Mathematics with a Microcomputer is a source book of ideas designed to encourage constructive mathematical use of computers by taking advantage of the opportunities they provide to stimulate and clarify mathematical thinking. It will be of particular interest to secondary mathematics teachers, and bright pupils (11-18+) should be able to use much of the material with minimal supervision. The authors expect that university students will also find the book interesting and useful.
Much mathematical activity involves creating algorithms and establishing proofs, and the authors hope this book will provide a useful introduction to these important mathematical processes. Each chapter concentrates on a single mathematical theme. A problem is introduced, an appropriate algorithm is developed, and questions are set that require readers to develop their own algorithms and write and run short programs. Though BASIC language programs are developed the short algorithms can easily be adapted to other languages or software such as spreadsheets and graph-plotters.
The mathematical background to each theme is developed rigorously and readers should make every effort to understand the arguments and tackle the problems posed. Most chapters can be read independently of each other, but where appropriate, cross-references have been given.
Topics covered include :
1. Factors, prime numbers, Eratosthenes' sieve, perfect numbers, Mersenne primes, Goldbach's conjecture, the infinity of primes, Fermat's 'Little Theorem'.
2. Euclid's Algorithm (highest common factors), lowest common multiples, 'the game of Euclid'.
3. Fractions, Egyptian fractions, Farey sequences.
4. Fibonacci sequences, the golden ratio.
5. Decimal expansions of fractions, recurring decimals.
6. Binary expansions of decimals - why computer calculations are not always exact.
7. Approximating decimals by fractions, continued fractions. pi , e , root of 2 , …
8. Square roots and cube roots.
9. Pythagorean triples. Fermat's last theorem.
10. Solving f(x)=0. xn+1=g(xn) , bisection method, method of false position, Newton-Raphson method.
11. Dividing without division.
12. Linear simultaneous equations - Jacobi method, Gauss-Seidel method.
13. Instant interest - consideration of (1+x/n)n. Nominal and effective rates of interest.
14. Logarithms - an interesting algorithm for finding log10x and hence of finding the logarithm of any (positive) number in any (positive) base. Mean Value Theorem. Taylor's theorem.
15. Taylor (& Maclaurin) series - sinx, cosx, ln(1+x), ln((1+x)/(1-x)), tan-1x, ex . Gregory's series.
16. Numerical integration - rectangles, trapezium method, Simpson's rule.
17. Random numbers - various problems including the Buffon Needle problem.
Answers and, where appropriate, model programs to the questions are included at the end of the book.
