Numbers

1. Investigate ``big'' numbers. What is a big number? The following examples might guide your investigation. A bank is robbed of 1 million loonies. How long would it take to move them? How much would they weigh? How much space would they take up? How big a swimming pool do you need to contain all the blood in the world? Is very big? What is the biggest number anyone has ever written down (check the Guiness book of world records over the last few years)? How did this number come about?
2. How do computer bar codes (the ones you see on everything you buy) work? This is an example of coding theory at work. Find others. Investigate coding theory - there are many books with titles like ``an introduction to coding theory'' (this is not about secret codes). References: [Gal1], [Gal2], and [Gal3].
3. Infinity comes in different ``sizes''. What does this mean? How can it be explained? References: [Kam] or [Hunt] or refer to any book on Set Theory.
4. It is easy to check if a number is divisible by 10 by looking to see if its last digit is a 0. Haw many other ``tests of divisibility'' can you find? Divisibility by 5 or 7 or 9? Why do they work? Reference: [Gard1].
5. Most computers these days can handle sound one way or another. They store the sound as a sequence of numbers. Lots of numbers. 40,000 per second, say. What happens when you play around with those numbers? eg. Add 10 to each number. Multiply each number by 10. Divide by 10. Take absolute values. Take one sound, and add it to another sound (i.e. add up corresponding pairs of numbers in the sequences). Multiply them. Divide them. Take one sound, and add it to shifted copies of itself. Shuffle the numbers in the sequence. Turn them around backwards. Throw out every third number. Take the sine of the numbers. Square them. For each mathematical operation, you can play the resulting sound on the computers speakers, and hear what change has occurred. A little bit of programming, and you can get some very bizarre effects. Then try to make sense of this from some sort of theory of signal processing. You will first have to discover how sound is stored.
6. Find out all you can about the Fibonacci Numbers, . In particular, where do they arise in nature? For example, look at the spirals on a pine-cone -- following the pattern of the cone, one spiral will go left, the other right. The cone will be covered by ``parallelograms'', the number of seeds on each side of the parallelogram will (always?) be two neighbouring Fibonacci numbers. For example 5 and 8. Similarly for pineapples, petals and leaves on plants.
7. What is the Golden Mean? Study its appearance in art, architecture, biology, and geometry, and its connection with continued fractions, Fibonacci numbers. What else can you find out?
8. Find out all you can about the Catalan Numbers, 1, 1, 2, 5, 14, 42, ...
9. Investigate triangular numbers. If that's not enough, do squares, pentagonal numbers, hexagonal numbers, etc. Venture into the third and even the fourth dimensions. Reference: [C&G].
10. Build models to illustrate asymptotic results such as Stirling's formula or the prime number theorem.
11. There is a well-known device for illustrating the binomial distribution. Marbles are dropped through the top and encounter a number of pins before dropping into cells where they are distributed according to the binomial distribution. By changing the position of the pins one should be able to get other kinds of distributions (bimodal, skewed, approximately rectangular, etc.) Explore.
12. Investigate the history of pi and the many ways in which it can be approximated. Calculate new digits of Pi - see the web site [Pi] to discover what this means.
13. Use Monte Carlo methods to find areas or to estimate pi. (Rather than using random numbers, throw a bunch of small objects onto the required area and count the numbers of objects inside the area as a fraction of the total in the rectangular frame).
14. Explore Egyptian fractions. In particular consider the conjectures of Erdös and Sierpinski: Every fraction of the form or , can be written in the form , where a < b < c, and a, b, and c are positive integers. See what you can discover. References: [Stew], [S&R].
15. Look at the ways different bases are used in our culture and how they have been used in other cultures. Collect examples: time, date etc. Look at how other cultures have written their number systems. Demonstrate how to add using the Mayan base 20, maybe compare to trying to add with Roman numerals (is it even possible?) Explore the history and use of the Abacus. References: [Bak], [Ifr].
16. There are several methods of counting and calculating using your fingers and hands. Some of these methods are still in common usage. Explore the mathematics behind one of them. Reference: [Ifr].