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### Squaring the Circle

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Date: 3/16/96 at 21:44:27
From: Deborah A. Lindsay
Subject: math question

Dr. Math,

Where did the phrase "squaring the circle" come from?
We found it in literature and wonder about its origins and
what it means.

Thank you.

(United States House of Representatives Page School)
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Date: 3/17/96 at 1:22:24
From: Doctor Jodi
Subject: Re: math question

The Greeks were great at posing mathematical questions that
no one could solve.  One of these questions was:

Using only a straight edge and compass, construct a square
with the same area as a circle. The Greeks tried to solve this
by construction as early as 200 BC.

"squaring the circle".

Now, we know that the constant that relates the area of a circle
and its perimeter to the radius, pi (usally represented by the
Greek letter of the same name), is a transcendental number.

Transcendental, you say, that's an interesting name for a number,
but what does it mean?

Well, a transcendental number is one that can't be expressed as
a ratio of two integers AND it can't be expressed as a terminating
decimal, like 1.09 or a non-terminating decimal with a pattern:
i.e. 0.3333333, etc. or 1.297297297297, etc.

In fact, we have calculated a few MILLION digits for pi and we
still find it to be transcendental.  But more importantly, people
have proven that pi is transcendental. I think that the first proof
of this was by Lindemann in 182 (described in Courant and
Robbins' _What is Mathematics_).

So that ancient problem which so many people have struggled
with, squaring the circle, is not possible with just compass and
straight edge.

I'm really curious about what the Page School is (a school for
would-be pages? for the children of House members?)

-Doctor Jodi,  The Math Forum
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Associated Topics:
High School Constructions
High School Definitions
Middle School Definitions

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