WEBVTT
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Hello.
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Welcome back to a
step in practice.
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We are dealing here
with rational numbers.
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In exercise 1, we are asked
to prove that square root of 5
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is not a rational number.
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Well, we proceed as
Francis showed us,
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that square root of 2 is
not a rational number.
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So we assume the opposite.
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We assume, by contradiction,
that the opposite is true.
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So assume that square
root of 5 belongs
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to the set of rational
numbers, and we are
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going to find a contradiction.
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Well, stating that the
square root of 5 is
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a rational number
means that it can
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be written as a quotient of
two natural numbers a and b,
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and we may assume that there are
no common factors in a and b.
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Without common factors.
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This implies that b times
square root of 5 equals a.
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So by taking the square, we get
that 5 times b to the square
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is the square of a.
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In particular, 5
divides a to the square.
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But if 5 divides
a to the square,
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necessarily 5 divides a.
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Otherwise, if 5 does not appear
into the decomposition of a, it
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cannot appear in
the decomposition
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of a to the square.
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So as I say, really 5 divides a.
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Well, but then 5 to the square
divides a to the square.
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Which is 5 times b square.
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And so 5 divides
b to the square.
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And again we get
that 5 divides b.
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But now, 5 divides, well,
a, and also divides b.
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This means that 5 appears
into the composition of a
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and in the composition of b.
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So they have a common factor.
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5 is a common
factor, to a and b.
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But this is a contradiction,
because we assumed that they
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had no common factors.
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So we get the contradiction.
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So it was not correct to
assume that square root of 5
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is a rational number.
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Thus square root of
5 does not belong
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to the set of rational numbers.
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In exercise 2, we
are asked to prove
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that 1 plus square root of
2 is not a rational number.
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Again, assume it is
a rational number.
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Assume that 1 to square root
of 2 is a rational number.
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Well, we get that square
root of 2 equals q minus 1.
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And q is a rational number,
1 is a rational number,
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so the difference is
again a rational number.
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But it is false.
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Because square root of
2, as Francis showed us,
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is not a rational number.
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So contradiction-- we
get a contradiction.
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So it was not correct to assume
that 1 plus square root of 2
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is a rational number.
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Thus 1 plus square root of
2 is not a rational number.
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Finally, in exercise
3, we want to show
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that 3 times square root of
2 is not a rational number.
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Again, assume it is
a rational number.
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If 3 times square root of 2 is
a rational number-- call it q--
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then we get 3 times
square root of 2
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equals q, rational number,
and thus square root of 2
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is the quotient
of q with 3, which
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is again a rational number.
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But this is a contradiction,
because square root of 2
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is not a rational number.
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Thus 3 times square root of
2 is not a rational number.
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And this ends exercise 3
and this step in practice.
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See you in the next step.
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