For October 30:

- Make up a problem involving the triangle inequality. (p. 47) Try to be different from the book!
- Explain matrix multiplication (2x2) and show why in general AB does not equal BA. Is there a multiplicative identity matrix? Do matrices have multiplicative inverses?
- Let [A,B]=AB-BA. Show that this operation is anti-commutative and non-associative.
- What is the relation between absolute value and distance? Rephrase the inequalities |x-2| < pi and |x+3| > 1 in terms of distance.

For November 6:

- Larson's
*Algebra II*has problems on p. 143 about the number of solutions to pairs of linear equations in 2 unknowns. Analyze 42,43,45,46,58,51,52 BY ROW REDUCTION of the associated 2 x 3 matrix. Explain your work. (I.e. if you use a calculator, explain your input and output). - There is a set of problems on 3 equations in 3 unknowns on pp. 181-182. As above, analyze 14,16,20,22,28,33 BY ROW REDUCTION of the associated 3 x 4 matrix.
- Make up a system of 3 equations in 3 unknowns where there is a 2-parameter set of solutions. It is hard not to have this fact obvious on cursory inspection of the system, but try.
- Make up a system of 4 equations in 4 unknowns where there is a 2-parameter set of solutions. Disguise the nature of the system with row operations if necessary.
- Explain why a system of 3 linear equations in 4 unknowns cannot have a unique solution. Argue by ROW REDUCTION.
- Is it true that a system of 3 linear equations in 4 unknowns must always have at least one solution? Argue by ROW REDUCTION.

For November 13:

Prepare derivations of the following laws:

- Law of cosines
- Law of sines
- Addition law for cosines
- Addition law for sines.

For November 20:

- Explain how a slide-rule performs multiplication.
- Explain how a piano keyboard is like a logarithmic scale. If the A below middle C has frequency 440 Hertz (cycles per second), what will be the frequency of the sound made by a bird which sings an A four octaves higher? Which is the most convenient base for logarithms in describing the pitch of a musical sound?
- Explain the Richter scale for expressing the strength of earthquakes.
- The intensity of sound is measured in decibels. "A decibel is
an arbitrary unit based of the faintest sound that a man can hear.
The scale is logarithmic, so that an increase of 10db means a
tenfold increase in sound intensity; ..." (from Time magazine).
- An ordinary conversation is 60 decibels. A rock music cocnert (near the amplifiers) is 120 decibels. How many times louder is the concert?
- The loudness of the water at the foot of Niagara Falls is a billion times louder that the least audible sound. What does that come to in decibels?
- We know what a decibel is; what is a bel?
- When do you think the quoted text was written?

For December 4

- Use your calculator to generate 400 random numbers between 0 and 1. Make and hand in a histogram of the distribution of these numbers by tenths (between 0 and .1, between .1 and .2, etc.)
- Use these numbers to simluate 20 repetitions of the experiment of tossing a coin 20 times (let [0,.5]=Heads, [.5,1]=Tails). Suppose the outcome of each experiment is p heads and q tails. Make and hand in a histogram of the distribution of p.
- Find, copy and submit an intelligent applicaiton of probability theory to the Florida Vote Problem. Explain carefully why you think it is intelligent.
- Write up an analysis of the Birthday Problem: how many people do you need in a room before the probability of two of them having the same birthday rises above .5?

Math Dept SUNY Stony Brook

tony@math.sunysb.edu

November 27 2000