Imagine a square room, of dimensions n metres by n metres, where n is a whole number. We have two identical square carpets, both having dimensions p metres by p metres, where p is also a whole number. The area of the room is twice the area of one of the carpets. We ask the question: what are the smallest possible values for n and p, for which this is true?
For example, let us try n is seven metres and p is five metres. Then the area of one of the carpets would be five times five, or 25 square metres, while the area of the room would be seven times seven, or 49 square metres. This area is almost twice 25 square metres, but not quite, so these values of n and p do not work.
Suppose, though, that the smallest possible values of n and p such that twice p times p is equal to n times n are available, and let the room and carpets have these dimensions.
We place these carpets on the floor of this room so that one corner of each carpet coincides with each of two opposite corners of the room. Since the area of both carpets together is equal to the area of the room, the area of the overlapping carpets must be equal to the area of the uncovered parts of the room. The overlapping part is a square of dimensions (2p – n) metres by (2p – n) metres, while the uncovered parts consist of two squares, each of dimensions (n – p) metres by (n – p) metres.
We have thus discovered that two identical squares, each of dimensions (n – p) by (n – p), together have the same area as a square of dimensions (2p – n) by (2p – n).
It is clear that (2p – n) and (n – p) are both whole numbers that are less than p and n. This is absurd, because p and n were assumed to be the smallest possible values that solve this problem. We conclude that no such whole number values for n and p exist.
The above proof argument is called Fermat’s method of descent, after Pierre de Fermat (1607 – 1665) who used it to prove several theorems. In this method, one starts from an assumed smallest solution, then demonstrates that an even smaller solution would exist, contradicting the assumption that the starting solution was the smallest. In this article, we have used Fermat’s method of descent to actually show that no square with a rational side length can have an area of two square units.
Alexander Farrugia is a senior lecturer at the University of Malta Junior College.
Sound Bites
• In a control system, a network of agents exchange signals with each other to produce an output from an input received by an external agent. A controllable graph is such a system where the input of the external agent, which communicates with the agents with equal sensitivity, is always able to produce a predetermined output. In the paper Alexander Farrugia, ‘On Strongly Asymmetric and Controllable Primitive Graphs’, Discrete Applied Mathematics, 211 (2016), 58-67, Fermat’s method of descent was used to demonstrate that one can remove agents from any controllable graph to end up with four agents connected by a path.
For more sound bites, listen to Radio Mocha: Mondays at 7pm on Radju Malta and Thursdays at 4pm on Radju Malta 2 (https://www.fb.com/RadioMochaMalta).
DID YOU KNOW?
• Pierre de Fermat was a lawyer by profession; doing mathematics was his hobby. His mathematics was imparted by letters, in which often his results were stated with scant proof.
• Fermat is famous for the assertion he made in 1637, nowadays called Fermat’s Last Theorem, stating that if n is a whole number, the sum of nth powers of two whole numbers cannot be equal to the nth power of a whole number except when n is 1 or 2. He had written this result on a copy of Diophantus’ book Arithmetica and cheekily added that he had a remarkable proof of the result but the proof was too large to fit in the margin of the book. The result was only proved 358 years later, in 1995, by Andrew Wiles.
• Fermat used his method of descent to prove that any prime number that is one more than a multiple of four (like five, 13, 17 and 29) can be expressed as the sum of two square numbers.
For more trivia, see: www.um.edu.mt/think.