How to Prove Pi Equals 2: A Mathematical Paradox
A unit semicircle
Take a unit semicircle (a semicircle with a radius of one unit). This semicircle will have a base of length 2 and a curved edge half the size of a unit circle's, hence equal to 1/2 × 2πr = 1/2 × 2π × 1 = π.
Splitting into two semicircles
Suppose we put two smaller semicircles on our base line. Each of these has base 1, so r = 1/2. The curved length of each semicircle is therefore 1/2 × 2π × 1/2 = π/2. Put together, this gives us a total curved length of 2 × π/2 = π, the same as before.
Two semicircles of radius 1/2
Splitting into four semicircles
Let's split these up again so we now have four semicircles, each with radius = 1/4. Each semicircle has a curved edge = 1/2 × 2π × 1/4 = π/4 and so together have a total curved length of 4 × π/4 = π again.
Four semicircles of radius 1/4
Splitting our line further
In fact, it doesn't matter how many semicircles we split our base into; the total curved length always equals π.
If we have n semicircles, each semicircle has a diameter 2/n, hence a radius of 1/n. Each semicircle therefore has a curved length of 1/2 × 2π × 1/n = π/n. The total curved length is then n × π/n = π.
If we have n semicircles, each semicircle has a diameter 2/n, hence a radius of 1/n. Each semicircle therefore has a curved length of 1/2 × 2π × 1/n = π/n. The total curved length is then n × π/n = π.
Splitting to infinity
The curious thing here is if we keep on splitting. Each split gets our semicircles looking more and more like a straight line, which suggests that the limit as we approach infinity is a straight line. We've shown that the total length of the curved edges always equals π and we know that the straight line has length 2, so π = 2.
Many semicircles on the diameter
The problem with this 'Proof'
But surely π = 3.141 ... not 2? Don't worry; π does indeed equal 3.141 ... . What we have done here is demonstrated how careful we need to be when dealing with limits, especially when we think about those limits pictorially.
The line of semicircles doesn't actually tend towards a straight line. It may look it at a certain scale, but as soon as we zoom in, we just get a row of semicircles again. It can be seen clearly from the first image (by comparing the diameter and the curved edge) that π > 2; hence, our example is not really a paradox, just bad reasoning.
The line of semicircles doesn't actually tend towards a straight line. It may look it at a certain scale, but as soon as we zoom in, we just get a row of semicircles again. It can be seen clearly from the first image (by comparing the diameter and the curved edge) that π > 2; hence, our example is not really a paradox, just bad reasoning.
Comments
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