This problem is from the 2015 Senior Mathematical Challenge published by the UKMT.
A fiendishly difficult problem
What is a complex number?
There was a time when negative numbers were not thought to be “real”. In the early days of mathematics, the subject was used to solve concrete problems where the answer would represent an area or a number of things. Therefore it was thought that negative numbers were an invention and not ‘real’. But now we do not question their “existence’. (Whether mathematics is invented or discovered is another question altogether.) We now find negative numbers indispensable in dealing with debt, temperature and so forth.
There came a time when another extension of the number system was desirable with the so-called complex numbers. These have no bearing to real-life in themselves however the mathematics which employs them can be used to solve problems in electronics, mechanics and has many other applications.
We simply define
[ i = sqrt{-1}]
and understand that complex numbers follow the usual rules of arithmetic. Thus we can solve a previously unsolvable quadratic equation such as
[x^{2} + 2x + 5=0]
and obtain the answer
[-1 + 2i] and [-1 – 2i]
If you substitute
[x = -1 + 2i]
into the equation above, you obtain
[(-1+2i)(-1+2i)+2(-1+2i)+5 = 1-2i-2i+4{i}^2-2+4i+5 = 1 – 4i -4 -2 +4i +5 = 0]
And the same can be shown for the other solution.
This field of mathematics was contributed to by many, not least by Rene Descartes, Leonard Euler, William Hamilton and others.
Mathematics in literature – Alice in Wonderland
This is an interesting topic. Probably the best author who wove mathematics into his stories was Lewis Carroll. You will find an interesting article here.
The article, from the New Scientist, describes how Lewis Carroll’s story was in fact making fun of the new mathematics of the 19th century. Lewis Carroll whose real name was Charles Dodgson was a mathematician at Christ Church College, Oxford. He was a traditionalist and took a dislike to the strange new mathematics of for example, complex numbers and other rapidly developing areas of mathematics.
Waves are everywhere
Look around you. That ripple when you lift your coffee cup, the sunlight streaming in through the window, the annoying noise your neighbour makes when he revs his car in the morning. All these are examples of the phenomenon of waves.
The mathematics of waves is modelled by the trigonometric functions, the sine and the cosine functions. They can also be modelled using complex numbers. waves are periodic. This means that the movement repeats itself in time and space. There is a characteristic frequency which represents the number of waves passing a point per second. Then there is the period of the wave which is the time between successive peaks or troughs.
A very important principle of waves is the principle of superposition. As a wave passes, particles move about fixed positions in a predictable way. The distance of tte particle from the resting position is the displacement. When two waves pass each other, their displacements add and after the waves have passed, they are completely unaffected.
Waves transfer energy but do not involve the large scale movement of matter. Next time you see the moon reflected in water or the sound of lightning, remember you are seeing physics in action.
The Oxford Mathematical Institute entrance examination
I’ve recently been tutoring a student who is taking this examination. It is very interesting and requires true problem-solving skills, something that A-level examinations have not been very good at. The format is of ten multiple choice questions and four longer questions. It covers the entire A-level syllabus but requires a great deal of intelligence to be able to spot neat methods to solve them. It is required to be taken by anyone applying for a mathematics based degree at Oxford or Cambridge.
It tests the student’s attention to detail, subject knowledge and flexibility in thinking. It is required for any student taking a scientific or mathematical degree. Problem-solving is at the heart of mathematics.
Advice on using your graphing calculator
The International Baccalaureate mathematics examinations require you to be familiar with a graphical display calculator. Here, I discuss the use of the GDC in Standard and Higher Level mathematics.
Common calculators used in the IB Diploma Program are the Ti-nspire (non-CA models or with CAS disabled), the Ti-84+ and the Casio FX-9750G. The IBO organisation publishes a list of acceptable calculators here.
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 of some type. If you did the MYP programme, then you may have used a GDC before. But if you took the GCSE or IGCSE, then it will be new to you.
I lay out some tips and thoughts below for using your GDC.
- 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 great tutorials on Youtube. See the list of links at the end of the article. For more complex queries, you may need to download the manual if you don’t have a hard copy.
- Do 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 off your neighbour. Put your name on it so you don’t lose it.
- Remember that in papers 2 and 3 of the AA course and in all papers of the AI course, you are going to need your calculator to do 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.
- 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.
- All GDCs have a ZOOM function to zoom in and out of regions of interest on the graph you have plotted.
- 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.
- 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.
- 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.
- 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.
- Practise makes perfect. Don’t leave learning calculator skills to the last minute.
Improve your algebra skills for A-level
As an A-level mathematics tutor, I notice that a large number of students beginning their A-level studies have remarkably poor algebra skills. This hard core of students (usually boys) are unable to manipulate fractions, confidently rearrange formulae, work with negative numbers or perform the necessary basic skills to progress in the subject. They repeatedly ask the same questions and try to learn methods by rote because they don’t have the skills to do otherwise. Teaching them can be very frustrating because the same issues are revisited again and again and little progress can be made until they learn basic skills which should have been learned at GCSE.
I should add that this is a minority of students but nevertheless a significant minority. If you recognise yourself in this category, do yourself a big favour and learn to walk before you try to run. Mathematics is a hierarchical subject and progression is made on the basis of having mastered prior skills.
Mathematics and physics resources
Most of these resources are free to use. If a link does not work, please let me know. I use them on a regular basis.
There are plenty of syllabus-specific resources available if you know where to look. These resources are designed by teachers in their spare time and are in some cases designed around a specific syllabus. They make learning fun and less of a chore. A student can find the resource that suits their own learning style. I use most of these resources in my own teaching, online and face to face. Some have video solutions, some have worked written solutions and some have topic notes.
GCSE mathematics
www.bland.in has a great set of GCSE questions arranged by topic. They are written in the style of the Edexcel A syllabus but can be used for any syllabus. Higher and Foundation tiers. I use these a lot in my GCSE teaching.
www.m4ths.com is a set of questions arranged by topic. Some of the worksheets have answers but not all. These are very nice because you can really choose the precise topic that you want to revise.
www.mathsnetgcse.com is an interactive vocabulary list for GCSE. Point the cursor at a word and it shows you the meaning. It’s also great for A-level students who need to brush up on terminology.
www.desmos.com is a free web-based graphing website. It is easy to use and intuitive. Plot graphs of data for a science report, enter functions with variables, integrate or differentiate numerically and so on. It has all the standard scientific built in functions. I find it indispensable for my teaching.
www.math-aids.com has loads of mathematics worksheets, dynamically created, by topic. Math-Aids.Com is a free resource for teachers, parents, students, and home schoolers. The math worksheets are randomly and dynamically generated by our math worksheet generators. This allows you to make an unlimited number of printable math worksheets to your specifications instantly.
www.mathworksheetsland.com claims that it has the largest set of maths worksheets available on the internet. You can decide if this is true.
www.edexcel.com is the edexcel exam board home page for mathematics where you can download past papers, mark schemes and syllabi.
www.examsolutions.net is an amazing site which uses video as the main medium of communication. Instructional videos are arranged by topic. The creator of this website has designed an app for revising A-level maths by unit. The apps are reasonably priced at £1.99 each.
BBC Bitesize has tailored its web pages to the syllabus that you are studying. These are a set of revision exercises and quick tests. In recent years, the BBC has invested a lot of money in educational resources.
A-level mathematics
I find that a lot of A-level students go into their first year of A-level very poorly prepared for the hard work that is to come. They often lack the general background knowledge that is essential to make progression or to understand properly the new material that they are undertaking. I thoroughly recommend that they review their subject knowledge in GCSE. Try the higher level questions in the first resource I mention above (www.bland.in).
www.examsolutions.net is a great website that has AS and A-level instructional videos arranged by syllabus. All the A-level syllabuses are covered.
The graphing tool mentioned above is very useful as a learning tool. Of course, you can’t use a graphing calculator at A-level in the exam but it is great for helping to understand what is going on as you learn a topic. It is useful for example to visualise how many solutions there are to an equation in a given interval, how numerical solutions to equations work and so on.
There is an online edition of a mathematics A-level text book available here if you want to try a different text book to the one that you have been given. It is good to use a variety of resources to get used to the different ways that a question can be posed.
Solomon Press does a great range of worksheets, some of which you can download here.
If you can’t find all the exam papers and mark schemes that you are looking for (sometimes there are restrictions), try xtremepapers.com. I make no warranty as to copyright infringement and you access this website and download at your own risk.
How about broadening your knowledge and trying some mathematics competition papers? You can access some here.
If you want to read up on the history of mathematics, there is a great place to learn about all the famous faces such as Pythagoras, Euclid, Fibonacci, Gauss, Liebnitz, and so on. I really think that we need to know more about them.
A-level Physics
Hyperphysics is a logically organised site where you can find the topic that you are interested in revising or learning about by viewing linked ‘mind map’ diagrams. All pages are cross-referenced so that if there is a term that you are unfamiliar with, you can simply click on it. There are diagrams, and many pages allow you to enter your own values into calculations to see the result. Diagrams are very clear and a lot of thought has been put into designing the site.
A large collection of animations is available here using java as the medium of animation. Your computer will prompt you to install Java if you do not already have the current version. These are very instructional and cover all topics at A-level. There are also associated videos to view on each topic.
A fantastic set of AS and A2 level worksheets are here. There are also answers. The questions are structured and generally quite short. They are very useful for testing your comprehension of a topic.
A site dedicated to AQA A-level Physics A is here. All the six units are covered including powerpoints, past questions, short questions and answers.
This site (A-level Physics Tutor) is very in-depth. There are youtube videos, course notes, worked examples, derivations of laws, and a list of books available to download.
GCSE Physics
GCSE.com has a useful glossary of terms.
BBC Bitesize has tailored its web pages to the syllabus that you are studying. These are a set of revision exercises and quick tests.
Khan Academy has its own recommended science links for GCSE here.
General mathematics resources
This resource has a very wide range of maths worksheets for all levels.
resourceaholic.com: Mathematics and physics resourcesMars Perseverance Rover Explorer
I’ve been following the progress of the Mars Perseverance Explorer which landed on Mars on 18th February. Its primary mission is to search for signs of past life on this arid world. There have been some marvellous images beamed back since the spectacular landing in February. Watch the 4K video below which was cleverly stitched together from many high resolution photos.
The landing alone was a technological wonder. Here is video filmed during the descent showing how complex the landing procedure was.
https://mars.nasa.gov/mars2020/multimedia/videos/?v=461
Keep up to date on developments with Nasa’s website.
Impossible maths problem? Not so hard!
Here is an interesting article on a maths question many Australian High School students were complaining about. I think that the question is rather contrived and therefore I would have avoided setting it. It’s not totally impossible though. It relies on knowledge that the median is the middle line in the box plot so that the mean can be calculated. Now you also need to calculate the regression line gradient so that requires knowing a point on the graph. This point is therefore (22 – 0.525, 684/20) i.e. (mean of x, mean of y).
b is therefore 2.0864 and using a value of x = 19, we find that y = -10.6063 + 2.0864 x 19 = 29 chirps per minute (to the nearest whole number).
https://interestingengineering.com/high-school-cricket-math-question-stumps-the-internet
If you are looking at ways to improve your understanding of mathematics, I can recommend some resources.