you should definitely tell your students about the discovery of irrational numbers by the greeks in the mystical pentagram. It goes something like this :
Ancient Greek mathematics (Pythagorean school) was chiefly based on geometry. They believed that all pairs of line segments are of commensurable length, that is, both segments are integer multiples of a common smaller segment. This is just saying that all numbers are rational.
Now take a pentagram inscribed into a (regular) pentagon and note the smaller pentagon inside the figure. For the Pythagoreans it was impossible to prove that the side lengths of the outer and the inner pentagon were commensurable. One can, in fact disprove this by contradiction.
make a drawing like the following
Using isosceles triangles you can show that
a0 = b0 + a1
b0 = a1 + b1
and
a1 = b1 + a2
Suppose a0 and a1 are commensurable then there exists a finite length r and integer numbers m0 and m1 with
a0 = r m0 and a1 = r m1. Using the sum identities above you find
b0 = a0 - a1 b1 = a0 - 2 a1
and
a2 = 3 a1 - a0 = r (3 m1 - m0) = r m2.
This means a0 and a2 are commensurable as well and even with the same r! There is no limit to inscribing more pentagrams into the figure and all side lengths an are commensurable with a0 and the same scale r by the reasoning above. Sice for any finite r there will be an n with ann cannot be an integer multiple of r, this contradicts the assumption of a0 and a1 being commensurable.
Note also that by similarity
b0/a1 = a0/b0 = (b0+a1)/b0
which defines the golden ratio.
The belief of the Pythagoreans must have been fairly shaken by this discovery. Some stories say this contradiction was attributed to special mystical powers of the pentagon. Others make it more colorful and say the one who first came up with this proof committed suicide or was killed for heresy.
Ancient Greek mathematics (Pythagorean school) was chiefly based on geometry. They believed that all pairs of line segments are of commensurable length, that is, both segments are integer multiples of a common smaller segment. This is just saying that all numbers are rational.
Now take a pentagram inscribed into a (regular) pentagon and note the smaller pentagon inside the figure. For the Pythagoreans it was impossible to prove that the side lengths of the outer and the inner pentagon were commensurable. One can, in fact disprove this by contradiction.
make a drawing like the following
Using isosceles triangles you can show that
a0 = b0 + a1
b0 = a1 + b1
and
a1 = b1 + a2
Suppose a0 and a1 are commensurable then there exists a finite length r and integer numbers m0 and m1 with
a0 = r m0 and a1 = r m1. Using the sum identities above you find
b0 = a0 - a1
b1 = a0 - 2 a1
and
a2 = 3 a1 - a0 = r (3 m1 - m0) = r m2.
This means a0 and a2 are commensurable as well and even with the same r! There is no limit to inscribing more pentagrams into the figure and all side lengths an are commensurable with a0 and the same scale r by the reasoning above. Sice for any finite r there will be an n with ann cannot be an integer multiple of r, this contradicts the assumption of a0 and a1 being commensurable.
Note also that by similarity
b0/a1 = a0/b0 = (b0+a1)/b0
which defines the golden ratio.
The belief of the Pythagoreans must have been fairly shaken by this discovery. Some stories say this contradiction was attributed to special mystical powers of the pentagon. Others make it more colorful and say the one who first came up with this proof committed suicide or was killed for heresy.
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