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Sure it's irrational! Just look!
May 9, 2012 6:29 PM   Subscribe

Geometrically the irrationality of the square root of 2 means that there is no integer-by-integer square whose area is twice the area of another integer-by-integer square. A visual proof that the square root of 2 is irrational (not found in previous visual proof post.)
posted by Obscure Reference (39 comments total) 20 users marked this as a favorite

 
previouser.
posted by Chuckles at 6:50 PM on May 9, 2012


Was sure this was going to go here (although that's not about irrationality so much as transcendal..ity.)
posted by DU at 6:52 PM on May 9, 2012


But is the square root of 2 really...necessary?
posted by kagredon at 7:26 PM on May 9, 2012 [1 favorite]


kagredon: Sure it is. Without it there would be a gap in the number line.
posted by aubilenon at 7:44 PM on May 9, 2012


There's quite a few proofs by contradiction of the incommensurability of a square and its diagonal. For example, in college we looked at one novel proof that had been produced by a student there about ten years earlier.

This particular proof is all about what squaring and square roots really mean, geometrically. And that's a good thing to try to understand because a lot of the mathemetical nomenclature we were taught in grade-school and high-school were abstracted away from the much more intuitive geometry and, typically, students learn this nomenclature and some of these important attributes by rote and not with any intuitive comprehension whatsoever.

It was actually deeply enabling for mathematical progress when western math released itself from the shackles it'd inherited from the Greeks of thinking rigorously in terms of the qualitative distinction, for example, between a length and an area and instead just abstracted to pure number. But the legacy of this is that now we think about squaring and square roots in a way that's just about some numbers rather than thinking about the sides of squares and the areas of squares and the areas of squares built upon the diagonal of another square and how all these things relate to each other. In many cases, that's good, just as it was good for mathematical progress. But in other cases, it makes it more difficult for students of mathematics to have any real intuitive understanding of what all this stuff means. I've known quite a few people who made it to the college calculus level but found that they didn't really enjoy math or feel they understood very much until they approached it anew from this foundational geometrical perspective (and then worked up again).

I prefer to discuss irrational numbers in terms of incommensurability because I think it makes it much more clear what's happening here. For example, the explanation in this post and its linked piece is unambiguous but not that easy to parse and its meaning, intuit.

Rather, I prefer to simply say that there's no measurement unit, no matter how small, that can both measure the side of a square and a diagonal without being just a bit too big (or small). No matter how small you subdivide one of these two lengths into some unit, that unit simply can't measure the other length exactly. Making it smaller won't help. The two lengths simply can't be measured by the same smaller length, no matter how small. They're "incommensurable" and that's why numbers (lengths) in these relations are called irrational — because, from Greek usage in Euclid and earlier, they can't be put into ratio. They're not called irrational because they're crazy or nonrational in the conventional contemporary common-language meaning of that English word; but because of this history and etymology.

However, the craziness connoted by irrational numbers is not totally anachronistic and irrelevant because the fact that these two fundamental lengths cannot be measured by each other was an astonishingly counterintuitive truth that profoundly challenged the Pythagorean mystics, and they kept it as a great secret of their cult. And, indeed, I think that incommensurability is still today deeply counterintuitive. From a common-sense perspective, it's hard to understand that you can't find some subdivided length of one of those lengths that's small enough to exactly measure the other.

Schoolchildren are often told that irrational numbers are never-ending decimal expansions, which is true. But there's many kinds of never-ending decimals that aren't irrational. Many numbers which can be expressed as the relationship of two integers are also never-ending when expressed as a decimal. So thinking of irrationals in these terms confuses more than it enlightens. However, if you can get your head around either or both of the facts that A) you can't express an irrational number as the relationship between two integers (as a fraction or ratio) and B) that two lengths which, in relationship, form an irrational number, are two lengths which cannot be used to measure each other (because, as I wrote, any unit made from one will never be able to measure the other, no matter how small), then you'll have the beginning of an intuitive understanding of what's happening here. Those two things are just different ways of expressing the same thing, of course.

The two most famous and obvious incommensurable pairs of lengths are the side of a square and its diagonal, and the circumference of a circle and its diameter. But the diagonal of the square is where its easiest to prove incommensurability.

Book X, Proposition 9 is the relevant proof in Euclid and it's quite important. There's some significance in the fact that Euclid waited all the way until Book X to deal with this stuff.
posted by Ivan Fyodorovich at 7:51 PM on May 9, 2012 [36 favorites]


The sides of the original pink and blue squares are a and b. The sides of the dark blue and small pink squares are 2b-a and a-b.

So this proof boils down to: if a/b = sqrt(2), then 2b-a/a-b does as well and has smaller denominator. (For example, if 17/12 = sqrt(2), then 7/5 = sqrt(2).)

Not that the picture doesn't illuminate things. I actually find it interesting; I was just curious how this corresponded to one of the well-known algebraic proofs.
posted by madcaptenor at 8:49 PM on May 9, 2012 [1 favorite]


Ivan Fyodorovich: "Schoolchildren are often told that irrational numbers are never-ending decimal expansions, which is true. But there's many kinds of never-ending decimals that aren't irrational. Many numbers which can be expressed as the relationship of two integers are also never-ending when expressed as a decimal. So thinking of irrationals in these terms confuses more than it enlightens."

Interestingly, if you write numbers as continued fractions rather than decimals then a number is irrational if and only if its representation fails to terminate (they also have interesting patterns for square roots and powers of e).
posted by Proofs and Refutations at 9:02 PM on May 9, 2012 [2 favorites]


madcatenor, interesting. And if you reverse the logic, making a and b the dark blue and small pink squares, then you are saying that if a/b = sqrt(2), then so is (a+2b)/(a+b), which turns into a nice iterative approximation
(basically making bigger and bigger squares). Let's start with assuming sqrt(2)=1=1/1.
Churning this repeatedly through (a+2b)/(a+b) we get the sequence: 1, 7/5, 17/12, 41/29, 99/70, 239/169, ... each time getting a better approximation of sqrt(2).
posted by fings at 9:24 PM on May 9, 2012


kagredon: Sure it is. Without it there would be a gap in the number line.

Curiously, this is more or less the same logic that gives us the square root of minus one. Completeness. Luckily it stops at the complex plane.
posted by iotic at 11:34 PM on May 9, 2012


iotic, the quaternions would like to have a word with you. I suppose the octonions would too, but they're nonassociative, so maybe not.
posted by nat at 11:52 PM on May 9, 2012 [2 favorites]


Iotic is right. Complex numbers are complete in the sense that they form an algebraically closed field.
posted by edd at 12:33 AM on May 10, 2012


We could also go to the one-point compactifucation of the plane - think of the complex numbers and the point at infinity as a sphere. Another way to fill in a gap.
posted by Elementary Penguin at 2:19 AM on May 10, 2012 [1 favorite]


s/compactifucation/compactification
posted by Elementary Penguin at 2:27 AM on May 10, 2012


Pfft. It's only a theory, and you can't disprove "Intelligent Counting".
posted by kcds at 3:23 AM on May 10, 2012


fings method looks like a method we were shown at school (it was a long time a go and my memory is hazy) that the Egyptians(?) used to find roots. Is this right?

So for example, the root of 16 is 4 and the root of 25 is 5, so the root of 20 is worked out as (5+4)/2 and iterated with the new value. I could well be misremembering, so if anyone knows what I am talkig about and if I am wrong, please chime in. (Or it will bug me now!)
posted by marienbad at 3:30 AM on May 10, 2012


Yes! Given a positive number a=x0, the recursive sequence xn+1=(xn+a/xn)/2 converges to √a.
posted by Elementary Penguin at 4:08 AM on May 10, 2012 [1 favorite]


Well, that us actually not exactly what you were talking about, but that should also work.
posted by Elementary Penguin at 4:09 AM on May 10, 2012


It's funny how rational the discussion has proven.
posted by Bathtub Bobsled at 4:54 AM on May 10, 2012


Schoolchildren are often told that irrational numbers are never-ending decimal expansions...

Ugh, are they really told that? Probably. Why are elementary school teachers so terrible at math?
posted by DU at 4:59 AM on May 10, 2012


We could also go to the one-point compactifucation of the plane - think of the complex numbers and the point at infinity as a sphere. Another way to fill in a gap.

Where could a guy who doesn't really understand Group Theory read more about this? Because it sounds awesome.
posted by nebulawindphone at 5:38 AM on May 10, 2012


Where could a guy who doesn't really understand Group Theory read more about this? Because it sounds awesome.

GOOGLE RIEMANN SPHERE
posted by Elementary Penguin at 5:49 AM on May 10, 2012 [1 favorite]


Schoolchildren are often told that irrational numbers are never-ending decimal expansions...

Ugh, are they really told that? Probably. Why are elementary school teachers so terrible at math?


Well, the way I was taught this long ago was that irrational numbers have non-terminating, non-repeating decimal expansions, which is true.

What I think is more interesting is that any repeating decimal expansion is necessarily the decimal expansion of a rational number.

Consider x = 0.142857142857..., for example (point 142857 repeating). If we multiply by 1,000,000 we get 1,000,000x = 142857.142857..., so by subtracting x from both sides we get 999,999x = 142857, and so x = 142857/999999, which reduces to 1/7. You can find the fractional representation of any repeating decimal expansion in essentially this way.

I don't want to grade calculus finals...
posted by Elementary Penguin at 6:40 AM on May 10, 2012 [3 favorites]


Ooh, that's interesting, Elementary Penguin. I'd never thought to work that out.

OK so (jumping ahead a couple of steps), to prove that any ratio of integers will come out as a repeating decimal, you could start by showing that the series

9, 99, 999, 999, ...

When decomposed into prime factors, picks up every prime - right? Hmm ... *thinks*
posted by iotic at 7:12 AM on May 10, 2012


Ah yes, now I remember there's an easier way. Still the repeated 9s decomposition thing is interesting. I suppose another way of putting it is that the decomposition of the series must pick up all primes, since every integer division has a repeating/terminating decimal.
posted by iotic at 7:16 AM on May 10, 2012


You need to deal with repeating decimals that don't start repeating at their tenths place, though. For example,

.01818...... = x
10x= .1818 and 1000x = 18.1818...
So 990x = 18 and x = 18/990 = 1/55.

On preview, further down the page on that Wikipedia page goes into more detail.
posted by Elementary Penguin at 7:21 AM on May 10, 2012


I know pi gets a lot of attention, but I had never understood it as well as I did until we watched a video in high school geometry explaining how it could be expressed as the number of sides required by a polygon to fit inside a circle while leaving no extra space.
posted by Demogorgon at 7:27 AM on May 10, 2012


Schoolchildren are often told that irrational numbers are never-ending decimal expansions

...that never repeat. If they include that part, then they're correct equating that to irrational numbers.
posted by Bort at 7:28 AM on May 10, 2012


"..that never repeat. If they include that part, then they're correct equating that to irrational numbers."

Yes, of course. I'm sure that this is often presented this way. What I think happens, though, is that the distinction between "never repeats" and "repeats" doesn't seem that important relative to the "never ending" part to the students and so that's what they remember.

This has a lot to do with a certain, dominant style of pedagogy. If you emphasize definitions of terms and certain easily identifiable attributes of the things you're presenting, that's what the students will learn, not what any of it really means. And you'll do that because it's easier to teach and easier to evaluate. You'll also do that, with better justification, if your emphasis is on quickly reaching, through an accumulation of a lot of preparatory rote learning, a facility at using a bag of techniques to solve certain frequently encountered problems. This is basically the state of almost all math education today, at all levels. That's math as something more like an engineering discipline rather than as a science or a humanity. I suppose I'd have less objections to this if we were honest with ourselves that this is what we're doing. Instead, though, we continue to imagine that students are really learning something fundamental about the world when they learn math; but they're not when they're being taught as a bag of tools.

Or, worse, as a means of enabling students to work through a checklist of problems on some standardized test. That's not even a generally applicable toolset, it's a toolset for passing standardized tests.
posted by Ivan Fyodorovich at 11:00 AM on May 10, 2012


iotic: "
OK so (jumping ahead a couple of steps), to prove that any ratio of integers will come out as a repeating decimal, you could start by showing that the series

9, 99, 999, 999, ...

When decomposed into prime factors, picks up every prime - right? Hmm ... *thinks*
"

You're skipping 2 and 5, natch. You will get all the other primes in there because:
Let's say a prime p never shows up in the factors of (string of nines). Then dividing by p will always give a remainder.

There are only p-1 possible (non-zero) remainders so the same remainder will eventually have to show up twice among 9, 99, ... 99999(p of these).
Subtract one from the other and you get a string of 9's followed by a string of 0's. The zeros are the result of 2's and 5's which we are excluding so we can further divide by the appropriate power of 10 to get rid of the trailing zeros. We now have a string of 9's guaranteed to be divisible by p.

(I think I just helped derail my own FPP!)
posted by Obscure Reference at 11:18 AM on May 10, 2012 [2 favorites]


Oh, very nice, thanks. And that's why maths is great.

Now what's the most intuitive proof that pi is irrational? (trying to help get the thread back on the rails, somewhat)
posted by iotic at 12:05 PM on May 10, 2012


Pi is trickier - for one thing, it doesn't have an easy algebraic definition. I believe you at least need the trigonometric numbers for a proof, but don't quote me.
posted by Dr Dracator at 2:41 PM on May 10, 2012


fings: Churning this repeatedly through (a+2b)/(a+b) we get the sequence: 1, 7/5, 17/12, 41/29, 99/70, 239/169, ... each time getting a better approximation of sqrt(2).

In fact those are the convergents of the continued fraction of sqrt(2), in a sense the best possible rational approximations. (Now I'm wondering if analogous proofs exist for other square roots. And my semester just ended, so I have time to play around with things ike this.)
posted by madcaptenor at 3:39 PM on May 10, 2012 [1 favorite]


I feel a little bored with flat surfaces. What sort of surface allows "squares" to have an rational relationship of their sides and diagonals?
posted by wobh at 4:13 PM on May 10, 2012


Consider a "square" drawn on the sphere by connecting the points at latitude λ and longitudes 0, 90 east, 180, and 90 west by great circles. (The angles won't be right angles, but they'll all be equal, and the lengths of each side are the same.)

If φ is zero degrees (that is, if your "square" is actually the equator), then the side length is one-quarter the circumference of the sphere, and the diagonal length is half the circumference of the sphere. So the diagonal-to-side ratio is two.

But it's "obvious" that the diagonal-to-side ratio is a continuous function of λ. And if λ is very close to 90 degrees (that is, we're drawing a small square around the North Pole), then we're essentially working in the plane and the diagonal-to-side ratio is the square root of two. So choose λ appropriately and you can get that ratio to be anything between sqrt(2) and 2, including all sorts of fun rational numbers like 3/2, 11/7, 34/21, and so on.
posted by madcaptenor at 4:46 PM on May 10, 2012 [4 favorites]


What sort of surface allows "squares" to have an rational relationship of their sides and diagonals?

You need a good definition for a square on an arbitrary surface - I'm thinking four sides of equal length, laying along geodesics, intersecting at right angles at four points to form a closed curve. Then you need to have geodesics passing through any two non-adjacent vertexes for diagonals, possibly of equal length. I'll let you call them wobh surfaces if you promise to put me in the acknowledgments of your differential geometry PhD thesis.
posted by Dr Dracator at 11:34 PM on May 10, 2012


Now what's the most intuitive proof that pi is irrational?

Unfortunately this turns out to be hard (even though it was thought to be true as far back as Aristotle). Most of the proofs I know of (here is the Wikipedia page Proof that π is irrational) involve calculus, although the first proof on that page, by Lambert, depends on the continued fraction of tan x.

It turns out to be easier to prove that e = ∑k=0 (1/k!) is irrational (again, there is a Wikipedia page with a proof) but I don't know that you'd call it intuitive.

Harder still in both cases is to prove that π and e are transcendental, which means that neither of them are the solution to any polynomial equation. All transcendental numbers are necessarily irrational (because the rational number a/b is the solution to the equation bx-a = 0), but not all irrational numbers are transcendental (For example, √2 is the solution to the equation x2-2=0).

All of this is discussed in a great book by Springer called Proofs from THE BOOK, which is a collection of "elegant" proofs of interesting mathematical facts. It is accessible to anyone who took some undergrad mathematics, and most of its super-calculus content is explained in marginal notes.
posted by Elementary Penguin at 5:18 AM on May 11, 2012 [3 favorites]


To correct an embarrassing mistake, I should add that transcendental numbers aren't the solution to any polynomial equation with integer coefficients. π is of course a solution to the equation x-π = 0.
posted by Elementary Penguin at 5:34 PM on May 11, 2012


You need a good definition for a square on an arbitrary surface

A square is a regular polygon with four sides. Regular means all the sides are equal and all the angles are equal. In euclidean geometry things work out that the angles must be 90 degree angles, but in other geometries (like the spherical example madcaptenor describes) that won't be possible.

I think there might be situations where a "square" by this definition is impossible - that's fine. We don't have regular 5-hedra in euclidean space, and that doesn't cause problems - until you need one for something.
posted by aubilenon at 11:47 PM on May 11, 2012 [3 favorites]


john baez on roots!
-Babylon and the Square Root of 2 [1,2,3]
-The Beauty of Roots (part 2 & 3) [1,2]
-Tidbits of Geometry [1]: "easy way to make a regular pentagon: just tie a simple overhand knot in a strip of paper... Did anyone ever build a mechanical gadget that lets you take fifth roots?"

also btw...
-Quantropy (Part 2/3)
-Symmetry and the Fourth Dimension (Part 2): "The idea is to start with something very familiar and then take it a little further than most people have seen."

more interesting/exciting stuff :P
Entropy of Non-Extremal Black Holes from Loop Gravity: "interesting paper by Eugenio Bianchi which claims to recover the Bekenstein-Hawking relation between the area of a black hole and its entropy, starting from a spin foam model of quantum gravity. This would be big news if it really works."

A Bayesian network: "Figuring out what causes what is tricky. Luckily we have tools to help us, like Bayesian networks. These have been developed by many people, but perhaps most famously by Judea Pearl, who in 2011 won the Turing Award 'for fundamental contributions to artificial intelligence through the development of a calculus for probabilistic and causal reasoning.' "

Entropy & probability theory: "Here Cosma Shalizi argues that in the Bayesian approach to probability theory, entropy would decrease as a function of time, contrary to what's observed. He concludes: 'Avoiding this unphysical conclusion requires rejecting the ordinary equations of motion, or practicing an incoherent form of statistical inference, or rejecting the identification of uncertainty and thermodynamic entropy.' "

Negative information: "In 2005 Michal Horodecki, Jonathan Oppenheim and Andreas Winter wrote a nice paper on negative information. I find it a bit easier to think about entropy. Entropy is the information you're missing about the precise details of a system. For example, if I have a coin under my hand and you can't see which side it up, you'll say it has an entropy of one bit."

Falling into a realistic Reissner-Nordstrom Black Hole: "What's it like to fall into a black hole? Here is a fairly realistic video made by Andrew Hamilton using a supercomputer simulation by John Hawley - though in reality you wouldn't hear music like this, thank god."

Debussy, Arabesque #1, Piano Solo (animation ver. 2): Watch Debussy's first 'Arabesque' unfold before your eyes and you'll see that this lush, dreamy, impressionist work is also geometry in motion, alive and organic as a fern yet precise as a crystal."
posted by kliuless at 2:34 PM on June 8, 2012 [1 favorite]


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