INTD 301 Problem Sets

Problem Set 1

The first problem set is due 1 March. This is finalised.

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, 111111111, 1(total of 63 ones)1, 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 7 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 6.4.4

6. Textbook 3.2.18-20 as one.

Here are some other textbook problems that I decided against, but considered 3.2.5, 3.3.7, 6.3.4, 6.6.9, 6.6.10, 6.6.13.

Problem Set 2

These problems are due on 22 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. - If you come to get it from me, I can give it to you before Thursday] 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.

[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?

6.6.13

Trigonometry, Logs, Exponentials, and Complex Problem Set

Due 19 April.

There is some uncertainty of different editions of the textbook. Here's some guidance. What I call Chapter 5 is called "The Triangle". It appears to be the same in both editions. What I call Chapter 8 is called "Building the Real Number System". It appears to be either chapter 6 or 8 in different editions. What I call Chapter 9 is called "Building the Complex Numbers". It appears to be either chapter 7 or 9 in different editions. What I call Chapter 12 is called "Trigonometry". It appears to be either chapter 11 or 12 in different editions. I hope this helps. Please always feel free to ask.

5.3.1, 5.3.9

as one question: (8.10.1.d,e,f, 8.10.10.b,f,g [say something interesting about g, at least]),

8.10.12 (for part b does the base matter? include this with the question.)

12.3.10 (also derive formulas for sin 3A and cos 3A), 12.8.1, 12.8.2

Present trigonometric proofs for as many special cases of SSA as you can discover. (one is done for you in the text - for HL - it also gives you a model for how to write such proofs).

Let f(x) = ln(1 - 1/x^2). Solve f(2) + f(3) + f(4) = ln(q) for q.

On your handout for trigonometric identities there are formulae for the radius of the circumscribed circle. Use the fact that the central angle is twice an inscribed angle to derive this result. Also, although I know that haversine is half of the versed sine (and vers ø = 1 - cos ø), when I look online I always find that haversine is sin^2(ø/2). Do we disagree?

**9.2.3** Talk to me about this one in class. Convince me there that you don't need to write it up.

9.4.10

Last Problem Set

Here's a beginning.

Create two data sets with each set having at least ten elements. Make it so that one has a larger standard deviation, but the _other_ has a larger mean absolute deviation. (To be clear, don’t compare MAD to SD for the same data, SD will always be bigger). Which of your data sets do you think is more spread out? Give evidence.

The probability of winning a certain carnival game is p = 0.3. Colby plans to play the game 3 times.

a. Make a tree diagram showing the possible outcomes for 3 plays of the game. For each outcome, compute p^, the proportion of wins.
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.

12.5.1
12.5.10

Prove log_3(5) is irrational (this is log base 3 of 5). Generalise as much as you can. Because this is due the day we'll discuss this more - hint: use unique prime factorisation.