[HN Gopher] How the square root of 2 became a number
___________________________________________________________________
How the square root of 2 became a number
Author : headalgorithm
Score : 104 points
Date : 2024-06-21 14:50 UTC (8 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| bandrami wrote:
| It's funny that everybody remembers Pythagoras for the right-
| triangle theorem, but that wasn't what made him important at the
| time. The right-triangle square equality had been known
| empirically for centuries. What he proved that was so completely
| earth-shaking is that for some right triangles A, B, C there is
| no rational Q for which QA = C. That was what was so important
| about his proof.
| moioci wrote:
| That would seem to imply that for A =/= 0, C/A is irrational.
| This seems counterintuitive.
| asolove wrote:
| What is counter-intuitive? In a triangle with sides A=1, B=1,
| then C=root(2), so C/A is irrational. That's what was so
| impactful about the discovery.
|
| Imagine not knowing about irrational numbers. You assume all
| numbers are just integers and fractional ratios between
| integers. It would be weird (terrifying?) that something as
| simple as a right triangle would require a whole category of
| numbers you can't express.
| Ekaros wrote:
| For some reason that feels so weird that it would be that
| late "discovery"... Once you define a square(sides same
| length) the length of diagonal is one of the first
| questions. And this being very weird number is something I
| believe someone must have thought about long before that
| point of time.
| asolove wrote:
| These are more or less the first people to think about
| geometry rigorously as an abstract system. Anyone
| previous would have just pointed to the hypotenuse and
| said "it's that length right there" and not asked a
| further question.
| dontlikeyoueith wrote:
| > more or less the first people to think about geometry
| rigorously as an abstract system.
|
| The first people whose thinking was preserved until the
| present.
|
| Which is still noteworthy, but a different thing.
| bandrami wrote:
| OK but at the time it was literally an open research
| question: given two reals A, B is there always a rational
| Q such that QA=B? Number theory as such was still in its
| infancy but I think it's impressive that this was
| _exactly_ the right question to ask and they understood
| how important it was.
| InitialLastName wrote:
| A lot of early math was done using geometry tools rather
| than symbolic representation.
|
| If you are drawing a diagram for a building and you need
| a distance equal to the diagonal of a square, you set
| your compass to the two points and use that distance. No
| need to determine that it can't be represented by a
| comfortable multiple of the sides.
| moioci wrote:
| My bad. I was thinking A, B, and C were integers.
| hinkley wrote:
| The 3/4/5 triangle is rational. The unit right triangle is
| not. You've dropped the "some" from the parent.
| bandrami wrote:
| For "most" right triangles, yes, C/A is irrational. In fact
| the triangles for which C/A is rational are vanishingly rare
| (though Pythagoras proved many important things about
| them[1])
|
| But before Pythagoras, it was still an open question if for
| any two reals A, B there might be a rational Q such that QA =
| B. Whereas we now know that for "most" reals there is no such
| Q, thanks to Pythagoras.
|
| 1: https://en.wikipedia.org/wiki/Pythagorean_triple
| seanhunter wrote:
| I'm not sure you've got that right. We know almost nothing
| about the historical Pythagoras and none of his actual writings
| survived. We only know of him through the Pythagorean
| brotherhood and other people (eg Plato) who he influenced.
|
| While the Pythagoreans did come up with many things (eg the
| first rigorously documented scientific experiments) my
| understanding was that it is known for certain that they
| definitely were not responsible for Pythagoras' theorem (or
| this rationality corolary you're talking about), and that the
| earliest formulation of it that is currently known is from
| Babylon where it's documented to do with sizing of farm
| plots[1] about a thousand years before Pythagoras. The proof of
| the irrationality of the square root of two was so terrifying
| to the Pythagoreans that the legend has it they threw the dude
| who produced it off a boat into the sea to drown because it was
| such a heresy (although I believe that is also known not to be
| true and the pythagoreans knew that root 2 was irrational).
|
| In that sense it's like Euler's number (first documented by
| Napier), Lambert's W function (Invented by Euler to solve a
| family of equations Lambert couldn't solve), Lagrange's
| notation for calculus (used by Lagrange yes but first also
| invented by Euler) etc etc.
|
| [1]
| https://www.researchgate.net/publication/222892801_Methods_a...
| lordnacho wrote:
| Stigler's Law of Eponymy
| pfdietz wrote:
| Let me guess: Stigler didn't invent that.
| bandrami wrote:
| The documentatary evidence is fragmentary but there is
| significant evidence that a 5th-century BC Greek
| mathematician proved the incommensurability of the side of a
| square with its diagonal (it was apparently trivially known a
| century later since it appears in Plato's "Meno"). Whether
| that person was named Pythagoras or Hippasus or something
| else is really neither here nor there since he was pretty
| clearly part of the Pythagorean tradition that got associated
| with one name.
|
| The point in any case is that incommensurability as a concept
| was not widely accepted at the beginning of the 5th century
| BC and was widely accepted at the end, and the name
| "Pythagoras" gets attached to the mathematicians who
| discovered that.
|
| But like for that matter Plato's name wasn't "Plato"; that
| was a nickname his wrestling coach gave him.
| dr_dshiv wrote:
| Speaking of Plato's wrestling coach...
|
| Pythagoras was also a successful wrestling coach. In fact,
| he coached the most winningest Olympic athlete of all time:
| Milo of Croton. Milo won 5 consecutive Olympics over a 20
| year period.
|
| Pythagoras himself was thrown out of the boys Olympics at
| age 16 for being too effeminate (long hair), but then
| entered the men's Olympics and won. Supposedly, he
| introduced some new kind of martial arts technique.
|
| This is documented in Thibodeau, 2019 "The Chronology of
| the Early Greek Natural Philosophers." Happy to share the
| refs. Ok, back to maths...
| javier_e06 wrote:
| Thank you so much!
|
| Like my teacher always says: "If at the end of the class,
| you haven't learned something new, come see me"
| n4r9 wrote:
| There is a PDF of your citation available at https://www.
| researchgate.net/publication/335965217_THE_CHRON...
|
| I searched for "long hair" and found only this:
|
| >The man with long hair at Samos': They say there was a
| Samian boxer with long hair who went to Olympia and won
| after being mocked by his opponents for looking like a
| woman; he became proverbial. Eratosthenes says that
| Pythagoras of Samos won with long hair during the 48 th
| Olympiad; Duris represents this as Pythagoras being
| excluded, challenging the men, and beating many of them.
|
| Like many stories concerning Pythagoras, I wonder if this
| was some local fable onto which his name later became
| plastered.
| thrownblown wrote:
| 11. Eratosthenes, Olympic Victors 3rd century via
| Favorinus, Varied History, via Diogenes Laertius, Lives
| 8.47 "Eratosthenes (according to what Favorinus reports
| in book eight of his Varied History), said this man [sc.
| Pythagoras] was the first to box using technique, in the
| 48th Olympiad, letting his hair grow long and wearing a
| purple robe; after being excluded from the boys' games
| and jeered at, he immediately joined the mens', and won."
| Olympiad 48: 588 to 584 BCE cf. Eusebius, Chronography,
| p. 93 Karst
| magicalist wrote:
| According to wikipedia, at least, that's a different
| Pythagoras of Samos
| https://en.wikipedia.org/wiki/Pythagoras_(boxer)
| User23 wrote:
| I do find it amusing that most if not all of the famous
| classical philosophers would stand up just fine to the
| "post physique" meme.
| kevinventullo wrote:
| Slightly off topic, but one of the coolest (to me)
| archaeological discoveries is from ~1700 BC, showing a
| calculation of sqrt(2) to the equivalent of six decimal
| places: https://en.m.wikipedia.org/wiki/YBC_7289
| ants_everywhere wrote:
| I haven't read the paper you posted yet, but just to clear up
| a common confusion, Pythagorean triples are not the
| Pythagorean theorem.
|
| The theorem is a logical statement about all right triangles,
| and it has a proof that the statement holds. Pythagorean
| triples are specific instantiations of the relation for some
| known triangles and probably would have served as evidence
| that the statement was even provable.
|
| Historically we probably had triples long before we had a
| proof of the theorem, just like many of the theorems proved
| by Euclid were probably already known as rules of thumb.
|
| Compare with an open problem today, like the Riemann
| hypothesis.
| EGreg wrote:
| This is exactly right!
|
| Oh, how I wish there was a book on the history of science and
| math. Like how they went from the Four Humours or Phlogiston
| and Spontaneous Generation and Luminferous Ether theories to
| what they had later. Like how scientists all thought the
| earth was 100 million years old for a couple centuries until
| the discovery of radioactivity.
|
| I want a book that would speak about how people made fun of
| Ignaz Semmelweis for washing hands in hospitals, until
| Pasteur in France and John Snow in England showed evidence
| for the germ theory of disease. How people used leeches and
| bloodletting, and when / why they stopped. (And maybe
| anecdotes like How Washington Roebling building the Brooklyn
| Bridge died from a gangrene because he thought pouring water
| over his wound was enough, and his son finished it)
|
| I want a book that would explain the experiments that led to
| the theories, like Michelson-Morley that challenged the
| Lumeniferous Ether model. Or how people first discovered
| X-rays and didn't know what to make of them.
|
| How, indeed, did people prove to others that atoms existed? I
| don't mean Democritus' theories 2500 years ago, I mean what
| made people convinced the world was made from atoms?
|
| And then the experiments that led to the standard model, how
| was it developed? The word Quark, where it came from, the
| reactions of scientists to Quantum theory etc.
|
| Our science comes shrinkwrapped, showing only the end result,
| not the history of thought and the places where (eg Andalusia
| in the 1100s or China in 20 AD). To me it is very interesting
| when people weren't sure yet but got the right result based
| on circumstantial reasoning, didn't have high-powered
| electron microscopes and still somehow realized about atoms
| and molecules (Gregor Mendel etc). Or how Darwin and
| Mendeleev developed theories genetics, why they so thoroughly
| discounted Lamarckian evolution and were they correct etc.
|
| I want a book that discusses the history including the
| ERRONEOUS theories, the time frames, the experiments that
| challenged prevailing theories, the controversies, and what
| made people find the correct theories.
|
| And maybe the people (eg Newton avoided women, and studied
| Hebrew to reconstruct Biblical history, Michael Faraday was a
| devout Christian etc.)
|
| Is there such a book? Can you guys recommend ??
| jtimdwyer wrote:
| I may be falling into a trap here by answering this, but
| you might enjoy The Structure of Scientific Revolutions by
| Thomas Kuhn.
| generuso wrote:
| It is not that we do not teach this anywhere at all, but we
| certainly do not teach this in the introductory science
| classes, where the goal is to provide an accessible path to
| the already available "shrink wrapped" conceptual
| framework.
|
| That is hard enough already. Showing the history of science
| properly is easily a hundred-fold more monumental of a
| task. It is a subject which is studied, and taught -- but
| for the specialists in the history and philosophy of
| science. There are many books dedicated both to cursory
| high level overviews of the history of science, and to some
| specific episodes in this history, going into nuances of
| what happened and how it happened.
|
| For example, for the history of mathematics, one can
| consult "The History of Mathematics: An Introduction"
| https://www.amazon.com/History-Mathematics-Burton-
| Professor-...
|
| Here is a much more specialized book, showing how much
| trouble relatively modern physicists had in correctly
| conceptualizing "heat" and "temperature": "The tragicomical
| history of thermodynamics, 1822-1854" https://scholar.googl
| e.com/scholar?q=history+of+thermodynami...
|
| Beware! The history of evolution of ideas in science and
| technology is a vast, vast and an extremely fascinating
| field, with many dangerous rabbit holes!
| wolfi1 wrote:
| I thought the Pythagoreans threw one of their members over
| board for proving the root of 2 to be irrational
| ants_everywhere wrote:
| This is a myth, it's covered in
| https://en.wikipedia.org/wiki/Hippasus
| bandrami wrote:
| There's multiple stories about this; one of the better-
| attested is that for a time the brotherhood swore each other
| to secrecy (with threats of drowning) about it because it ran
| against the Parmenedian epistemology of the time.
| mehulashah wrote:
| I never learned of these formal definitions in high school
| mathematics. Nor in the lower level college ones that I took.
| There's a beauty to this perspective-- irrational numbers are
| what rationals are not.
| bubblyworld wrote:
| I think that's at the heart of mathematics - deceptively simple
| definitions that capture the essence of something.
| crdrost wrote:
| Yes, but it also messes with a lot of normal intuitions. Some
| examples:
|
| "Because rationals are dense --between any two rationals there
| are infinitely many other rationals--there are actually vastly
| more spaces between rational numbers, than rational numbers
| themselves. These spaces-between are the real numbers."
|
| "Because every finite text document can be converted to UTF-8
| and thus then an integer, it is only possible to describe 0% of
| the real numbers between 0 and 1 with text."
|
| "Since most numbers are indescribable, there are
| (discontinuous) functions which have the value 2 for almost all
| numbers, but any number that you actually can describe and try
| to evaluate the function on, gives 1 and not 2."
|
| You start to appreciate that logic itself is this Lovecraftian
| eldritch-horror abomination, and that we only live in the Bliss
| of Sanity because we live in ignorance, never staring into its
| depths lest the abyss stare directly back into our souls.
| tomrod wrote:
| > You start to appreciate that logic itself is this
| Lovecraftian eldritch-horror abomination, and that we only
| live in the Bliss of Sanity because we live in ignorance,
| never staring into its depths lest the abyss stare directly
| back into our souls.
|
| Oh poppycock. We are the eldritch horror. We are the universe
| experiencing itself. Humans are space orcs, if Reddit is to
| be believed.
| markusde wrote:
| > describe 0% of the rational numbers between 0 and 1
|
| I think you mean irrational :)
| Edwinr95 wrote:
| No, the statement holds perfectly fine for the rationals.
| LudwigNagasena wrote:
| There is a 1-to-1 mapping between integers and rationals.
| crdrost wrote:
| The basis for this statement holds, but what the
| statement implies does not.
|
| That is, the numbers are a subset of the rationals, but
| it does not follow that we can't describe a rational with
| a number. In fact the rationals between [0, 1) have a
| well known numbering, [ 0/1, 1/2, 1/3,
| 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6,
| 1/7, 2/7, 3/7, 4/7, 5/7, 6/7, 1/8, 3/8, ... ]
|
| where one increments the denominator and then goes
| through all numerators but keeps only numerators which
| have GCD 1 with the denominator (since if they share a
| factor they were already listed).
| Perseids wrote:
| What you were probably thinking of is that 0% of the
| irrational numbers between 0 and 1 can be described by
| language as single entities. Or phrased differently: If you
| had a magic machine that could pick a random real number
| between 0 and 1, with 100% probability you would get a
| number that no finite phrase / definition / program / book
| could define. That is because everything we can abstractly
| define is part of a countable set and the set of irrational
| number (and real numbers) is uncountable.
|
| For that reason, quite a few mathematicians view the real
| numbers as a useful, but ultimately absurd set. Much more
| sane is the set of computable numbers, that is the set of
| numbers for which you can find an algorithm that computes
| the number to arbitrary precision. (More formal: A number x
| is computable if there exists a Turing machine that gets as
| input a natural number n, terminates on all inputs, and
| outputs a rational number y such that |x-y|<10^-n .) Every
| number you ever thought of is computable, but as a
| mathematician, working with the set of computable numbers
| is much more tedious than working with real numbers.
| tshaddox wrote:
| > Much more sane is the set of computable numbers, that
| is the set of numbers for which you can find an algorithm
| that computes the number to arbitrary precision.
|
| But perhaps still not as sane as one may hope. It would
| be very sane to be able to compute, for any two numbers,
| which one is larger (or whether they're equal), but sadly
| this is not computable for the computable numbers.
|
| > Every number you ever thought of is computable, but as
| a mathematician, working with the set of computable
| numbers is much more tedious than working with real
| numbers.
|
| I mean, I've thought of noncomputable reals like Chaitin
| constants.
| staunton wrote:
| > it is only possible to describe 0% of the rational numbers
| between 0 and 1 with text
|
| It is possible to describe 100% of the rational numbers with
| text. You describe the numerator, make a space, then describe
| the denominator. The length of the text document depends on
| the number described and can be arbitrarily long.
| crdrost wrote:
| Sorry, phone autocorrected "irational" to "rational" rather
| than "irrational." fixed!
| andrewla wrote:
| For what it's worth, this is only true in Cantor's horrifying
| paradise. In the world of the intuitionists, none of this is
| true. Reject Cantor and embrace Brouwer and you can once
| again live in a world without these horrors, and all you lose
| is absurd statements about things true "almost everywhere"
| that are never true, and crazy results like Banach-Tarski
| that get an impossible result by doing two impossible things
| to set it up.
| m3kw9 wrote:
| How do you prove irrational numbers doesn't repeat down the line?
| IngoBlechschmid wrote:
| That's a great question, and the answer is by direct inspection
| that repeating digits cause the number to be rational.
|
| For instance, 0.123123123... is checked to be the same as
| 123/999, a fraction -- hence rational. Similarly,
| 0.abcdabcdabcd... is the same as abcd/9999. This works for
| repeating blocks of digits of any length.
| arvindh-manian wrote:
| To add on to this, the question then can become why can't the
| number start repeating after a certain point (e.g.,
| 3.14133333333...). But then we can represent it as a sum of
| 3.141 and 0.000333333..., i.e., two rational numbers. Then we
| can construct a fraction that represents the number.
| empath75 wrote:
| and it can also be shown that all rational numbers repeat
| because there's only so many remainders possible for a
| given denominator (all the numbers from 0 to n-1), and as
| soon as you repeat a remainder, you necessarily have to
| repeat everything from the last occurrence of that
| remainder.
| BobaFloutist wrote:
| Ok, a different question: How do we know that several orders
| of magnitude past the digits we've calculated so far, Pi (or
| e, or 2^1/2, or any irrational number) doesn't start
| repeating (or end), and turn out to be rational.
|
| If an irrational number has to have infinite digits without
| repeating (or stopping), and we can't calculate infinite
| digits, how can we ever know that a number is actually
| irrational? What if it's just an absurdly specific rational
| number, and we just haven't gotten to the end?
| pfdietz wrote:
| There are proofs that pi and e are irrational. These proofs
| aren't entirely trivial (especially for pi), but they're
| not very long. There are longer proofs that neither number
| is algebraic (solution to a polynomial equation with
| rational coefficients.)
|
| https://en.wikipedia.org/wiki/Proof_that_e_is_irrational
|
| https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irration
| a...
|
| The "third" fundamental constant of mathematics, Euler's
| constant, is thought to be irrational but there is as yet
| no proof (so it's conceivable it could be rational!)
| xigoi wrote:
| You don't need to know the digits of a number to prove that
| it's irrational. You just need to demonstrate that it can't
| be expressed as the ratio of two integers.
| arvindh-manian wrote:
| You may enjoy reading through this:
| https://mathoverflow.net/questions/32967/have-any-long-
| suspe...
| ordu wrote:
| If a number start repeating from some point, you can do a nice
| trick: multiply this number by 10^n choosing n to be a length
| of a period (effectively shifting the decimal points n places
| to the right). Then subtract:
|
| 10^n _x-x
|
| infinite periods will cancel each other due to the subtraction,
| you'll number with a finite amount of digits, say y, so:
|
| 10^n_x - x = y or x = y/(10^n - 1).
|
| y is rational (finite amount of digits), 10^n - 1 not just
| rational but integer, so x is rational.
| ordu wrote:
| According to Van der Warden "Science awakening"[1] Ancient Greeks
| treated numbers as some kind of "dirty" (Real) model of a
| platonic Ideal of a quantity. Numbers were invented by filthy
| traders and accountants while wise philosophers used geometry to
| reason about quantities.
|
| This attitude to numbers can be felt even now when we are taught
| to solve straightedge and compass construction problems. Greeks
| had no issues dealing with square root of 2 with geometry or
| "geometric algebra" how Van der Warden names it.
|
| [1] https://archive.org/details/scienceawakening0000waer
| diffxx wrote:
| I happen to be reading Poincare's Science and Hypothesis right
| now and he introduced a way of defining the square root of two
| that I found enlightening. In less articulate form: consider two
| sets, one of which contains all numbers whose square is less than
| two and one of which contains all numbers whose square is greater
| than two. The square root of two is then the symbolic name for
| the element that divides those two sets.
|
| Poincare says it better in the book though.
| __MatrixMan__ wrote:
| I think you're describing a
| https://en.wikipedia.org/wiki/Dedekind_cut
|
| (edit: oh, I see the article makes that clear. Well here's a
| link anyway)
| postoplust wrote:
| That's also the definition given in the article, attributed to
| Richard Dedekind.
| andoando wrote:
| You can do the same with all numbers, and you get the
| construction of the surreal numbers.
| andrewla wrote:
| I'm with Dedekind on this one -- Cantor's work, by and large, was
| hot garbage and led and continues to power some of the most
| naval-gazing mathematics ever invented.
|
| Dedekind had a great idea that at its heart was a constructive
| notion of what a "number" was in terms of our ability to
| approximate it. That's the core of intuitionism, and after a long
| dark interval has finally come back into prominence in modern
| mathematics.
| sandworm101 wrote:
| >> constructive notion of what a "number" was in terms of our
| ability to approximate it.
|
| That's the key. The problem isn't whether the number exists or
| not. It does exist as a point on the number line. The issue is
| our inability to describe its position using our number system.
| Adopt a different numbering system, a different language for
| describing locations on the number line, and one can avoid the
| debate altogether.
| Tainnor wrote:
| It's really hard to see the history of the sciences especially
| in the last couple of centuries as anything but a resounding
| success of modern mathematics. Whatever qualms some people may
| have about classical mathematics, nobody has shown it to entail
| a contradiction, nor has any practical result obtained in
| physics, engineering or anywhere else been shown to be
| erroneous for mathematical reasons (modelling errors, of
| course, happen all the time, no matter what mathematics you
| use).
|
| All the issues such as Banach-Tarski disappear once you apply
| mathematics to real-world things. Meanwhile, classical
| mathematics remains insanely practical.
|
| People like you, who call Cantor's work "hot garbage", are
| giving constructive mathematics, which by itself can be a very
| useful additional way of doing maths, a bad name. Cantor's
| diagonal argument, for example, doesn't just disappear in a
| constructive framework.
| andrewla wrote:
| I would argue that all the success of mathematics in modern
| times has been the result of constructive branches of
| mathematics. Where has anything useful been achieved from a
| non-constructive premise?
|
| Physicists and engineers are notoriously imprecise with their
| use of mathematics -- "all functions are integrable" and "all
| matrices are invertible", etc., and that's where all the
| real-world uses of mathematics have yielded results.
|
| I would love to hear examples where non-constructive
| techniques have yielded anything of interest. There have been
| places, for sure, where mathematicians working in those
| spaces have emerged with real and interesting work (Turing
| and von Neumann come to mind) but their work ultimately fits
| well within the bounds of intuitionism.
| tombert wrote:
| Irrational numbers have become a constant nuisance for me in
| Isabelle. I really wish that the Greeks' initial hypothesis that
| everything could be expressed in rationals was actually correct.
| abtinf wrote:
| Why is it better to invent a weird new class of numbers
| (irrationals) rather than just identify that there is something
| wrong with how we think about this issue?
|
| Put another way, why don't we reject out-of-hand the notion that
| sqrt(2) cannot be calculated, given that right isosceles
| triangles do exist in reality and their hypotenuse has a definite
| length?
|
| Put yet another way, why not just say sqrt(2) equals 1.41 (or
| however much precision you need) + some infinitesimal amount?
| neeleshs wrote:
| We probably have not discovered the unified theory of numbers
| yet, and these are all patches to the current system
| function_seven wrote:
| > _some infinitesimal amount?_
|
| What does that even mean? If we're rejecting the notion of an
| irrational, then the statement, "some infinitesimal amount"
| might as well be "some gorkly boggleboop".
|
| Sure, we can approximate to whatever precision is required for
| building a wall or calculating an orbit, but math itself would
| be hobbled by trying to make discoveries with the handicap of
| only allowing rationals.
|
| > _given that right isosceles triangles do exist in reality and
| their hypotenuse has a definite length?_
|
| I might be agreeing with you in a sideways manner, but right
| isosceles triangles _don 't_ exist in reality. Nor do any of
| the simple shapes like squares and circles. We have physical
| things that approximate those ideal shapes, but even the most
| precise triangle will not have a perfect right or 45 deg angle.
| Nor will the real-world hypotenuse be precisely sqrt(2). These
| physical items are made of a countable amount of molecules each
| of which is in some quantized state. Hell, the length of each
| side of the most perfect triangle we can make will be in
| constant flux.
|
| So for practical everyday purposes, sure. We can't work
| directly with irrationals, and there's no need to. But for
| making new discoveries in math, we must work out how to deal
| with "weird new" classes of numbers, like 0, or the negatives,
| or the complex, etc.
|
| Each one of those classes of numbers has survived because it
| has proven useful. If you can identify the "something wrong
| with how we think about this issue", you would probably win a
| big old prize for that :)
| xigoi wrote:
| How would that help you? Being able to reason about irrational
| numbers is useful.
| autoexec wrote:
| > The ancient Greeks wanted to believe that the universe could be
| described in its entirety using only whole numbers and the ratios
| between them -- fractions, or what we now call rational numbers.
| But this aspiration was undermined when they considered a square
| with sides of length 1, only to find that the length of its
| diagonal couldn't possibly be written as a fraction.
|
| Let me try it! If I make a square with sides 1 inch long, then
| measure the diagonal with a tape measure I get... 1 and 6/16ths!
| Only whole numbers and the ratios between them involved there, so
| I guess that's all the Greeks needed after all.
| xigoi wrote:
| You can't do math by measuring real-world objects.
| autoexec wrote:
| > You can't do math by measuring real-world objects.
|
| Don't we all start out doing math by counting/measuring real-
| world objects? It was all "Sally has x apples" and rulers
| when I started school but some kids do math using other tools
| like cubes and cuisenaire rods which are also used to do math
| through measuring/counting real world objects.
| xigoi wrote:
| Mathematics is a tool that can be used to describe the real
| world, and mathematical discoveries are often motivated by
| practital uses. However, mathematical structures are not
| directly influenced by the real world; they can just model
| it.
| autoexec wrote:
| > Mathematics is a tool that can be used to describe the
| real world,
|
| And that's what the article says the Greeks believed they
| could do using whole numbers and fractions. In the case
| of a real world square with sides of length 1 it seems
| that you can get away with describing the length of its
| diagonal in those terms which made it seem odd that it
| was what caused them to abandon their belief/aspiration.
| Maybe the author just described their dilemma
| poorly/strangely.
|
| I'm not at all suggesting that math has to be limited to
| the real world or that irrational numbers don't have
| their place. We're certainly better off with them.
| paulpauper wrote:
| lol finally a quantamag math article where I can follow the math
| Razengan wrote:
| Something I always love to ponder: Would aliens who perceive
| reality vastly differently than humans come up with different
| number systems?
|
| For example, for the longest time we thought of numbers as 1
| dimensional and refused to consider 2-dimensional numbers. Even
| know we try to shunt them off to the side as much as possible
| ("complex", "imaginary") even though they model reality more
| closely.
|
| Might a hypothetical alien race _begin_ with 2D numbers? or
| something entirely different?
| yen223 wrote:
| I have pondered if a hypothetical race of liquid or gaseous
| aliens living in a fluid world invented maths that were not
| rooted in counting numbers, what would that look like.
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