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DR. .JAMES GRIME: So today we're
going to talk about one
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of the great unsolved problems
in mathematics that went back
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to the Ancient Greeks, thousands
of years ago, which
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was eventually solved in 1882.
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And the problem is called
squaring the circle.
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And you may have even heard of
the problem as a metaphor for
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something that's impossible
to do.
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The question is, can you make a
square with the same area as
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Now you have to understand
the rules.
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In Ancient Greece they
didn't have algebra.
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So they could only
construct numbers
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using lines and circles.
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So you could only make things
using a straight edge--
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like a ruler, but not a
measured ruler, just a
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So lines and circles,
those are the rules.
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Those are the rules the Ancient
Greeks had to work by.
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Using those rules, can you
construct a square with the
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same area as the circle?
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Let's have a look at
what you can do
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with rules and compasses.
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You can add numbers.
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Here's a line, and
it has length a.
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And I add another line
of length b.
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And the whole thing
is a plus b.
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So you can add numbers together
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quite easily with lines.
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You can subtract numbers,
as well.
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If I start with b, and then I
mark off a length, which I'm
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going to call a, then this
bit here, that's going
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to be b minus a.
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We can multiply.
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If I draw a little triangle
here, this has length 1.
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And this has length a.
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I'm going to scale
up the triangle.
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I scale it up, so now this has
length b, the big triangle has
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length b, then this has been
scaled up as well, so that
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this has a length a times b.
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Same idea, but it's the
reverse of that.
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If I took a bigger triangle, if
that has a length of b, and
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the long edge here has a length
of a, if I scale it
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down so now this has a length
of 1, then this has been
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scaled down as well.
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And you scale it down so
it has a length of
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So you can divide by scaling
triangles as well.
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And there's one more
thing you can do.
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