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49 results in How to Think Like a Mathematician

Page 2 of 3

  • 19 - A study of Pythagoras' Theorem

  • from III - Definitions, theorems and proofs
  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp 126-136

  • Summary

    Reason is immortal, all else mortal.

    Pythagoras, Diogenes Laertius (Lives of Eminent Philosophers)

    Pythagoras' Theorem is probably the best-known mathematical theorem. Even most non-mathematicians have some vague idea that it involves triangles and squaring something known as the hypotenuse.

    Because the ideas in the previous chapters on ‘How to read a theorem’ and ‘How to read a proof’ are so important we will apply them to this famous theorem to see them in action. So in this chapter we will pull apart the theorem and its proof, we'll see a converse for it and also a generalization.

    Statement of Pythagoras' Theorem

    As you are a budding mathematician, you probably have a better idea than a non-mathematician of what the statement is, but here it is again.

    Theorem 19.1

    For a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other sides.

    Exercise 19.2

    Use the ideas from Chapter 16, How to read a theorem, to analyse the theorem. Compare your analysis with the one given below.

    Study of the theorem

    We now analyse the theorem as though we were meeting it for the first time. Obviously we would check what all the words mean, for example, what is a hypotenuse? This is fairly obvious, but what about the other techniques in Chapter 16? We shall apply them now.

  • 3 - Writing mathematics I

  • from I - Study skills for mathematicians
  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp 21-34

  • Summary

    We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover up all the tracks, to not worry about the blind alleys or describe how you had the wrong idea first, and so on.

    Richard Feynman, Nobel Lecture, 1966

    As a lecturer my toughest initial task in turning enthusiastic students into able mathematicians is to force them (yes, force them) to write mathematics correctly. Their first submitted assessments tend to be incomprehensible collections of symbols, with no sentences or punctuation. ‘What's the point of writing sentences?’, they ask, ‘I've got the correct answer. There it is – see, underlined – at the bottom of the page.’ I can sympathize but in mathematics we have to get to the right answer in a rigorous way and we have to be able to show to others that our method is rigorous.

    A common response when I indicate a nonsensical statement in a student's work is ‘But you are a lecturer, you know what I meant.’ I have sympathy with this view too, but there are two problems with it.

  • (i) If the reader has to use their intelligence to work out what was intended, then the student is getting marks because of the reader's intelligence, not their own intelligence.

  • (ii) This second point is perhaps more important for students. Sorting through a jumble of symbols and half-baked poorly expressed ideas is likely to frustrate and annoy any assessor – not a good recipe for obtaining good marks.

  • 26 - Techniques of proof V: Contrapositive method

  • from IV - Techniques of proof
  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp 180-184

  • Summary

    If it isn't hurting, it isn't working.

    Prime Minister John Major's comment after being told that his government's policies were hurting the country

    The contrapositive method is very confusing to grasp initially – however, one student told me that she liked using it as it made her feel that she had done something clever. I think this feeling comes from the fact that the method is very indirect.

    We saw in Chapter 8 that the statement ‘A ⇒ B’ is equivalent to ‘not B ⇒ not A’. The contrapositive method is the use of this equivalence. The indirectness and feeling of being clever is that in using it to prove ‘A ⇒ B’ we start with not B (and proceed to show that not A is true).

    Since this idea causes a lot of confusion for beginners we shall start with some revision of Chapter 8 and show, via a different proof to the one in that chapter, that a statement and its contrapositive are equivalent. We shall then see the contrapositive in action.

    Revision of the contrapositive

    Definition 26.1

    The contrapositive of the statement ‘A ⇒ B’ is

    ‘not(B) ⇒ not(A)’:

    Examples 26.2

  • (i) ‘I am Winston Churchill implies I am English’ has contrapositive ‘I am not English implies I am not Winston Churchill.’

  • (ii) ‘If I am Jane, then I am a woman’ has contrapositive ‘If I am not a woman, then I am not Jane.’

  • […]

  • 32 - Putting it all together

  • from VI - Closing remarks
  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp 243-247

  • Summary

    Nothing has such power to broaden the mind as the ability to investigate systematically and truly all that comes under thy observation in life.

    Marcus Aurelius, Meditations

    In the preceding five parts of this book there is a lot to digest and each part focuses on a different way of thinking like a mathematician. Putting them all together and applying them effectively takes time and practice. It is not easy to think like a mathematician; there is no set of rules to follow, though many techniques have been given. Doing mathematics is certainly more of an art than a science.

    Therefore, in this chapter I will pick out some interesting points from earlier chapters and see how we can integrate the methods.

    Getting started

    The biggest hurdle with any mathematics problem is getting started. I set weekly exercises for my students and when a student says ‘I am stuck on question X’ my standard reply is ‘Well, what have you done so far?’ Their reply is often ‘Nothing. I can't get started. I don't even know where to start.’

    The advice in many Study Aid type books is to look for previous worked examples. However, my students don't need to have this pointed out to them, they have probably already looked in their notes for a similar problem. And in fact this study book advice may be the cause of their difficulties.

  • 21 - Some common mistakes

  • from IV - Techniques of proof
  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp 149-154

  • Summary

    To err is human. To really mess up takes a computer.

    Anon.

    The direct method of proof is probably the most basic and will be used in the other methods. Despite this there are a couple of pitfalls that are easy to fall into. Two of the most common are assuming what had to be proved and incorrect use of equivalence. We shall investigate these in this chapter. And since it would be nice to gather common mistakes together in one handy chapter rather than having them distributed throughout the book some other mistakes are included. Also brought in is an explanation of why we can't divide by zero – a mistake you probably already know about but may not have been given a reason why. I have seen all these errors made and, like most mathematicians, have made them myself.

    Don't assume what had to be proved

    Probably the most common mistake in proofs is assuming what had to be proved. Suppose that we had to prove statement P. If we assume it is true, then it is not surprising that we can deduce it is true; P ⇒ P would seem to be very obviously true. Another error in this vein is that P is assumed to be true and this is used to deduce something that is true and so it is concluded that P is true. This is of course an incorrect argument.

  • 2 - Reading mathematics

  • from I - Study skills for mathematicians
  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp 14-20

  • Summary

    Don't believe everything you read.

    Anon

    Obviously you can read and probably you have been taught reading skills for academic purposes as part of a study skills course. Unfortunately, mathematics has some special subtleties which often get missed in classes or books on how to study. For example, speed reading is recommended as a valuable tool for learning in many subjects. In mathematics, however, this is not a good method. Mathematics is rarely overwritten; there are few superfluous adjectives, every word and symbol is important and their omission would render the material incomprehensible or incorrect.

    The hints and tips here, which include a systematic method for breaking down reading into digestible pieces, are practical suggestions, not a rigid list of instructions. The main points are the following:

  • You should be flexible in your reading habits – read many different treatments of a subject.

  • Reading should be a dynamic process – you should be an active, not passive, reader, working with a pen and paper at hand, checking the text and verifying what the author asserts is true.

    • The last point is where thinking mathematically diverges from thinking in many other subjects, such as history and sociology. You really do need to be following the details as you go along – check them. In history (assuming you don't have a time machine) you can't check that Caesar invaded Britain in 55 BC, you can only check what other people have claimed he did.

  • 8 - Finer points concerning implications

  • from II - How to think logically
  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp 69-74

  • Summary

    He's suffering from Politician's Logic. Something must be done, this is something, therefore we must do it.

    Antony Jay and Jonathan Lynn, Power to the People, episode of Yes, Prime Minister

    The English language is full of ambiguities. The humorous quote above exploits this and works because the word ‘something’ has two different meanings in the sentence. With mathematical language we aim to remove such ambiguities. In this chapter we will first look at a particular problem that can occur in everyday usage with the use of the if/then structure. We shall see how this example leads us to consider what are called the inverse statement and the contrapositive statement. This shows that although using everyday examples can be very illuminating we do have to be careful as the English language can play tricks on us.

    The inverse: a common mistake

    The most common initial mistake that people make in logic arises from an everyday usage we learn at a young age. For example, an adult says to a child

    ‘If you don't tidy your room, then you won't get ice-cream.’

    Our instinct, like the child and parent, is to interpret this as a contract: if the child tidies their room, then they get ice-cream. In other words

    ‘If you tidy your room, then you will get ice-cream.’

    Yet the original statement does not say that. It only says what will happen if the child does not tidy their room.

  • 27 - Divisors

  • from V - Mathematics that all good mathematicians need
  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp 187-195

  • Summary

    You know my methods. Apply them.

    Sherlock Holmes in Arthur Conan Doyle, The Hound of the Baskervilles

    The set of integer numbers is among the simplest objects in mathematics. These numbers have a whole theory devoted to them, called, not surprisingly, number theory. In this chapter we will consider a small part of this theory. The aim is to provide you with mathematics that you will often need but, more importantly, we will apply the earlier methods to see how they illuminate the material.

    Divisibility

    What basic properties do numbers have? What can I do with them? Well, I can add and subtract then, multiply and divide, take to the power just to name a few. The best set of numbers to begin playing with is the integers, denoted ℤ. Adding any two integers produces an integer. Subtracting any integer from another produces an integer. Multiplying any two integers produces an integer. Dividing any integer by another produces an integer… sometimes. This means we have found an interesting property concerning division of integers. Being mathematicians we produce a definition to isolate this idea and start to investigate it.

  • 6 - Making a statement

  • from II - How to think logically
  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp 53-62

  • Summary

    When dealing with people, let us remember we are not dealing with creatures of logic. We are dealing with creatures of emotion, creatures bristling with prejudices and motivated by pride and vanity.

    Dale Carnegie, How to Win Friends and Influence People, 1936

    When I tell people that I am a mathematician it is often not very long before they bring up the subject of how logical mathematics is. Sometimes they consider this a downside to the subject as it makes it so much harder. After all, with logic you have to be right whereas in other areas of life opinions matter more. And it's easy to have an opinion – even if you can't back it up with evidence.

    Logic is in essence quite simple despite its reputation for difficulty. A small number of simple rules exist. We need only to apply these and reduce complicated statements with them to achieve success.

    In the next few chapters we shall concentrate on the logic used by mathematicians in their day-to-day work rather than on deep conundrums and paradoxes or on technical material such as predicates, compound statements and so on. I will, however, explain truth tables, which are rarely used by mathematicians in their everyday work, as they do provide a lot of clarity for beginners.

    Mathematics is the business of proving mathematical statements to be true or false. So, first let's look at statements.

  • 13 - Summary of logic

  • from II - How to think logically
  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp 96-96

  • Summary

    A summary of ways of writing statement A implies statement B

  • A implies B.

  • A is true implies B is true.

  • A ⇒ B.

  • If A, then B.

  • If A, B.

  • B if A.

  • A only if B.

  • A is sufficient for B.

  • B is necessary for A.

  • Either A is false or B is true.

    • Ways of writing statement A is equivalent to statement B

  • A is equivalent to B.

  • A ⇔ B.

  • A if and only if B.

  • A is necessary and sufficient for B.

  • 25 - More sophisticated induction techniques

  • from IV - Techniques of proof
  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp 175-179

  • Summary

    Simplicity is the ultimate sophistication.

    Leonardo da Vinci

    In this chapter we investigate more sophisticated versions of induction. There are three variants we shall be most interested in.

  • (i) We use a different initial case. Rather than show that A(1) is true we show, for instance, A(7) or A(15) is true. Thus A(n) is true for all n ≥ 7 or all n ≥ 15 respectively.

  • (ii) We change the inductive step to ‘A(k − 1) and A(k) imply A(k + 1).’ This requires us to have as initial case that A(1) and A(2) are true.

  • (iii) We change the inductive step to ‘A(j) true for all 1 ≤ j ≤ k implies A(k + 1) true.’ We use initial case A(1) true or some other initial case like (i) above.

    • All three can be referred to as the Principle of Mathematical Induction and, in addition, the latter two are sometimes called the Principle of Strong Mathematical Induction.

    First variant

    We do not need to start with n = 1 as the initial case. For example, for statements A(n) the first few cases may be false. If we can show

  • (i) A(r) is true for some r ∈ ℕ, and

  • (ii) A(k) ⇒ A(k + 1) for all k ≥ r,

    • then the statement is true for all A(n) with n ≥ r. Observe that our main change to induction is really the initial step – the change of the range of values in the inductive step is minimal.

  • Preface

  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp ix-xii

  • Summary

    Question: How many months have 28 days?

    Mathematician's answer: All of them.

    The power of mathematics

    Mathematics is the most powerful tool we have. It controls our world. We can use it to put men on the moon. We use it to calculate how much insulin a diabetic should take. It is hard to get right.

    And yet. And yet … And yet people who use or like mathematics are considered geeks or nerds. And yet mathematics is considered useless by most people – throughout history children at school have whined ‘When am I ever going to use this?’

    Why would anyone want to become a mathematician? As mentioned earlier mathematics is a very powerful tool. Jobs that use mathematics are often well-paid and people do tend to be impressed. There are a number of responses from non-mathematicians when meeting a mathematician, the most common being ‘I hated maths at school. I wasn't any good at it’, but another common response is ‘You must be really clever.’

    The concept

    The aim of this book is to divulge the secrets of how a mathematician actually thinks. As I went through my mathematical career, there were many instances when I thought, ‘I wish someone had told me that earlier.’ This is a collection of such advice. Well, I hope it is more than such a collection. I wish to present an attitude – a way of thinking and doing mathematics that works – not just a collection of techniques (which I will present as well!)

  • 28 - The Euclidean Algorithm

  • from V - Mathematics that all good mathematicians need
  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp 196-207

  • Summary

    An algorithm must be seen to be believed.

    Donald Knuth, The Art of Computer Programming, Vol. 1, 1999

    The Euclidean Algorithm is a very powerful device. We will apply it to three problems:

  • Finding the gcd of two numbers.

  • Finding integer solutions to equations like 32x + 17y = 45, i.e. those of the form ax + by = c.

  • Finding additional hypotheses so that when n|ab we can say something about whether or not n divides a or b.

    • We will show how the proofs are created and how they are polished for the final version. Hopefully, it will show that there is a huge difference between the way a proof is created and how it is presented.

    The Division Lemma

    The obvious way to find the gcd of two integers is to factorize them – we know we can do this by the Fundamental Theorem of Arithmetic (Theorem 25.5) – and find what common prime factors they have. Multiplying these together gives the gcd. For example, 440 = 23 × 5 × 11 and 1300 = 22 × 52 × 13. The common factors are 22 and 5 so the gcd is 22 × 5 = 20.

    The problem is that factorization is hard for large numbers. Instead of this brute force method we shall give a more useful method. We start with a lemma.

  • 7 - Implications

  • from II - How to think logically
  • Kevin Houston, University of Leeds
  • Book:
    How to Think Like a Mathematician
    Published online:
    05 June 2012
    Print publication:
    12 February 2009, pp 63-68

  • Summary

    Mathematics consists of propositions of the form: P implies Q, but you never ask whether P is true.

    Bertrand Russell

    Russell's quote above is extremely incisive. Modern mathematics is indeed made up of statements of the form statement P implies statement Q. That is, we have ‘If statement P is true, then statement Q is true also.’ Usually, however, this structure is hidden, mainly to make mathematics more comprehensible – it would be hard to read if we always wrote it that way.

    The second part of Russell's quote is also true but a lot more subtle. One could argue that the statement ‘The Moon is made of cheese implies the Moon is a tasty snack’ is true because if the Moon was cheese, then it would be tasty. The point is that the statement makes sense and is true yet it has nothing to say on whether the Moon really is made of cheese or whether it really is a tasty snack. All it says is that if it is cheesy, then it is tasty. It is worth bearing this example in mind as we proceed.

    Instead of Russell's P and Q we will, in general, use A and B to denote our statements.

    ‘If …, then …’ statements

    Most mathematical statements are of the form

    ‘If statement A is true, then statement B is true.’

    They may be heavily disguised but when you break them down, that is what you will find. This type of statement is called an implication.

Page 2 of 3