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A (wrong) proof that the square root of 2 is equal to 2

A (wrong) proof that the square root of 2 is equal to 2

That post is a wrong "proof". We know it because it "proves" that the square root of 2 is equal to 2, and thus, elevating both sides to the power 2, that 2=4!

We do it by the means of successive geometrical constructions, in order to obtain a broken line that "approximates" a straight line of length the square root of 2.

First, we obtain the square root of 2 as the diagonal of a rectangle triangle of equal sides 1.

Second, we split the sides 1 of the triangle by 2 each, and construct a square of side 1/2 in the right angle of the triangle. That square touches the diagonal to the middle, and we can follow a broken line from one end of the diagonal to the other one. That broken line has 4 segments of length 1/2, and thus has a total length 2.

Moreover, we have 2 rectangle triangles of sides 1/2, so that we can repeat the operation of splitting their sides by 2 and drawing squares at their right angles. This results in a broken line along the diagonal, made of 8 segments of length 1/4, thus of total length 2 as well.

We have now 4 rectangle triangles of length 1/4, and we can repeat 4 times the operation on these triangles, and so on.

Given a step k of the construction, we have a broken line of a given number N(k) of segments of length L(k), thus being of total length N(k)*L(k), and we have N(k)/2 rectangle triangles.

If we repeat the operation on all the rectangle triangles, we get the double 2*N(k) of segments in the broken line, each segment being of length half the previous length L(k)/2. Thus the total length is equal to (2*N(k))*(L(k)/2)=N(k)*L(k): the total length of the broken line is unchanged!

Thus, all the broken lines obtained by one of these iterative operations has the same length as the first one, that is 2.

So that the total length of the "limit" of these broken lines, as k goes to infinity, is of length 2.

But the broken lines all get along the diagonal of length the square root of 2, and they get closer and closer to that diagonal. As a matter of fact, the "distance" from the broken lines to the diagonal go to 0 as the number of steps go to infinity.

Thus, we could think that the diagonal of length the square root of 2 is the "limit" of the broken lines of length 2 for all of them, couldn't we? And thus that the "limit" of a sequence made of 2's is the square root of 2, so that the square root of 2 is equal to 2!

But this is wrong: the square root of 2 is not equal to 2, it begins with 1.41…!

So what is the problem?

It is because we can not "go to infinity" with that construction so easily.

Indeed, the broken lines have a number of segments going to infinity, each one with a length going to 0. The total length of the "limit" is then the product of the infinity by 0, and this is an undetermined form.

Saying it another way, the "limit" curve has an infinity of angular points, resulting in a length being more than the length of the straight line it follows.

This is a strange property…

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