MATH1025
EXERCISES V
Rational and irrational numbers
Hand in your solutions to your tutor on Friday November 7th. Your answer
to question 3 should be prepared using laTeX.
a
1. Show that the decimal expansion of a rational number in lowest terms
b
where a and b are positive terminates after at most three decimal places if and
only if b is a factor of 1000. Which ones terminate in exactly three places?
a
2. Show that the decimal expansion of a rational number in lowest terms
b
where a and b are positive eventually recurs with the same digit repeated if
and only if b divides 9.10k for some k, but not 10k .
3. Prove that Q is ‘densely’ ordered, that is, that for any rationals x < y
there is a rational z such that x < z < y.
4. By using the fundamental theorem of arithmetic, show that a rational
a
number in lowest terms (where a, b > 0) has a rational square root if and
√
√b
only if a and b are both integers.
√
5. The crucial point in the proof by contradiction that 5 is irrational
was to show that a2 divisible by 5 ⇒ a divisible by 5. By following a similar
method, prove that
a2 even ⇒ a is even,
a2 divisible by 3 ⇒ a divisible by 3.
√
Hence give a proof by contradiction that 3 is irrational.
6. Give a proof from the fundamental theorem of arithmetic that
irrational.
√
3 is
√
7. Follow through the through the proof by contradiction that 2 is irrational, with 2 replaced by 4 throughout, and explain which step breaks down.