebraic proofs I've seen leave me with little more than the desire to re-check the proof, because I'm not at all certain that something equivalent to the F.T.A. hasn't been implicitly assumed at some point in the proof.
Now it may be "just me" — I've always had an easier time following analytical proofs than abstract algebraic ones — but just thinking of the necessary prerequisites — homotopy between maps defined by complex polynomials vs. what? Galois theory? — I don't think it's just me.
Incidentally, complex numbers are another case where, as with the irrationals, a poorly-chosen name has made simple and quite generally useful ideas seem esoteric to those not already familiar with the subject.
In theory. In reality you have an integer number of atoms in one half and another integer number in the other.
There are no irrational numbers to measure, and squares made from "continuum material" do not exist in reality.
I thought that because nature is made of quantized things, that the opposite is true - all numbers are really rational, and it's irrational numbers that don't exist except in human imagination.
One way to look at why an irrational number cannot describe a physical object-
Suppose you had an object with a variable position in one dimension that could be described with an irrational number. Then that single object can, in principle, store an infinite amount of information in the decimal expansion of that number simply by positioning it and measuring its position.
Systems of coordinates don't have to linear, they don't even have to be monotonic. The underlying object is unchanged by a human making a choice of a set of coordinates in which to describe it, nor by switching to a different system of coordinates.
Your keyboard's "Q" key does not know when you think of it in terms of you-centric Cartesian-ish or spherical-ish coordinates. It doesn't know when someone nearby thinks of your "Q" key in a different set of coordinates, or with a different coordinate-origin, and it doesn't matter to the "Q" key or its behaviour even if those coordinates aren't relatable by e.g. a Lorentz transformation. A physicist might care about that. The engineers who blueprinted the keyboard might care about that too, for electronics-timing reasons, for example. The person nearby you might not.
It is that you are using a real valued coordinate system -- chosen by you, rather than by some law of nature, and certainly not by the "Q" key or any of its molecules or subatomic components -- that lets you encode the coordinates of its depressed position as containing some decimal representation of your login password. You are free to choose coordinates in which that information vanishes, just like you can choose a system of coordinates with the origin on the depressed Q key in which the entire works of Shakespeare can be found in the base-36 expansion of real-valued coordinates at some small spot in your left fovea.
> an irrational number cannot describe a physical object
Use a system of coordinates in which the centre of mass of your pineal gland is at the coordinate origin and the tip of your nose is at (x, \pi, z) at all times.
Simply bisecting a square creates an irrational number. If you refuse that then you must refuse that the sides of a square are integers.
By your logic, a fraction 0.1 1/10 is infinite because it can be expressed as an infinite series of nontrivial powers of pi or even in base 3.
2. This is not my logic nor what I wrote about someone else's argument. I didn't say that all infinite decimals store infinite information. Obviously you can talk about how much information is stored in a given infinite string but I didn't go there and it's not relevant. It doesn't matter whether you think all real numbers store the same amount of information or not.
I described the claim that if you could physically realize an arbitrary real number you could store unlimited information in the tiniest piece of matter. Which goes against the intuitive idea based on experience that more matter is required to store more information.
Position isn't quantized; energy is.
An object having a position described by an irrational number does not store infinite information. The object isn't storing anything at all; it's just sitting at that location.
The person doing the decimal expansion has to store an infinite amount of information. What if the person just uses an infinite-series expression?
Another question - what if I just shift my reference frame to be one irrational unit offset from yours? Does the information encoded by object-positions in my reference frame now change?
https://physics.stackexchange.com/questions/32665/why-positi...
Does a slide rule or an abacus store information, or does it just "sit there"?
Yes, an abacus stores information, but not an infinite quantity; the information is only a few bits, and it's interpreted that way because we take a very low-precision view of the state of the abacus by dividing the rungs in half, mapping each side to 0 or 1. This is the key; the units of position used for measurement.
The object itself can be at any 'true' position (including an irrational position) but no measurement device has enough precision to say for sure; so I think the answer is that objects can take any position, but the fact that measurement-devices are limited makes it a moot point.
And even if we could, is there a law of physics that would contradict? (I'm actually asking, I have no idea)
Area of a platonic circle is a real number, area of an actual circle is discrete, but varies depending on how you draw the border.
You can count atoms and get your volume of the tree in integers that way. Bonus points for being temperature and pressure independent (more so than a volume of the spherical tree in vacuum anyway).
I guess they call those numbers irrational because they are never represented physically and thus are a pure figment of imagination. :)
The real numbers are much stranger than they first appear, and a lot of early pedagogy is designed so that you don't look at this too hard.
After all, almost all reals are transcendental (but interestingly we only have proved a small handful of them).
Most numbers people have used in "regular life" are algebraic, obviously, but they are a countable subset so occur almost never, in the set.
https://en.wikipedia.org/wiki/Weierstrass_function
https://en.wikipedia.org/wiki/Conway_base_13_function
It's a standard example trotted out in elementary real analysis courses along with Cantor sets, etc. If you have a good lecturer, they might make you try and come up with an example like it as an exercise before you've seen it...
A good part of becoming competent at analysis is building up a grab bag of examples like this you can throw at new situations.
x sin(1/x) is continuous and non-differentiable at 0, but I wouldn't say it has a corner there.
A sharp corner to me means a point where the left and right derivatives both exist but are different.
It is similarly used as an example, for a function that is continuous but not absolutely continuous.
cfL https://en.wikipedia.org/wiki/Cantor_function
It's even worse, almost all reals aren't even computable (let alone transcendental)
This holds for whatever system you use which defines a (possibly partial) function from [ the set of finite (but unboundedly long) sequences of characters over a finite alphabet], to real numbers
Or equivalently, from the natural numbers to the real numbers.
The dart will bounce off the metal framey bit and stick into the forehead of local barfly, 'Very Angry Ron'. Then when you graph the result, violent discontinuities will appear in the function, across a wide range of scales.
These already have their own special set in category theory and are sometimes referred to, within the field, as; 'Assorted and Unreasonable Associates of Very Angry Ron'. That is, the other side of a reasonably big field, ideally with some cattle in the way.
Is that the same as making sense? Well to paraphrase a physicist, you don't so much understand it as get used to it.
One of the fun things about teaching this material is that usually students have been through years of calculus prior to it, and you get to watch the moment when they realize all of this stuff has been "hiding in plain sight".
Does the followong seem right to you?
Intuition often comes from relating a new thing to something known. So maths, as an abstraction of reality, initially has many sources of intuition.
Later maths never has exactly the same patterns as earlier maths (it's already abstracted; so same patterns would be the same thing, though it cam build-on). Eventually, it doesn't relate to anything known, and you have to create that familiarity from scratch.
That's hard... but if some stuff became known in the first place, why not this too? (One counter is that the other stuff was instinctively known, e.g. 3D space, or at least our minds are pre-shaped to know it, e.g. language).
Assuming math is open-ended, there'll always be new stuff that doesn't relate.
You can learn to prove things about them rigorously, and memorize a lot of properties, but very few people will ever develop an accurate intuition for them.
Edit: A great way to unlearn bad intuition is through the study of counterexamples. A good starting point might be chapter 1 of Counterexamples in Analysis [1].
[1]https://pdfs.semanticscholar.org/a4e7/eb352e4c44bf75d8fabaf7...
For me it was once I really understood what a Cauchy sequence of rational numbers is, and how that is a real number. Let's see if I can explain that.
In math we have the following kind of construction in lots of places. We take some simple system, we construct some way of representing things from a more complex relationship. And then define some sort of equivalence. The new
This is a mouthful but you've seen it. Take the construction of the rational numbers from the integers. A rational number is just a pair of integers (n, m) with the second one not zero. It represents n/m. However there is an equivalence, 1/2 is the same number is 2/4. The equivalence is that (n, m) = (n', m') if and only if n * m' = n' * m.
You finish by defining operations as (n, m) + (n', m') = (n * m' + n' * m, m * m') and (n, m) * (n', m') = (n * n', m * m'). This looks like a mouthful, but it is exactly the rule that you're used to.
So we've seen this kind of construction before. (You do the same when constructing the integers from the natural numbers.)
So constructing the reals from the rationals is done as follows. Intuitively a real is a sequence of rationals that is converging. And two sequences of rationals are equivalent if they should converge to the same thing.
Where "converging" means that you have a sequence of rationals (x_1, x_2, x_3, ...) such that if we pick n, m "big enough", then x_n - x_m will be as close to 0 as we want. Or in usual Calculus notation, for every epsilon > 0 there is an N such that for every n and m both bigger than N, abs(x_n - x_m) < epsilon.
And (x_1, x_2, x_3, ...) should "converge to the same thing" as (y_1, y_2, y_3, ...) if (x_1 - y_1, x_2 - y_2, x_3 - y_3, ...) converges to 0. Or in usual Calculus notation, for every epsilon > 0 there is an N such that for every n bigger than N, abs(x_n - y_n) < epsilon.
You define operations pairwise. So (x_1, x_2, x_3, ...) + (y_1, y_2, y_3, ...) = (x_1 + y_1, x_2 + y_2, x_3 + y_3, ...).
Here is a sanity check. If you have a decimal representation, that gives us a sequence of rationals converging to that real, (3, 3.1, 3.14, 3.141, ....). Switch from base 10 to base 2, and you get a different sequence, but it is the same real. And the old chestnut, 1 = 0.99999... repeating is easy to verify.
And now work your way through the following axioms:
The algebraic axioms are easy.
1. There is a well-defined binary operation + such that x+y is always defined.
2. + is commutative, so x+y = y+x.
3. + is associative, so (x+y)+z = x+(y+z).
4. There is an additive identity 0 such that x+0 = x for all x.
5. Every x has an additive inverse called -x such that x + (-x) = 0
6. There is another binary operation called .
7. is commutative, x * y = y * x
8. * is associative, (x * y) * z = x * (y * z).
9. The distributive property holds. x * (y + z) = (x * y) + (x * z).
10. There is a multiplicative identity 1 different from 0.
11. Every x other than 0 has a multiplicative inverse 1/x such that x * (1/x) = 1.
And now the order axiom.
12. Every number is exactly one of positive, negative or 0. Or, more formally, there is a set P closed under addition and multiplication such that for all x, exactly one of three things is true: x is 0, x is in P, or -x is in P.
And then the tricky one. Completeness.
13. If X is a non-empty set of reals with an upper bound, it has a least upper bound. (For example the set of x such that x^2 - 2 < 0 is non-empty, it has an upper bound, and therefore it has a least upper bound. Which happens to be sqrt(2).)
To see that the order axiom holds, let x_1 be a rational number below the value of something in X, and y_1 be a rational number above an upper bound. And now we construct two sequences as follows.
At each step if (x_n + y_n) / 2 is an upper bound, then x_(n+1) = x_n and y_(n+1) = (x_n + y_n) / 2. Else x_(n+1) = (x_n + y_n) / 2 and y_(n+1) = y_n.
We can prove three things.
1. (y_1, y_2, y_3, ...) converges to an upper bound.
2. No upper bound can be below what (x_1, x_2, x_3, ...) converges to.
3. Both sequences are equivalent, they represent the same real.
The conclusion is that that real has to be the least upper bound.
If you can really get that, then congratulations! You understand the reals!
However I personally found it very helpful in real analysis to be able to take any question about the reals back to how it relates to this construction. This greatly helped my intuition.
Dedekind cuts have a special case at all of the rationals. Which is weird. And the whole construct the completion using equivalence classes of sequences construction is one you'll encounter a bunch of times in topology. So it is worth learning it properly.
Plus if you're into that kind of thing, you can also construct the p-adics this way. :-)
https://arxiv.org/abs/1204.4483
It is a linguistic coincidence that we wound up with words that has another reasonable meaning.
It's not a coincidence. The root of both ratio and rational come from latin ratio, gerund of reri, 'to calculate, to reckon, to think'.
To the ancients, just like us, computation, thinking, and reasoning are linked. Irrational numbers were literally numbers that 'could not be reckoned' in the normal sense.
You can even go further with this analogy. The halting function is not computable because there is no finite procedure that tells whether any program halts. But this objection to infinity is also what the ancient Greeks had to continuity (Zeno's paradoxes) and irrational numbers, as there is no finite procedure that results in an exact answer. In fact, there is no bounded algorithm that can distinguish sqrt(2) on all possible numeric inputs.
Now in practice this isn't a problem, but it reveals the problem with finitism.
If you choose to represent numbers as infinite streams of digits, then obviously you can't compare them for equality in a finite amount of time.
The issue is: is it possible to use real numbers in a way that sidesteps this problem? In general, it's impossible -- most real numbers are uncomputable! -- but for some useful subsets (beyond the rational numbers) it's possible.
Classical math takes the attitude that absolute truth exists, and we can reason about reasoning fairly freely. In particular I can ask a question like, "Does this program halt?" and it will have a well-defined answer. Even if I don't know what it is. The set of programs that halt is a well-defined set, even if there is no procedure for that can always determine if a given program is in the set.
Constructivists do not accept this point of view. To a constructivist, a question has 3 possible answers. True, false, and unknown. Talking about whether a program "really" halts when nobody has verified it one way or another is nonsensical. A construction that requires knowing something we can't find out, even in principle, is not a valid construction.
Now in this point of view, we can carry out the construction of the real numbers as follows. A Cauchy sequence is a program that produces a sequence of numbers along with a proof that it converges to 0. Two programs define the same number if their sequences converge. Easy, peasy.
But consider the following. A program that conducts a search for a proof or disproof of the Riemann conjecture, at each step of the search giving (-0.5)^n. If it finds a proof or disproof, it will continue giving (-0.5)^N where N is the step where it found that answer. If it doesn't, it continues.
Now this is a Cauchy sequence. It converges to something. But to what? Is it positive or negative or 0? If there is a proof or disproof it will not be 0. It might be positive or negative. If neither proof nor disproof exist, it will be 0.
To a classical mathematician, there must be an answer, we just don't know what it is. To a Constructivist, this is a question whose answer is unknown and therefore undefined. This number therefore cannot be categorized as positive, negative, or 0. Exactly because of the problem that you state, we have no way in guaranteed finite time to figure out whether this sequence becomes constant or forever approaches 0.
I advocate learning constructivism. Not because it is useful - it is not. But because it shows that many things that mathematicians confidently claim do not actually follow by pure reasoning and cannot be proven. For example the existence of numbers that cannot be written down. To a constructivist, all numbers can be written down. We just cannot always tell them apart!
There's finite, exact representation for 1/3 in whole numbers, namely itself, but as far as I know (?) there isn't one for sqrt(2) unless, say, your choice of representation is the root of some polynomial. Is there a bounded procedure that shows whether any two polynomials represent the same set of roots?
Getting back to the original etymological question, my point is that irrational in the sense of 'can't be reckoned, computed' is very close to how we would see it. The Greeks simply had a different conception of computation, one grounded in finitism and constructing things geometrically.
Yes. Two polynomials share a root if their resultant vanishes.
You don't have to express sqrt(2) as an infinite string of digits. All the algebraic numbers can be expressed by a finite number of symbols.
“Not logical”, which by definition does not apply to math.
“Not a ratio”, e.g., a number that cannot be written as the division (ratio) of two whole numbers.
It should be no surprise to anyone, that calculus deals with the latter, but the title would allude to the article dealing with the former.
It was "alogos" (later "arritos"), with the meaning "inexpressible", "which cannot be spelt out" - in the sense that you can never fully write out e.g. the square root of two the way you can a natural or a rational number like 2/3 or 1/4.
Unfortunately "alogos" also meant "illogical", so that's where the confusion stems...
> Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible.
[1] https://en.wikipedia.org/wiki/Irrational_number
Because there are uncountably many more irrational numbers than there are rational numbers.
https://en.wikipedia.org/wiki/Computable_analysis
Whether there's an infinity between the naturals and the reals depends on the continuum hypothesis, which is independent of the usual axioms of set theory, so you can pick.
In short, it isn't just that we have more computables than non-computables; there are categories of numbers like "normal" that nominally contain "all the numbers" but we can barely even name a handful.
And, yeah, that's one of my favorites in the STEM genre. I can't wait until my son (now nine) is ready to watch along with me.
The real numbers are a bit spooky.
Basically, since the reals were created by René Descartes, in a misunderstood attempt at numerical sarcasm, the number line has become infested with conceptual prions, or 'priorns' as the tabloids call them nowadays, that are worryingly both everywhere and conversely, far too small to detect.
If these priorns get into the brain tissue of a working mathematician, the neural substrates responsible for higher order mathematical abstractions and preferences for knitwear, can start to come up with really stupid number definitions, such as binary reals where every even nth digit is the nth digit of pi and every odd nth digit is from the result of an idealised coin flip.
If this should unfortunately occur to a mathematician in your area, the kindest thing to do is to try and find them a role in quantative trading and tell them that they are going to be working on something really important. The banking sector subsidises daycare office environments providing these fictional job roles, as a social service for those infected by priorns, as a way of providing thanks to the wider mathematical community for inventing all the numbers that banking relies on.
It is unfortunate to have to resort to such deceit, however by that stage in the pathology, mathematicians become constantly irrational and fintek is therefore provided as a safe and enclosed habitat where they cannot damage anything particularly critical to wider society. They may even thank you for the opportunity, the poor deluded things.
In particular, the reals mean you accept measure theory, which in one of its key results says you can build two spheres of radius x from one sphere of radius x, and no holes. The computables don't go there.
There are infinitely many computable numbers, but only countably many -- no more than of natural or rational numbers. That turns out to be enough for everything sane you want to do.
Of course you need altered versions of the key theorems of calculus, because the computables are not continuous in the "real" sense. The numbers blur a bit, instead, to cover the gaps, much as water manages to behave like a fluid despite being made of nothing but discrete particles. The differences are a PITA but keep you on the straight and narrow. You get the same answers for everything that makes sense, and no answer for things that don't.
All the supposedly real numbers you will ever encounter are computable (too). Roots, pi, e, anything representable with a Taylor series. So you don't really give anything up.
Reals make a good enough approximation, which means you don't need to go hungry. Pretend you're using reals. Nobody needs to know. Everybody else is, too. The sane, anyway.
Is “garbage in, garbage out” so unreasonable?
You cannot accept or reject measure theory. There are no axioms in measure theory; there are just the definition of a measure and theorems about measures.
You're actually talking about the Axiom of Choice. If AC is true, then you can construct subsets of the reals that aren't measurable, and if AC is not true, then you can have a model of set theory where all subsets of the reals actually are measurable.
Take a derivative? that's a limit. Take an integral? That's a limit. And limits of seemingly rational expressions are sometimes irrational, case and point the limit of
(1 + 1/n)^n
as n approaches infinity is euler's constant, e.
So if you're taking limits with rationals you're going to find yourself with undefined limits (because irrational numbers are no longer defined) breaking things constantly.
Formally, the irrationals are dense but not complete.
That means they're dense, but continuous is something else.
You sort of can't get arbitrarily close. And that's the whole reason calculus doesn't work for rationals.
Rationals fundamentally have "gaps", and correspond 1:1 with integers.
”There are also representations like
{ 0, 1, 2, 3, … | } = ω
{ 0 | 1, 1/2, 1/4, 1/8, … } = ε
where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number.”
So, ε is a gap just above zero, π-ε a gap just below π, etc.
(The difference is that, when stepping from real to real, you won’t accidentally step on a surreal in the way you’ll frequently hit irrationals when you step from rational to rational)
Maybe you have to leave those gaps unfilled.
I’ve never seen derivatives or integrals, but https://math.stackexchange.com/questions/112492/integral-of-... claims one can define them.
Is there an obvious concrete example? I've never spent proper time studying Real Analysis, and I confess I have the same intuition as OP: That you could use rationals to approximate irrationals to arbitrary precision.
What's special about real numbers is that, if you have a sequence of reals for which the distance between two consecutive elements approaches zero, then there exists a real number that's the limit of the starting sequence.
This isn't the case with rational numbers, eg. the sequence 1.4, 1.41, 1.414 ... (EDIT: these are increasing approximations of sqrt(2)) satisfies the hypothesis but there's no single rational number this sequence approaches.
This property is called completeness. The real numbers are a complete topological space, whereas the rationals aren't.
TL;DR: Every real number can be approximated by rationals. GP was being careless with language.
A consequence is that you can divide the rationals cleanly into disconnected parts. For example A = the set of rationals < sqrt(2) and B = the set of rationals > sqrt(2)
https://en.wikipedia.org/wiki/Completeness_of_the_real_numbe...
Edit: spelling
Edit: the infinitesimal belongs not in the real numbers, but in the hyperreal numbers, if you want to learn more
It is an interesting pattern that anyone who insists upon their "rational" or "sane" systems in the face of reality tends to actually deeply unhinged.
Can you build something like calculus on a number system weaker than the reals, but stronger than the rationals? What's the weakest you could go?
To find (=construct) the extreme value, for example, you probably won't be able to find a general procedure; it has to be treated case by case. Now you'll appreciate how easy real numbers are. It gives the assurance that the extreme values exist.
I would not have thought it would interest Hacker News readers, but here's more about computable numbers: https://observablehq.com/@neobourbaki/computable-numbers
https://en.wikipedia.org/wiki/Surreal_number
> ...the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers...
Invented by Conway while studying Go.
:)
https://en.wikipedia.org/wiki/Renormalization
He also put forward a compelling argument for hedonism, preserved by Aristotle.
Because you never end up with whole numbers both ways?
That's not true, and that is the point. To prove the main theorems of calculus, you don't need complex numbers, but you do need irrationals.
When you first learn calculus, you do one or two epsilon-delta proofs, and then your teacher gets a little hand wavy about limits and you move on to the real work of derivatives and integrals, cause the limits stuff intuitively makes sense. When you continue on in Real Analysis or Topology of the Real Line, you discover that your intuition lied to you, and concepts like open and closed sets and intersections and accumulation points are important and are in general non-obvious.
Counterexamples In Analysis
https://www.amazon.com/Counterexamples-Analysis-Dover-Books-...
(Pdf version) https://pdfs.semanticscholar.org/a4e7/eb352e4c44bf75d8fabaf7...
That's really not the point, it really isn't. In the middle it says:
"The theorems of the calculus that work for the reals but fail for the rationals, despite the very different claims they make, are all equivalent to each other!"
There's something quite deep going on with the reals, and it may be that you don't realise it, that you do realise it and don't care, or whatever, but it's not as simple as you seem to be implying.
Agreed. I did not want to click it, because I knew the author could not prove that Calculus made no sense. But due to the high amount of upvotes, I wanted to see just what were his arguments. The title turned out to be just a pun; which also happens to be clickbait-y, though I don't think that was an accident. As the author said, nothing in the post is particularly deep or surprising to anyone who has taken a course on real analysis, or even Calculus with some amount of rigor (which is unfortunately getting increasingly rare). This kind of subtle clickbait is, IMO, even more harmful than the obvious kind: whereas one can simply ignore the latter, in this particular case I couldn't figure out whether the title was a clickbait or not, and I had to waste time reading it to find out it's not what I thought it was, and it's not what I was interested in reading.
Complex analysis was definitely interesting. I hope there was an award for whomever figured out the easiest way to integrate some real-to-real functions was by side-stepping into imaginary numbers