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- 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?
- 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].
- 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.
- 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].
- 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.
- 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.
- 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?
- Find out all you can about the Catalan Numbers, 1, 1, 2, 5, 14,
42, ...
- 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].
- Build models to illustrate asymptotic results such as
Stirling's formula or the prime number theorem.
- 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.
- 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.
- 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).
- 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].
- 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].
- 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].
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