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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?


marked as duplicate by GEdgar, Thomas Andrews, Cameron Buie, Amzoti, user53153 Mar 8 '13 at 22:53

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  • $\begingroup$ This is an abstract duplicate of this popular question, and indeed a direct duplicate of this question and this question. $\endgroup$ – Zev Chonoles Mar 8 '13 at 22:24
  • $\begingroup$ 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. $\endgroup$ – André Nicolas Mar 8 '13 at 22:26
  • $\begingroup$ @ZevChonoles Thanks, I didn't see those. You can close my question as a dup then. $\endgroup$ – Doorknob Mar 8 '13 at 22:27
  • $\begingroup$ In particular, see the accepted answer for the "duplicate" question 12906. $\endgroup$ – GEdgar Mar 8 '13 at 22:27

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.

  • $\begingroup$ Thanks for the simple explanation without any complex math terms! $\endgroup$ – Doorknob Mar 8 '13 at 22:31
  • $\begingroup$ Hi, I still have problem in this. A line is made up of dots. If we shrink the triangles small enough into dots, why cannot it be a straight line? $\endgroup$ – Chin Huan Dec 28 '16 at 5:46
  • $\begingroup$ @ChinHuan you cannot shrink them small enough. Dots are infinitely small in 2 dimensions and lines are infinitely small in 3 dimensions. So there will never be a combination of lines that can be small enough to be a dot. $\endgroup$ – Jess Bowers Jul 23 '18 at 12:25

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.

  • $\begingroup$ Um... I don't understand this. Please clarify? $\endgroup$ – Doorknob Mar 8 '13 at 22:27

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