Welcome to this first issue for 2008. I hope you enjoy the articles and problems. The articles by Michael Deakin and Peter Brown in this issue both relate to proof: Dedkind’s proof that there are an infinite number of objects and the proof by mathematical induction.

Michael A. B. Deakin

During my student days at the University of Melbourne I first encountered the passage I want to share with you. It was brought to my attention by a fellow-student, who found it interesting and unusual, as I did then and still do today.

P.G. Brown

The method known as mathematical induction is generally thought to have been introduced by Pascal (circa 1654), although a contrapositive form called the 'method of infinite descent’ was used by Fermat a little earlier. The name ‘mathematical induction’ was first used by De Morgan.

N. J. Wildberger

There are two really *fundamental theorems in metrical geometry*. One of them you already know - it is Pythagoras’ theorem. The other one is the Triple quad formula, which you probably don’t know.

Stephen Wright

The number of students studying higher level mathematics in Australian high schools is declining.

Michael A. B. Deakin

This column is prompted by some correspondence with K R S Sastry, who for many years has been active in Mathematics, particularly Geometry. He has worked in his native India and also in Ethiopia, and has contributed prolifically to Mathematics journals over many years.

**Q1231** Given a>0 , prove that

(√(a+a(√a+....+(√a))) [n times] < (1+(√4a+1))/2

**Q1221** (submitted by Frank Drost, Research Associate, School of Mathematics and Statistics, UNSW. Edited.) Complete the mathematical equations below by inserting the least number of mathematical symbols from the table

Each year Parabola celebrates the winners of the annual UNSW School of Mathematics Competition and each year there are standout performances from many Sydney Schools including James Ruse Agricultural High School and Sydney Boys High.

David Angell

If you have already begun studying complex numbers at school, you have probably been taught that it makes no sense to say that one complex number is less than another. However, there are various plausible ways in which we might attempt to do just that. Is it really true that none of them works?

Gerry Sozio

Numerical integration enables approximations to be found for ∫baf(x)dx
where the integral for f(x)
cannot be written in terms of elementary functions. Integration is the process of measuring the `signed area' between the curve y=f(x)
and the x
axis in between the end points x=a
and x=b.

John Steele

Many of you reading this article will be aware of the problems the world faces over energy supply: can we rely on fossil fuels (oil and coal), or should we look again at nuclear power? By nuclear power we usually mean fission, the break up of heavy atoms (uranium) to lighter ones.

Michael A. B. Deakin

I could kick myself! I have to begin this column by confessing to a stupid mistake. Here is what happened. I was surfing the net when I came upon a website that held great interest for me.

Problem 1. You are given nine square tiles, with sides of lengths 1,4,7,8,9,10,14,15 and 18 units, respectively. They can be used to tile a rectangle without gaps or overlaps. Find the lengths of the sides of the rectangle, and show how to arrange the tiles.

**Competition Winners – Senior Division**

**Q1241** Show that Simpson's Elementary Rule

∫^{b}_{a }f(x)dx ≈ ((b−a)/6) (f(a)+4f((a+b)/2)+f(b)) is an exact equality for the quadratic function

f(x)=Ax^{2}+Bx+C.

**Q1231** Given a>0 , prove that

(√a+(√a+...+(√a))) [n times] < (1+(√4a+1))/2

Welcome to the final issue of *Parabola* for 2007. The focus of this issue is on practical applications of mathematics.

Michael A. B. Deakin

The cartoon reproduced below first appeared in The New Yorker. It so caught the fancy of the Mathematical Association of America that they acquired the rights to it

Bruce Henry

One of the most difficult problems to be faced by developed countries throughout the world over the next fifty years is the ageing population problem. This problem is essentially a result of the "baby boom generation" (people born in the period 1946 to 1962)

Bruce Henry

The Earth’s climate is the result of myriad interactions between the Earth’s atmosphere and its surface, which is composed of oceans, land masses and ice-caps.

Greg Doherty

The success of Google is predominantly due to its page rank algorithm. All web crawlers can index each page for the terms contained in each page.

**Q1251** Show that the product of 4 consecutive integers is always one less than a perfect square.

**Q1241** Show that Simpson's Elementary Rule

∫^{b}_{a }f(x)dx ≈ ((b−a)/6) (f(a)+4f((a+b)/2)+f(b))

is an exact equality for the quadratic function

f(x)=Ax^{2}+Bx+C.