Math Is Fun Forum

riemann-man

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(Dis)Proof pi = 2 (!)

The proof is intuitively wrong, but I just can't figure out where.

(Circumference = 2pi*r)

1. Consider a rod of length L = 2. Draw a semicircle around it, which has radius R=1 and arclength C= pi
2. Now draw two small semi circles, one going from the midpoint of the rod to the top and to the bottom. Each of these smaller circles has R=1/2 and C=pi/2, making the total length of the arcs = pi
3. Then a 4-partitioned rod with 4 arcs would each have R=1/4 and C=pi/4, with a total arclength of pi.
4. Repeat this process with a limit to infinity; so with infinitely many semicircles you can approximate the sum of the arclengths to be the actual length of the rod. This makes it seem pi (sum of infinite C) = 2 (original length of rod)

!!!

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: (Dis)Proof pi = 2 (!)

I don't really know the answer, but I have another example:

Consider a straight staircase 1 m long and 1 m high, consisting of many steps. If you're a very small ant, the distance from the beginning of the staircase to the end will be 2m (1m up and 1m forward). This is true no matter how many steps there are. However, as the number of steps tends to infinity, it should approximate a flat slope, which, by pythagoras, has length √2.
Our conclusion is that 2 = √2

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: (Dis)Proof pi = 2 (!)

I have seen that one identity. It incorrectly proves that Pythagoras is wrong. a + b = c!

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Devantè

Real Member

Registered: 2006-07-14

Posts: 6,400

Re: (Dis)Proof pi = 2 (!)

The error lies in your 4th step. All you've done is proven that the arc lengths of a convergent series of paths does not necessarily converge to the arc length of the limit path.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: (Dis)Proof pi = 2 (!)

I believe I have a partial solution to the problem. Consider this. With one semicircle, the curve touches the straight line at two points (the end points). With two semicircles, it touches the straight line at three points (the end points and the midpoint). With four semicircles, it touches the line at five points. And so on …

Now if you let the process tend to infinity, it is clear that the endpoints of the semicircular arcs will only touch the straight line at only countably many points of the straight line. It may be impossible to visualize what the infinite curve will be like as those countably many points will be densely spread over the straight line (like the rational numbers on the real line). However, the straight line itself contains uncountably many points! Hence, it is a false assumption that in the infinite limit, the curve will become the straight line itself. It won’t – it will remain longer than the straight line.

Last edited by JaneFairfax (2010-10-09 02:02:51)