Mathematics and Physics Blog

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Fermat’s Last Theorem and Andrew Wiles

Any maths student will be familiar with the fact that there are pairs of integers whose squares add up to the square of another integer, for example 3^2 + 4^2 = 5^2.

Examples of these so-called Pythagorean Triples have been known for millenia.

Fermat’s LastTheorem posited that there are no integers for which a^n + b^n = c^n where n is an integer greater than or equal to three.

There is a fascinating documentary by Simon Singh about the proof by the English mathematician  Andrew Wiles and his very lengthy proof in 1995.

It may be found here.

Pierre de Fermat was a famous mathematician who lived in the 17th Century in southern France. He is best known for Fermat’s principle that explains how light travels and Fermat’s Last Theorem in number theory, which he described in a note at the margin of a copy of his book Diophantus‘ Arithmetica.

Fermat’s Last Theorem is possibly the most well-known theorem in mathematics. It was suggested by Fermat, and indeed he said that he had a proof for it but this was never published. A theorem without a proof is a strange thing indeed – not a theorem but a conjecture – a mathematical law which has not been proven.

It took over three hundred years and seven years of work for a British mathematician, Andrew Wiles, based at Princeton University in the USA to solve the problem.

The idea of Fermat’s Last Theorem can easily be understood with a few examples and a calculator. Challenge students to find a case where n is greater than two. They may well not believe that such cases don’t exist.

The documentary lasts 50 minutes and first explains what Pythagoras Theorem is. It then extends the idea to any power to a whole number and explains the hint by Fermat that he had found a proof that there are no integer solutions to the equation

x^2 + y^2 = z^2 for n>2.

It then discusses quite clearly how a problem in one field of mathematics can be translated into a different problem in another area of mathematics. So it was that the original problem was translated into a different problem to which a solution needed to be found. Andrew Wiles, through a flash of inspiration, which he describes vividly, came to this solution.

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Ten tips on using the graphical display calculator

So, you’ve just been handed a brand new graphical display calculator (GDC) for your IB maths course. There’s a good chance that you have been given a Texas Instruments Ti-84+ or a Casio FX-9860. If you did the Middle Years Programme, then you may have used a GDC before. But if you took the GCSE or IGCSE, then it will be new to you.

Here are some top tips on how to familiarise yourself with the GDC and make it work for you:

1. Don’t expect your teacher to show you all the features of the GDC

If you don’t understand how to do a particular operation, there are some great tutorials on YouTube (see the list of links at the end of this article). For more complex queries, you may need to refer to the manual, which can be downloaded if you are without a hard copy.

2. Take it to class every day

Although there will be times when you don’t need it, you don’t want to be borrowing one from your neighbour. And remember to put your name on it so it doesn’t get lost! Your classmates will almost all have identical calculators!

3. Use it!

Remember that in paper 2 (Standard Level or Higher Level) and both papers (Studies) you are going to need your calculator to tackle many of the questions. Don’t try to do long-winded calculations by hand when there is a quick method using the GDC. You won’t get extra credit and you increase your chance of making mistakes. You need a different way of thinking when tackling calculator questions. All good IB textbooks identify whether a question is intended for the calculator or not.

4. Grasp the WINDOW

First and foremost, the graphing calculator can solve equations and inequalities graphically for you. But like drawing any graph, you need to tell the calculator the range of values for the x and y axes. This is called the WINDOW. If you don’t get the WINDOW right, you won’t see any curve on your display.

5. Use the ZOOM function

All GDCs have this function to zoom in and out of regions of interest on the graph you have plotted.

6. Set the mode

For calculations involving angles (sine, cosine, etc.), you need to know if you are working in degrees or radians. Make sure you know how to change the mode of the calculator. In IB, you are usually working in radians. The sine of ten degrees is not the same as the sine of ten radians.

7. Harness the power of the GDC

The statistical functions on the calculator are very powerful. Make sure that you learn how to enter data sets, display a scatter diagram and work out mean, median and other common statistical functions.

8. Understand the various operations

For example, don’t confuse the ‘subtract’ and the ‘minus’ operations. These are distinct and not accessed by the same key. Minus for entering a negative number. Subtract for taking away.

9. Beware of raising a negative number to a power

Put all negative numbers in brackets first if they are to be raised to a power. Try both ways – you will see what I mean.

10. Practise makes perfect

Don’t leave learning the calculator skills to the last minute.

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People I admire

Some of these people are known for their academic brilliance and others for their bravery in standing up for what they believe is right. There is no particular order to the list that I present below. It is just as it comes.

Paul Erdős was a brilliant and eccentric Hungarian mathematician who had a long and productive career. He spent most of his adult life living out of a suitcase and worked ceaselessly. He was truly single-minded in his devotion to the subject.

Paul Erdös Paul Erdös

Albert Einstein revolutionised physics in the twentieth century. He was responsible for shaking up its foundations and introducing relativity. The idea behind special relativity theory is very simple. The laws of physics must be the same as seen by any observer in the universe.  He was also a respected social commentator. He believed in compromise as the best approach to any problem.

Albert Einstein Albert Einstein

Continue reading “People I admire”

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Teaching in Kuwait

I taught at a school in Kuwait for a year  from 2003. This was an interesting experience culturally. I had not lived in a Muslim country before. I taught mathematics up to A-level at Kuwait English School. This is one of the better international schools in the country. Some of my classes were difficult and it was hard work. We started the day at 7 am which was the time we had to be on site. Lessons started at 7.30 and finished by the early afternoon. This had its advantages as we could then have the afternoon free. To be honest I was often so tired that I just went home and slept.

I lived in a block about 10 miles from the school which was where the school rented some basic apartments for staff. I hired a car each term, which worked out surprisingly cheap. I think petrol worked out at about 10 US cents per litre. The driving in Kuwait is an experience not to be missed. The standard of driving is amongst the worst I have seen anywhere in the world. The main technique could be called ‘point and push’. It was very scary. There were horrendous accidents every day on the highway which ran south from Kuwait City.

CNV00094 A famous landmark in Kuwait City

The weekends were on Thursdays and Fridays. We called this the virtual weekend. Saturdays and Sundays were normal days. One weekend, we went out for a drive in the desert to see if we could find the so-called tank graveyard which was where the US military had dumped all the Iraqi tanks and trucks that it had attacked during the Iraqi retreat from Kuwait in the first Gulf war. There were hundreds of tanks in row upon row as well as armoured vehicles, unexploded ordnance and so on. It was blisteringly hot in the desert and I only had a rough idea of the location as we were following directions given to us by other teachers. But we found the site in the end.

CNV00101 View from my window – lovely

This huge amount of scrap metal rusting in the desert is worthless. The reason is that depleted uranium, a byproduct of uranium enrichment, is used in the armour piercing shells that the Americans used. The result is a lot of contaminated metal. It will lie there for many years to come.

At the top of the post, you see pictures that we took on that day. Unbeknown to us, there was a military base nearby and we were soon seen by some soldiers who came over and arrested us. We were questioned in broken English and our cameras were confiscated but not before I had removed my camera’s card. We had to report the next day to the army headquarters to collect these items. We were politely questioned by a well-spoken Kuwaiti officer and given tea.

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Tulbagh to Pietermaritzburg

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Here you see me sitting in the driver’s seat. The Land Rover is well and truly stuck in the mud. I was travelling with my friend Nontando from Bulawayo. We had just dropped off Maurice at the farm. We visited the local Drostdy Hoff winery where we picked up some rather nice white wine. After leaving Tulbagh, Nontando and I headed down to Cape Town. We were planning on meeting up with Nontando’s other half, Tim Cherry, and her young four-year-old son, Andy. After a few days in Cape Town, we then followed the Garden Route via George, Knysna, and Port Elizabeth. Our destination was ultimately going to be Durban before driving back to Zimbabwe via Beitbridge.

At Pietermaritzburg, I paid for Nontando to do a tandem skydive. I was also doing some parachute jumps myself and had just started my advanced freefall course. Unfortunately, she threw up on the way down inside her jumpsuit, as recommended by the tandem master!

DSCF0010

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Tour in a 1960s Landrover

Maurice and Mike Berry at Tulbagh.

In July and August of 2001, I drove from Bulawayo to Cape Town and back in a friend’s old 1960s Landrover. I had worked previously in Zimbabwe as a teacher and had several very good friends in Bulawayo. Mike Barry, pictured, was the son of John and Severine Barry and had spent all his life in Bulawayo. One of the family’s workers, Maurice, also pictured, was unwell with a heart condition and I was asked if I could help take him down to the family’s farm at Tulbagh, about 100 miles north of Cape Town, where apparently he could get better treatment. We left Bulawayo and headed out towards Plumtree on the Botswana border. But not before spending several days trying to get the old Landrover roadworthy. I had been told by the owner that it was serviceable but this was far from the truth. It would not start, had a flat spare, almost bald tyres, brake lights that did not work as well as poor brakes. The rear fuel tank was faulty, the fuel pump was on the blink, the electrics were decidedly dodgy and you could see the road through a big hole in the floor. The windscreen wipers did not work either.

Once we got the worst of the problems fixed, It took several days to drive down to Tulbagh and we ran out of diesel a few times. The fuel gauges did not work and we did not realise at the time but fuel was draining from the rear tank. We slept in the vehicle on the way down. Here we see Maurice and Mike about to pull our Landrover out of the mud on their farm in the hills surrounding Tulbagh. The farm had extensive vineyards and a beautiful old farmhouse in the Cape Dutch style. The little town of Tulbagh had some fine cafes where you could sample the delights of milk tart, an Afrikaans speciality.

My friend Rachel, wife of Mike, was from England, She had been a VSO volunteer at the same time as me in Bulawayo. They were trying to make a success of a bed and breakfast business they were running.

I heard later that Maurice died some weeks after we left Tulbagh for Cape Town.

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Teaching in Denmark

Clearing snow in Hungary in 2011

This is my son Patrick in Sønderborg‎, Southern Denmark, taken in October 2011.

I taught at a college in this small provincial town in Southern Denmark for just over a year. It was called EUC Syd. I found the transition from working in Slovakia (where I had lived previously) to cold and wet Denmark difficult. Part of the reason was the high cost of living and my reduced income. The other factor was getting used to the Danish rules and the culture of expecting foreign workers to find out important facts by osmosis. The attitude was one of unhelpfulness. I found this very frustrating. The language barrier was also another factor and the high tax regime.

EUC Syd  would fall broadly under the category ‘community college’ in the UK. It is mostly for vocational students but also has a technical high school. There was being set up an IB department (International Baccalaureate) here but there were certain people in the school who seemed quite resistant to the idea that they should cooperate with the setting up of this school within a school. This caused quite a lot of difficulties.

lex-EUC

The academic coordinator was a Czech man who had lived for thirty years in Denmark. He had recruited me and had been very optimistic about the opportunities the new department would afford. However the first cohort of students was too small. We only had eleven students to begin with and I had had no hand in recruiting them. They were mostly unmotivated, lazy and bored by the whole concept of education. I don’t know if it was something in the water but they were rude and surly too on the whole. Getting homework done was like getting blood out of a stone. Forget deadlines. Plagiarism warnings went unheeded as everyone copied off each other for coursework assignments. We went down from 11 students to four by the year’s end and two of these were students coming in from other colleges. I really felt that I was wasting my time.

The behaviour of a lot of the students in the college left a huge amount to be desired. They became aggressive when challenged. Some of them used to enjoy kicking a full water bottle as hard as they possibly could down the entire length of the corridor. Once, this happened right in front of the principle and he did nothing. They also enjoyed pushing each other on office chairs as fast as they could. After I began complaining, a spate of door knocking began on my classroom door – by the invisible man. I found the group behaviour amongst the students the strangest. They had an unnerving habit of all turning and staring at someone at the same time. It was very odd. Danish culture to me seems a warning against those who would say give young people all the freedom they want.

I was also struggling financially and I could not afford to own a car. Most of the other staff had two breadwinners. I am a single parent with a teenage son.

When something needed doing urgently, the attitude was ‘who gives a fuck.’ Not even the management seemed to care. No one seemed to try to plan ahead of time. My first pay packet was taxed 55 % and I had to inform the school that unless they taxed me correctly, I would have to leave. I needed my son’s residency permit in order to register him for school in August 2011 and to apply for educational support for him and I had to plead to get this piece of paper in time. No one had thought that I might need it. On some days, I would be the only one in the office 5 minutes before lessons started. One childish teacher began to accuse me regularly of not putting away lab equipment and would even chastise me on having an untidy desk. I would definitely advise anyone against working in Denmark.

The holidays were a lot shorter than in the UK and in Slovakia. In Denmark, the school year starts in the first week of August and continues right through to just before Christmas but nothing ever seems to get done for the amount of time kids spend at school.

After 17 months of this, I decided to leave because I could see that it was a lost cause. I did not make myself popular because I vented my frustrations in the staff room. Other staff did not seem to care or were too soft on the students. Communication was very sloppy and my feeling was that many IB teachers were not following the IB guidelines in grading and assessing material and in assessing group coursework. Staff were always happy to make excuses for the students.