This week’s
post is going to be about a problem called squaring the circle. The problem
dates back to the time of the Greeks. The problem itself is not the difficult
part, the difficult part of the problem is that one of the restrictions of the
problem is to solve it only using the technology available to the Greeks. First, this post will cover the problem itself. Secondly, the post will
cover how pi relates to the problem. Finally, the post will cover how pi was
proven to be transcendental.
Squaring
the circle is a problem about finding a circle with the same area as a square.
At first this problem may not seem hard, but not impossible. After a few minutes
of thinking, it might become apparent that a circle of radius one, and a square
of side length root pi would have the same are. That would be correct, but the
Greeks did not have the square root of pi. The difficult part of the problem is
that, the problem can only be solved with the abilities of the Greeks. That
means there is no algebra, there are straight edges, and compasses. Straight
edges, and compasses limit the ability of available math. With this Greek
Technology the four basic functions of mathematics, addition, subtraction,
multiplication, and division, can be performed. Addition and subtraction are
straight forward. Adding is putting two line segments together, and subtraction
is removing a line segment. Multiplication, and division require some scaling
with similar triangles, but are also possible. There is one more operation that
can be performed with a straight edge and a compass, and that is square
rooting. Imagine a line of length x, and then a distance of one is added to it,
then a semi-circle is drawn to the end of the line. If a line is drawn
vertically from where the line x and line of length one meet, up to the
semi-circle, the length of the line is the square root of x. Looking back at
one of the solution to the problem, one might ask then why could the Greeks not
square root pi? The answer is that pi is not a constructible number, a
constructible number refers to a number that can made with a compass, and a
straight edge.
Pi is a
rather important number in mathematics, but at the same time it is equally
complex, and mysterious. Pi is a transcendental number. The name for
transcendental comes from the idea that pi is beyond other numbers, that it has
transcended to a high plane of understanding. Or in other words, it cannot be
expressed as an algebraic number. This is why the Greeks could not solve the
problem. Since pi cannot be an algebraic number, then automatically it cannot
be a construable number. However, pi was not known to be transcendental until
the late 1800’s. Even though pi was known, and could be calculated, it was
still unclear whether or not it was an algebraic number. It was proven to be a
transcendental number by a combined understanding from Euler, and a proof of
e’s transcendentalism.
Two
important facts about the number e were proven, e is transcendental, and e to any
algebraic term is transcendental. This was proven via contradictions. What that
means is that if e was algebraic, certain properties would apply. Since those
properties do not apply, e is transcendental. Now the same thing is done with
pi, but there needs to be a relationship between e and pi. This is where
Euler’s Identity comes to the rescue. Euler’s Identity is e raised to the (pi)
x (i) is equal to negative one. Going back to the proof of e raised to an
algebraic power, if pi is algebraic then e raised to it, should result in a
transcendental number. Negative one is not transcendental, so by contradiction
it must not be algebraic, so if pi is not algebraic, then it is transcendental.
Going back
to the problem itself, this means that there was not a way for the Greeks to
find pi. Even if the Greeks tried to use a square with a side length not having
pi in it, the radius circle would have to multiple by a ratio of pi to get a
whole number answer. Pi is crucial to this problem, and so it was proven that
under the given rules of the problem, Squaring the circle is an impossible
problem to solve.
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