This problem set is due 20 February. It may see some
changes before then.
1. Write these counting numbers in base -10:
1, 2, 11, 22, 33, 44, 99, 155, 266, 377. The
division algorithm requires the divisor, b, > 0, but not the
dividend. What does the division algorithm produce as a
quotient and remainder for -23 divided by 7? Create a way
to extend the division algorithm for negative integer divisors,
b < 0. Prove your result.
2. Consider the polynomial: 4x3 + 3x2 + 5x + 3.
Change this into a polynomial in (x - 2). (An incorrect
answer in the correct form is 4(x - 2)3 + 3(x - 2)2 + 5(x - 2) +
3.) How does this question relate to converting between
bases?
3. Factors of some large numbers can be found by
writing the numbers as polynomials. Use your knowledge of
polynomial factors to find as many factors of each number as you
can: 1000002000001, 1(50 zeroes)2(50 zeroes)1 [this number
has 103 digits], 111111111, 1(total of 63 ones)1 [this number
has 63, not 65 digits], 827827, 123123123123123123.
4. Items from the handout of inverse proofs:
Prove the following, in a similar fashion to the way we did in
class on 4 February: (-a)(-b) = ab [Be careful not to use
(-1)(-1) = 1. This proves that, not the other way
around You may assume #1-3, prove all else you need to get
there.]
Also prove the division algorithm for fractions is
true [this is not easy to typeset here, it's #10 on the
handout], i.e. (a/b) / (c/d) = ad / bc. [Again, you may assume
the results we proved in class, prove all else you need to get
there.]
5. Textbook 8.4.4.
6. Textbook 3.2.18-20 as one.
7. Textbook
8.6.13.
8. Pick one of 3.2.5, 3.3.7,
8.3.4, 8.6.9, 8.6.10. (spread out).
These problems are due on 11 March. They are finalised now.
3.2.13-14 as one problem
Here's one question:
work entirely in base 7. Express your answer as a base 7
fraction (not necessarily in lowest terms). 3.12 -
2/13. Please note: 3.12 is a septimal.
It is like a decimal, but is base 7, not in base 10.
[Hint: Polynomials are easier than numbers - it's a mantra
- it's a way of life.]
From handout in class: 21 -
25. [Make sure I give you this handout.] Hint on
23: Prove If N(w) is prime, w is irreducible. Then
use this when it helps. Hint on 24: Suppose
that w is the smallest norm element that cannot be factored into
irreducibles, prove that it can be. [There is no such thing as a
'norm element', that makes no sense, so this must mean "the
element with smallest norm". Be careful about that and
remember to think about words. If they don't make sense
as you read them - you may be misreading them.]
[Full credit on this question
will be 2 points - if you attempt you will earn two
points. If you have a solution you will receive 1 point
extra, if it is correct you will receive two points
extra.] Find the monic [leading coeffcient = 1] polynomial
f(x) of lowest degree with integer coefficients such that cube
root of 2 + square root of 2 is a root of the equation f(x) =
0. Make a graph to find out whether any of the numbers
obtained by negating one of the two terms seem to also be
roots. Carry out the algebra to prove which of these three
other numbers are roots.
Consider the equation x3 + px - q = 0, where
p and q are prime numbers. Show that 1 is the only
possible rational root. Show that if 1 is a root, then we
must have q = 3 and p = 2. What are the remaining roots if
1 is a root?