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?
a. Make a tree diagram showing the possible outcomes for 3 plays of the game. For each outcome, compute p^, the proportion of wins.Finish this question which I believe you began during class:
b. Complete a table to show the probability distribution of p^.
c. Find P(p^< p) and P(p^>p). Is the sample proportion more likely to underestimate or overestimate the population proportion?
d. Use the probability distribution from b. to find E(p^).
e. Is p^ and unbiased estimator of p? Explain why or why not.
f. Find the variance of the distribution of p^ from b. Compare your result to what you would get from the formula Var(p^) = pq/n.
Also please complete one presentation of a derivation of a distance formula from a point (p,q) to a line Ax+By+C=0. Using any of the methods from the handout or your own method will all be fine.Suppose that we currently have a test for a serious disease, say tuberculosis, which has 100% reliability, but is very expensive to perform. A new and much less expensive test comes along and we want to determine how effective it is in determining if a person has tuberculosis. One way of doing this is to test 1000 people with the more expensive test to determine how many of the people have the disease. Then test these same people with the new method and see in how many cases it properly predicts the disease. Let us imagine that we have done this with the following results. According to the completely reliable expensive test 8% of the 1000 people have the disease. Of those who had the disease, the new test indicated such in 98% of the cases. Of those who didn抰 have the disease, the new test indicated such in 98% of the cases. Thus, the test is what we call 98% accurate. (a) What is the probability that a person chosen at random from this 1000 people test positive? (b) What is the probability that a person will test negative? (c) What is the probability of a person having a false positive? (d) What is the probability that a person who tested negative actually did have the disease (i.e. a false negative)?
a. For practice, write an equation that describes the set of all points a distance 7 from (3,4). Graph that to see if your software is doing what you want.
b. Now use the formula from the previous problem and graph the parabola with focus (3,4) and directrix x+2y+3=0. Because it will be changing in the next parts, this is the set of all points such that the distance to (3,4) = the distance to the line x+2y+3=0.
c. Next graph the set of all points such that the distance to (3,4) = twice the distance to the line x+2y+3=0.
d. And graph the set of all points such that the distance to (3,4) = half the distance to the line x+2y+3=0. (these should all three be very easy to make by merely changing numbers).
e. Graph the set of all points such that the sum of the distance to (3,4) and (5,6) is 7.
f. Graph the set of all points such that the difference of the distances to (3,4) and (5,6) is 1.
g. Graph the set of all points such that the distance to x+2y+3=0 is equal to the distance 3x+2y+1=0. That should be pretty boring.
h. Graph the set of all points such that the _product_ of the distances to x+2y+3=0 and 2x-y=0 is 5.
i. One more, for good measure, graph the set of all points such that the product of the distances to x+2y+3 = 0 and 3x+2y+1=0 is 5. I know what I expect for all of the others, but I'm not sure what this one produces. I think I know what it looks like, but I think it's the only one on this this list that is _not_ a conic section.