Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

This question already has an answer here:

The Pythagorean theorem has already been proved and it is a basic fact of math. It always works, and there are proofs of it. But I have found a problem.

Say you want to get from point A to point B.

an image

Here is a way to do it, where red is vertical movement and grey is horizontal movement.

another image

Now say you split the path up like this. Note that it is the same length, as you can see from the color of the lines:

another image again

You can continue to do this... (note that the path still continues to stay the same length):

yet another image

And if you continue forever, the path will become diagonal.

yet another image again

But now there's a problem. This is contradicting the Pythagorean theorem:

so many images!

I know the Pythagorean theorem is true and proven, so what is wrong with this series of steps that I went through?

share|improve this question
    
This is an abstract duplicate of this popular question, and indeed a direct duplicate of this question and this question. –  Zev Chonoles Mar 8 '13 at 22:24
    
Length is a tricky notion. If you have two curves $y=f(x)$ and $y=g(x)$ that are visually indistinguishable from each other, ther area under one curve, from $x=a$ to $x=b$, is very close to the area under the other. But their lengths, as your work shows, can be quite different. –  André Nicolas Mar 8 '13 at 22:26
    
@ZevChonoles Thanks, I didn't see those. You can close my question as a dup then. –  Doorknob Mar 8 '13 at 22:27
    
In particular, see the accepted answer for the "duplicate" question 12906. –  GEdgar Mar 8 '13 at 22:27
add comment

marked as duplicate by GEdgar, Thomas Andrews, Cameron Buie, Amzoti, 5pm Mar 8 '13 at 22:53

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

2 Answers

up vote 3 down vote accepted

By splitting the path you have essentially created lots of little triangles. You still need to apply Pythagoras' theorem to each one. If you do, then you will get the correct answer.

share|improve this answer
    
Thanks for the simple explanation without any complex math terms! –  Doorknob Mar 8 '13 at 22:31
add comment

The problem here is that the limit of the lengths is not the length of the limit. One has assumed that the sequence of lengths $x+y,x+y,x+y,\ldots$ converges to the length of the hypotenuse in the fake proof.

share|improve this answer
    
Um... I don't understand this. Please clarify? –  Doorknob Mar 8 '13 at 22:27
add comment

Not the answer you're looking for? Browse other questions tagged or ask your own question.