Serious math question: Why is the diagonal in a rectangle shorter?

Hey folks I need your help with a math question. I have learned (and never dared to question) that a diagonal in a rectangle is shorter. So if I have a rectangle with a side length of 10x10 cm walking the diagonal will just cost me 14.14..cm instead of 20 cm around the sides. So I've learned it and so I can measure it with a ruler.
But tonight I though about the 'why is it so' and couldn't sleep anymore.

Please take a look at this picture (quick drawing with paint so please excuse the hand drawn style):

If I walk along the brown line I will still have to go 20cm although I'm already moving kind of diagonal through the rect. If I cut this into half again and move along the red line it will still be 20cm. Now I can play this game nearly endless and always will end with 20 cm although I'm reached a nearly perfect diagonal.

If there is no physical limit (atoms/quarks...) I can do this infinitely and never reach the point where 20cm 'converts' into 14.14cm right?

And if there is a physical limit (I can't split the atom..) than isn't this also a limit for me walking this line? Why can I walk the diagonal if I have to move inside our physical world? Where is the point that my infinite stairway becomes a diagonal? Is it just a hypothetical model that we can't understand and simply is correct because it is so?

You are flirting with the idea behind integral calculus with the concept of infinitely small intervals but Pythagoras gets to the answer a bit simpler.

If your seriously having trouble visualizing why the shortest path between two points is a straight line, why don't you cut three sticks, 10cm 10cm and 20cm. Try to lay it out. Then it should be obvious, it will no longer be an angle with 90 degree corners, but a straight line. ...seriously...

Sorry, I spoke to soon. I see what your asking...

reply to post by VekTorVik


actually,,with the higgs-bosun,,the shortest path between two points is a straight line,

may have too include,,or wavey. ie not straight.

,the shortest path, that is.
ya deep.
but now mathematically, i am assuming,,,
and physic's confirmed.

the argument you are offering isn't a logical argument, because your approaching it from what all first year calculus students say.....your diagram is not a true representation of the issue, it is just a series of smaller and smaller trips around the outside of the rectangle.....which by definition is point A to point B to point C.......the trip in question is between point A to point C. Your concept is just making the sides smaller, but still travelling along those sides which if continued will equal the original sides, if you were to take one stride to B from A and then from B to C, that is two distinct linear steps, as opposed to taking one straight stride from A to C.

Originally posted by UnixFE
Hey folks I need your help with a math question. I have learned (and never dared to question) that a diagonal in a rectangle is shorter. So if I have a rectangle with a side length of 10x10 cm walking the diagonal will just cost me 14.14..cm instead of 20 cm around the sides. So I've learned it and so I can measure it with a ruler.
But tonight I though about the 'why is it so' and couldn't sleep anymore.

Please take a look at this picture (quick drawing with paint so please excuse the hand drawn style):

If I walk along the brown line I will still have to go 20cm although I'm already moving kind of diagonal through the rect. If I cut this into half again and move along the red line it will still be 20cm. Now I can play this game nearly endless and always will end with 20 cm although I'm reached a nearly perfect diagonal.

If there is no physical limit (atoms/quarks...) I can do this infinitely and never reach the point where 20cm 'converts' into 14.14cm right?

And if there is a physical limit (I can't split the atom..) than isn't this also a limit for me walking this line? Why can I walk the diagonal if I have to move inside our physical world? Where is the point that my infinite stairway becomes a diagonal? Is it just a hypothetical model that we can't understand and simply is correct because it is so?

edit on 3-7-2012 by UnixFE because: (no reason given)

What am I missing here? A shape with sides of 10x10 is a square. Right?

I have learned that a diagonal in a rectangle is shorter.

Shorter than what?

reply to post by jiggerj


a square is a rectangle,,,


I think he means straight diagonal is shorter then the steps,,,,,, but is wondering why if he shortens the length of each step,, untill they are barely steps,,, but like a jagged wave almost a straight line,,, why its not shorter then the original steps...



i think its because each jagged step,,,, the shorter you make them,,,, the more steps you make,,, and they will still equal the distance,,, because the more kinks the more deviations of straight they are taking,,

reply to post by pointr97


I concur. After thinking about it for a minute, by traveling the perimeter you are increasing the distance by something like 28%. That could be considered an error, and even when you have smaller angles to span the distance the error is still there, and is cumulative, creating the difference in distance.

Most people call it a square, however square fits the definition of a rectangle, 4 right angles opposite sides are parallel and the same length.

But you and I would probably always call it a square.

Originally posted by Phage
You are flirting with the idea behind integral calculus with the concept of infinitely small intervals but Pythagoras gets to the answer a bit simpler.

Thats exactly what i though

reply to post by Phage


When I was 9 my Dad told me it was impossible to get from point A to point B because there was no way you could ever cross half the distance as there was no way to ever measure every half. Or something. Same thing? Dog is making me drink. I don't know where he got the gun, but he sure knows how to get some pizza.

reply to post by UnixFE




Am I missing something here?

reply to post by Wide-Eyes


Yes

The error in the OP's theory is that it suggests that one must travel in a direction other than a straight line to the target....that one must travel in a different vector from the destination, then turn and travel in a different direction....That it is not possible to travel in the same vector, that is the error of the original OP, when that is grasped the OP can get some sleep

You are creating horizontal and vertical lines on which to travel. If you have big/longer ones, then halve their size creating double the horizontal/verticals, it's the same math, you just have twice as many horizontals and verticals, halve them again and they still equal the same distance even though you've doubled the number. You're never taking a diagonal or non perpendicular route. You're creating the same linear amount with different sized horizontal and vertical pieces.

reply to post by Domo1


I don't know how much beer you've had but even after ten pints, I can still walk in a straight line. Thus, the distance from A to C is shorter than wobbling around in a zig-zag pattern. I don't get it, walk in a straight line dammit!

I'm stamping this one "BUSTED"...

(do I really have to have more than one line, or is that just something people do?)

If you're talking about intentionally zig-zagging then I gues it's like angles of a triangle always add up 180degrees. Guess that's what Phage meant about Pythagoras.

Originally posted by Wide-Eyes
reply to post by Domo1

 

I don't know how much beer you've had but even after ten pints, I can still walk in a straight line. Thus, the distance from A to C is shorter than wobbling around in a zig-zag pattern. I don't get it, walk in a straight line dammit!

Hey let's bake his noodle some more . What if your ziggy-zaggies are an escalator? You're not stepping but standing still while traversing the distance. How far are you really traveling?

reply to post by VekTorVik


The one line rule is more aimed at off topic posts, i.e 'lol'. BTW, who gave you that picture of me for your avatar?