Math 117: Honours Calculus I
Fall, 2013
Assignment 1
September 6, due September 18
1. Prove that
√
3 is irrational.
Suppose that there existed integers
p
and
q
such that
p
2
= 3
q
2
.
Without loss of
generality we may assume that
p
and
q
are are not both divisible by 3 (otherwise we
could cancel out the common factor of 3). We note that
p
2
is divisible by 3.
Express
p
= 3
n
+
r
where
r
= 0, 1, or 2. Then
p
2
= 9
n
2
+ 6
nr
+
r
2
. If
r
= 1 or
r
= 2,
then
p
2
is not a multiple of 3. The only way that
p
2
can be divisible by 3 is if
p
is
itself a multiple of 3. (Alternatively, consider the prime factorization of
p
. Since 3 is
prime, the only way it can be a factor of
p
2
is if it is also a factor of
p
.)
Hence 9
n
2
= 3
q
2
, or 3
n
2
=
q
2
. Replacing
p
by
q
in the above argument, we see that
q
is also divisible by 3. This contradicts the fact that
p
and
q
are not both divisible
by 3.
2. Let
r
and
s
be rational numbers.
(a) Is
r
+
s
necessarily a rational number? (Prove or provide a counterexample.)
Yes, let
r
=
p/q
and
s
=
m/n
, where
p,m
∈
Z
and
q,n
∈
N
; we may always write
p
q
+
m
n
=
pn
+
mq
qn
.
(b) Is
r
−
s
necessarily a rational number?
Yes, we may always write
p
q
−
m
n
=
pn
−
mq
qn
.
(c) Is
rs
necessarily a rational number?
Yes, we may always write
p
q
m
n
=
pm
qn
.
(d) Is
r/s
necessarily a rational number?
No, consider
r
= 1,
s
= 0. There is no rational number
r/s
.
3. Let
x
and
y
be irrational numbers. Prove or provide a counterexample:
(a) Is
x
+
y
necessarily an irrational number?

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