Honors Calculus I - Irrational Numbers - Math Worksheet With Answers

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MATH 1823
Honors Calculus I
Irrational Numbers
Due in class on Friday, Sept 8, 2000
Classical belief. The Pythagorean school believed that you could obtain any number (measure-
ment) you like by taking the ratio of two whole numbers (integers). That is, they believed that all
numbers were rational. On one level, this might seem to be a reasonable belief. You see you cant
make all possible measurements using whole numbers (integers), since the gap between successive
whole numbers is 1. We can introduce the fractions
2 to make these gaps all be 1/2. In general,
if we have two fractions, no matter how close together, then their average will be a third fraction
whose distance from the first two will be exactly half the distance that they were apart.
Show that the average of
and
is another rational number.
Note that
+
+
=
2
2
and that (
+
) and 2
are whole numbers with 2
= 0.
Draw a picture of these three numbers, starting from the assumption that
.
Therefore, since the gap length can be made to go to zero, there must be no gaps at all between
the rational numbers. That is, every measurement (number) we want is rational.
Some problem numbers. Here you see that there are some numbers which are not rational.
1.
Show that
2 (the length of the diagonal of a square of side length 1) is not a rational number.
First we prove a neat fact.
Fact 1. If the square of a whole number is even, then the whole number must also be even.
Proof. Let
be a whole number such that
is even. We have to rule out the fact that
is odd. If
were odd, then we could write
= 2
+ 1 for some whole number
. Thus,
= (2
+ 1) = 4
+ 4
+ 1 = 2(2
+ 2 ) + 1 which is clearly odd. Thus
cannot be
odd.
Now we are ready to argue that
2 cannot be rational. We argue by contradiction. Suppose
that
2 is rational. This means that we can write
2 =
where the fraction
of whole numbers is in its lowest terms. That is,
and
have no
factors in common. In particular, this means that we cannot have both
and
being even.
Now square both sides of the equation above to get 2 =
and so
= 2 . Now the
right hand side of this last equation is clearly even. Thus the left hand side, namely
, is
also even. But
is a whole number, and so Fact 1 tells us that
must be even.
Write
= 2 for some whole number . Substituting into
= 2
yields 4
= 2
and
hence 2
=
. Now the left side of this last equation is clearly even, and therefore the right

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