Honors Calculus I - Irrational Numbers - Math Worksheet With Answers Page 3
ADVERTISEMENT
1
2
3also has a remainder of 1 and so is not a multiple of 3.
In conclusion, the only way
can be a multiple of 3 is for
to have been a multiple of 3.
Done!
Now we give the proof of the irrationality of
3. As in case 1, we argue by contradiction.
Start off by assuming that we can write
3 =
where the fraction of whole numbers
and
is in its lowest terms. In particular,
and
do not have a factor of 3 in common.
Now squaring gives 3 =
or 3
=
. The left side of the last equation is clearly divisible
by 3. therefore the right side must also be divisible by 3 (since it’s equal to the left side!).
This means
is divisible by 3. Fact 3 above tells us that
must be divisible by 3. Thus
we can write
= 3 for some whole number . Substituting into the previous equation gives
3
= (3 ) or 3
= 9
or
= 3 . Now the right side of this equation is divisible by 3, and
so also must the left side be divisible by 3. Thus
is divisible by 3, and fact 3 above ensures
that
is divisible by 3. Thus
and
have a factor of 3 in common. This is a contradiction.
So our original assumption (that
3 could be expressed as a fraction) must be false. That is,
3 is irrational.
5.
Do a net-search to find answers to the following questions. Say which search engines you
used, and which keywords you used!
(a)
Is
rational?
Answer. No,
is irrational.
(b)
Is
rational?
Answer. No,
is irrational.
(c)
Are there more rational numbers than irrational numbers?
Answer. No. In fact, there are many more irrational numbers than rational numbers.
We saw in class that you can “list” all the fractions. If you could list all the irrational
numbers too, then you could list all the numbers. But we also saw in class that you
cannot “list” all the numbers. Thus it must be impossible to list all the irrational
numbers.
Conclusions. Now, something must be flawed with the Pythagorean reasoning at the top of the
page.
Where is the gap? Can you reconcile the two sections of this page [one: gaps between
rationals can be made to go to zero, and two: there exist irrational numbers]?
Just because the gap between numbers tends to zero, does not mean that the numbers fill
up everything. We can see this clearly by looking at the fractions of the form
where
is a
whole number and
0 1 2 3
. Again, the argument in the first paragraph tells us that the
gap distances between such numbers can be made arbitrarily small, but we clearly don’t have the
number 1 3. So, even though, we have a collection of numbers with arbitrarily small gap distances,
we don’t have everything. The same thing can happen (and does!) with the rational numbers.
ADVERTISEMENT
0 votes
Related Articles
Related forms
Related Categories
Parent category: Education