Year 1996 - Volume 32
2020 - Present

First let me apologise for the lateness of this issue. After many years of faithful service, Dr. David Tacon has felt a need to have a break from Parabola and so has retired from the job of editor.

Peter Brown
The introduction, this century, of computers into mathematics has certainly revolutionised the subject, and led to many new areas of mathematics previously unimagined.

Rod James
One of the main aims of Mathematics is to invent ideas and notation which will help understand the real world.

C. Cox
Doubtless the above heading will be to most of our readers quite cryptic and meaningless. It is the purpose of this article both to explain what it means, by defining the terms “almost all” and “transcendental”, and also to outline how it may be proved.

For anyone who has access to the part of the Internet called the World-Wide Web, there is a vast amount of mathematical material available.

Q.966 Prove that
(n 1)−1/2(n 2)+1/3(n 3)−⋯±1/n(n n)=1+1/2+1/3+⋯+1/n ,

Q.957 Solve the three simultaneous equations
ab/(a+b)=1/2 , bc/(b+c)=13 , ac/(a+c) = 19.

Once again we apologise for the lateness of this issue. Please bear with us while we sort out the bugs.

Frank Reid
It is a well known fact in cricket that the new ball when bowled by a fast bowler will often swing in its flight on the way down the pitch to the batsman.

Peter Coutis
In high school we learn some interesting mathematics and develop some (potentially) very useful skills. But how exactly are these skills and techniques applied to the real world?

Solutions - Junior Division
If x is a real number, [x] denotes the largest integer less than or equal to x ; for example, [π]=3 .
Find all positive real numbers x,y satisfying the equation
[x][y]=x+y .

SENIOR DIVISION
Equal first prize
MAH Alexandre, North Sydney Boys’ High School. STITT Daniel Ian, Sydney Grammar School.

Q.975 For which real numbers x is it true that
[5x]=[3x]+2[x]+1 ?
Here [x] denotes the greatest integer less than or equal to x ; for example, [π]=3.

Q.966 Prove that
(n 1)−1/2(n 2)+1/3(n 3)−⋯±1/n(n n)=1+1/2+1/3+⋯+1/n ,

In this issue, we bring you four articles which illustrate the diversity of Mathematics. In the article on Fermat’s Last Theorem, Professor Michael Cowling has shown how a long-standing problem about integers has recently been solved by looking at the graphs of functions which appear to be quite irrelevant.

Frank Reid
David Rowe, a Year 12 student at Barker College, rang us recently with the following question. We thought that it would be of interest to our readers.

David Sharpe
Most of us will have a tree in our homes over Christmas. To the mathematician, a tree is a special type of graph.

Carlos Alberto da Silva Victor
Our aim is to find a simple formula for the area of a polygon whose vertices are
A1(x1,y1),A2(x2,y2),⋯An(xn,yn), joined in that order.

Geoffrey Coombs
It was recently found that some versions of the Intel Pentium processor possessed a flaw in the floating point divide unit. This generated considerable interest in the algorithm used in this chip to perform the division of two real numbers.

Michael Cowling
Some 3000 years ago, the ancient Egyptians knew that the triangle with sides 3 , 4 and 5 is a right-angled triangle. And of course, they also knew the related fact that 9+16=25 , i.e., that 32+42=52 .

Q.985 For what values of the positive integer n is
1. 5n+2
1. 7n+2
a perfect square?

Q.975 For which real numbers x is it true that
[5x]=[3x]+2[x]+1?
Here [x] denotes the greatest integer less than or equal to x ; for example, [π]=3 .