# Pythagorean "Paradox" (right-angled triangle). [duplicate]

Consider an isosceles right-angled triangle as shown in the figure (top left). The length of its hypotenuse is $$c$$. The figure distinguishes both legs of the triangle, however, from now on let's assume, since it's an isosceles right-angled triangle, that $$b=a$$. Now, let's build a stair on the hypotenuse with steps of height $$\frac{a}{n}$$, and width $$\frac{a}{n}$$. If $$n=1$$ we have the picture of the figure (top centre), in which $$d=a$$ and $$e=a$$. Here the length of the stair is $$d+e=2a$$.

Notation: I will refer to each step in the stair as $$s_k$$ for some $$k\in\mathbb N:0.

If we continue doing the same procedure, we have that for some $$n\in\mathbb N$$ that the lenght $$\ell(n)$$ of the stair is:

$$\ell(n)=\sum\limits_{k=1}^n \text{lenght}(s_k) = \sum\limits_{k=1}^n2\frac{a}{n}=2a.$$

If we build a stair with infinitely small steps, why don't we end up with a straight line? Because if we did, we would be saying that $$c=2a$$, and by the pythagorean theorem we know that $$c=\sqrt{2}a$$. I appreciate your thoughts.

• This is the same idea as in: math.stackexchange.com/q/12906/73324 Nov 17 '14 at 3:16
• Agreed, it's the same idea. I'd also put out there that the distinction between $c = 2a$ and $c = a\sqrt{2}$ is analogous to the difference between taxicab distance and euclidean distance. If you haven't looked into taxi-cab geometry, it's relevant here. Nov 17 '14 at 3:25
• Pretty cool! Didn't know about that. Nov 17 '14 at 3:30
• @NickH: I believe your answer is most complete (taxicab vs Euclidean). Could you please copy it into an "answer" so that OP can accept it or we can upvote? Nov 17 '14 at 3:37
• "why don't we end up with a straight line". In Euclidean Geo, a (straight) line is the shortest distance between 2 points. The path that passes by a and b has a total length of 2a. The hypotenuse, otoh, has length of sqrt(2)*a, known from Pythagoras Theorem (we can prove from basic principles). Thus the longer 2*a distance won't meet the definition of a (straight) line because it is provably not the shortest distance here. Oct 30 '16 at 13:06