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.
The introduction, this century, of computers into mathematics has certainly revolutionised the subject, and led to many new areas of mathematics previously unimagined.
One of the main aims of Mathematics is to invent ideas and notation which will help understand the real world.
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
$$\left(n\atop1\right)-{1\over2}\left(n\atop2\right)
+{1\over3}\left(n\atop3\right)-\cdots\pm{1\over n}\left(n\atop n\right)
=1+{1\over2}+{1\over3}+\cdots+{1\over n}\ ,$$
Q.957 Solve the three simultaneous equations
$${ab\over a+b} = {1\over 2} \quad , \quad {bc\over b+c} = {1\over 3}\quad, \quad {ac\over a+c} ={1\over 9}.$$