Why is the diagonal of a square not "2"

[oca]

But take a piggy-back ride on Gozilla and let him go in a direct line and you end up with root(2).

Thanks jug, I can visualize that too

More seriously, thanks everybody for all your explanations, it seems my problem is just to accept that this is not a "pixel world"...
[edit]
but I play guitar not violin so it's not easy
[/edit]

[Uberdude]

oca, something for you to ponder: if it were a pixel world, which way are the pixels aligned? Which way is the x direction and y direction? Related to the earth? What about in space? It reminds me of the famous Michelson-Morley experiment which showed that there wasn't this 'aether' as a fixed background reference frame through which the earth moves, rather like your background grid of pixels.

Also when talking about mathematical objects we are not concerned with the physical world so we don't worry about issues like the quantisation of space but instead focus on the 'platonic' world of an ideal square, perfectly straight lines etc.

[Aidoneus]

FYI, Taxi Cab Geometry Introduction: http://jwilson.coe.uga.edu/MATH7200/Tax ... xiCab.html

[oca]

if it were a pixel world, which way are the pixels aligned? Which way is the x direction and y direction? Related to the earth? What about in space?

Well, by "pixel world" I mean a world with quantisation in space, state of energy or whatever...
I world where that statement :

for any adjacent 2 points you can construct another point between them.

would not be true... because you can be in state A or state B but not in between...

Also when talking about mathematical objects we are not concerned with the physical world so we don't worry about issues like the quantisation of space but instead focus on the 'platonic' world of an ideal square, perfectly straight lines etc

I like abstraction, but sometimes, I'm not sure abstraction reflects our univers... and somehow I think we have got the wrong path at some point... because often when we try to explain something, that leads to complexity rather than simplicity... allways more laws, more theories... answers lead to more questions...

I'm surprised that so many "key numbers" like pi, e, gold number, root(2) are all irational numbers...
I think we are missing something. like if our world is just an imperfect "projection" of something else...

Sorry, my english is a bit limited for that kind of discution...

[Aidoneus]

I like abstraction, but sometimes, I'm not sure abstraction reflects our univers... and somehow I think we have got the wrong path at some point... because often when we try to explain something, that leads to complexity rather than simplicity... allways more laws, more theories... answers lead to more questions...

I'm surprised that so many "key numbers" like pi, e, gold number, root(2) are all irational number...
I think we are missing something. like if our world is just an imperfect "projection" of something else...

Perhaps you will find this old blog of mine concerning discrete (digital) versus continuous concepts interesting: http://www.britannica.com/blogs/2006/10 ... x-answers/

[DrStraw]

Perhaps you will find this old blog of mine concerning discrete (digital) versus continuous concepts interesting: http://www.britannica.com/blogs/2006/10 ... x-answers/

That was interesting. For years I have felt that I am teaching my students an unrealistic view of the world. In mathematics all real numbers have an existence. Pi is a number which has an infinite, non-repeating decimal expansion. But this is not really meaningful. I forget the actual number, but if we accurately know Pi to less than 100 decimal places we can still calculate the diameter of the known universe to an accuracy of less than the diameter of an electron. For all practical purposes Pi = 3.14159265358979 is way more than accurate, so why do mathematicians say that is wrong if an exact value is required?

It is partly because we are all taught that ourselves and get into the habit of thinking that real numbers are real. But in reality they are no more real than imaginary numbers. Of course, at the other extreme, kids are taught to rely on calculators and that any number of decimal places is okay. They cannot understand why they get the wrong area by using Pi = 3.14 is they are asked for two decimal places of accuracy.

There are times when it is frustrating to be a mathematician!

[Aidoneus]

There are times when it is frustrating to be a mathematician!

Yeah, as an undergraduate discovering Gabriel's Horn disturbed me. But not as much as the Banach-Tarski paradox, and the realization that the axiom of choice and the way that mathematicians have handled infinity since Cantor has led to the rejection of Aristotle's fundamental distinction between actual infinities and potential infinities. While I mostly kept it to myself in graduate school...I have always sympathized with Kronecker and, especially, Poincare.

[Thunkd]

Think of a triangle as two different routes of getting from A to B.

$$B
$$ +-----------+
$$ | . . . . b |
$$ | . . . . S |
$$ | . . . . S |
$$ | . . . . S |
$$ | a S S S S |
$$ +-----------+

Click Here To Show Diagram Code

[go]$$B
$$ +-----------+
$$ | . . . . b |
$$ | . . . . S |
$$ | . . . . S |
$$ | . . . . S |
$$ | a S S S S |
$$ +-----------+[/go]

It's true that B is four steps to the right of A and four steps above A. But if you take 4 steps to the right, those steps only take you to the right... they don't take you up at all. So now you have to take 4 steps up to get to B. That's not terribly efficient.

If you instead turn slightly and walk directly towards B then every step takes you both up and to the right. You're heading directly towards your goal instead of starting off heading parallel to it. Which means it's a shortcut. Of course it'll be shorter.

Try this experiment. Nail three nails into your Go board in the shape of a triangle (just kidding, don't actually do that!) Now take a piece of string and tie one end to A. And then run it along the bottom of the triangle and turn at the nail and run it up to B. Cut the string there. Now return all the string to A and stretch it directly to B. You'll see that the string that was needed to reach A going right and then up is much longer than you need when you go directly from A to B.

[Aidoneus]

That was interesting.

Now you've done it--encouraged me!

Perhaps you will find this old blog of mine of some small interest, also: http://www.britannica.com/blogs/2007/01 ... innocence/

BTW, I was Britannica's first blogger, as part of my duties as an editor.

[Polama]

Here's another one: Half of any positive number is still positive. The limit of ((((1/2)/2)/2)...) is 0. Therefore 0 is positive, and all the fun that entails.

The key idea these are all addressing is that the limit of a sequence is not necessarily a member of that sequence. It can be something else that the sequence never quite reaches, so it doesn't necessarily have the same properties as the sequence. So the length of every path in your sequence is 2, and the limit is indeed a straight line, but that straight line has length sqrt(2). It seems odd at first, but remember that any member of your sequence still has an infinite number of iterations in which to lose that extra ~.6

Your observation does demonstrate that any simple finite grid would have the property of a zig-zag being equal to two straight lines, so our universe is clearly not quantized in that fashion.

If you prefer a pixel universe, though, it could just be a more complicated one. After all, it's easy to simulate euclidean motion on a computer screen. All circles could be jagged, just too fine for us to detect =)

[Aidoneus]

If one wants to look at infinite series...Consider any conditionally convergent infinite series. Then by Riemann's Rearrangement Theorem, the terms can be reordered in such a way to make the sum be any number. Do you get that? While 1 + 2 = 2 + 1 (commutative property), this does not hold true for conditionally convergent series. Allowing actual infinities generates (infinitely?) many paradoxes.

FYI, see links starting here http://mathworld.wolfram.com/Conditiona ... gence.html

[lemmata]

If one wants to look at infinite series...Consider any conditionally convergent infinite series. Then by Riemann's Rearrangement Theorem, the terms can be reordered in such a way to make the sum be any number. Do you get that? While 1 + 2 = 2 + 1 (commutative property), this does not hold true for conditionally convergent series. Allowing actual infinities generates (infinitely?) many paradoxes.

FYI, see links starting here http://mathworld.wolfram.com/Conditiona ... gence.html

That isn't really a paradox if you think about the definition of infinite series: It is the limit of a sequence of real numbers, where the sequence is defined by a particular partial sum formula.

An infinite series is not defined as the sum of all numbers in a particular sequence independent of permutations. If this were the definition of an infinite series, then the sum of a conditionally convergent sequence would be undefined.

In other words, Gabriel’s Horn could be completely filled with a finite quantity of paint, but no amount of paint would suffice to paint the horn’s surface.

This is not quite a paradox either. Volume and area have incomparable units. The Horn already contains a subset called the "surface"---the part of the horn that makes up the "surface" and has zero volume---that can be used to paint the surface. In mathematical idealizations, surfaces are infinitesimally thin. Real paint is not.

Mathematics is about an internal consistency of abstract ideas defined by humans, rather than physical reality. Our difficulties with accepting mathematics of infinities often arise when we confound the reality that generates some of the ideas for mathematics with mathematics itself, which is often an idealization. Platonic perfect circles help us think about real world imperfect objects similar to perfect circles, but we shouldn't be shocked that perfect circles of our imagination sometimes behave differently from the imperfect objects of reality that inspired them. People can understand this logically but still be emotionally discomforted by it. I wonder what part of our evolution makes it so.

(The Britannica blog posts were very interesting. I didn't realize that these existed)

[Uberdude]

... Gabriel’s Horn ... Remarkably, the volume of the resulting three-dimensional figure is infinite, while the area of the two-dimension surface of the horn is finite.

Wrong way round there. Wikipedia gets it right.

[Aidoneus]

Mathematics is about an internal consistency of abstract ideas defined by humans, rather than physical reality. Our difficulties with accepting mathematics of infinities often arise when we confound the reality that generates some of the ideas for mathematics with mathematics itself, which is often an idealization. Platonic perfect circles help us think about real world imperfect objects similar to perfect circles, but we shouldn't be shocked that perfect circles of our imagination sometimes behave differently from the imperfect objects of reality that inspired them. People can understand this logically but still be emotionally discomforted by it. I wonder what part of our evolution makes it so.

(The Britannica blog posts were very interesting. I didn't realize that these existed)

Yes, I understand, and this was my real point. One needn't embrace Robinson's non-standard analysis to realize problems exist in applying math too literally to the real world. (No one can deconstruct and reassemble a real world object in the manner of Banach-Tarski.) Confusing models, mathematical or otherwise, for reality leads to paradoxes.Take painting Gabriel's Horn as an example, I know the argument about the volume of any paint application, but it is just a simple way of conveying an image and is not meant to be taken literally, while the actual mathematical volume and surface area are finite and infinite, respectively--and this imperfect match between model and reality, I suggest, is what leads to the perception of paradox or should I say cognitive dissonance. Of course, such observations are not original with me. Many smarter, more creative thinkers have explored some of the issues. (Owen Barfield, Saving the Appearances, is a relatively easy starting point if anyone is interested. But many other philosophers have tackled the subject.)

BTW, I am glad if you found any of our blogs of interest. I especially had fun writing How to Cheat at Chess (before I retired!): http://www.britannica.com/blogs/2006/11 ... -at-chess/.

[DrStraw]

Here's another one: Half of any positive number is still positive. The limit of ((((1/2)/2)/2)...) is 0. Therefore 0 is positive, and all the fun that entails.

The key idea these are all addressing is that the limit of a sequence is not necessarily a member of that sequence. It can be something else that the sequence never quite reaches, so it doesn't necessarily have the same properties as the sequence. So the length of every path in your sequence is 2, and the limit is indeed a straight line, but that straight line has length sqrt(2). It seems odd at first, but remember that any member of your sequence still has an infinite number of iterations in which to lose that extra ~.6

Your observation does demonstrate that any simple finite grid would have the property of a zig-zag being equal to two straight lines, so our universe is clearly not quantized in that fashion.

If you prefer a pixel universe, though, it could just be a more complicated one. After all, it's easy to simulate euclidean motion on a computer screen. All circles could be jagged, just too fine for us to detect =)

That reminds me of the snowflake curve: an infinite length surrounding a finite area.