Download 3 pt

Math 312.3
ASSIGNMENT #1
(due January 23, 9:55 AM)
PROBLEM 1.1
PROBLEM 1.6
(3 pt)
(3 pt)
PROBLEM A
(3 pt)
√
Prove that 3 is an irrational number.
Note. In the text-book such facts are deduced from Theorem 2.2. I√skipped this
theorem and provided a simple straightforward proof of irrationality of 2. I ask you
to follow this proof in Problem A.
In the following problems use the fact that the sum, difference, product
and ratio of rational numbers is a rational number.
PROBLEM B
(3 pt)
√ √
√ √
√
Prove that 1 + 3, 3 − 1, 1 − 3, 2 3, 3 + 3 are irrational numbers.
√
Hint: Prove by contradiction: assume that 1 + 3 = r, a rational number and
come to a contradiction to the result of Problem A. Proceed the same way to prove
the remaining statements.
PROBLEM 2.4
(2 pt)
Advise. Prove the statement in Problem 2.4 by contradiction.
PROBLEM C
(2 pt)
Give examples
1) when the sum of two irrational numbers is a rational number.
2) when the sum of two irrational numbers is an irrational number.
3) when the product of two irrational numbers is a rational number.
4) when the product of two irrational numbers is an irrational number.
Hint: use the results of Problems A and B.
Note the difference: As mentioned above, the sum and the product of two rational
numbers are always rational numbers.
See the second page!
1
PROBLEM 3.1 (a,b)
(4 pt)
2