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Alien Encounters - Questions and Exercises for Act 1

Gavin Hitchcock
  1. Is Wallis' speech at the beginning a good model for teaching, apart from the old-fashioned style of expression? Can you re-express (or re-enact) it in a more modern style?
  2. Express in modern notation what Arnauld calls "a basic principle of multiplication of two factors, which is that the ratio of unity to the first factor is equal to the ratio of the second factor to the whole product."
  3. Which properties of negative numbers does Arnauld use in his argument? Express in your own words the "contradiction" he arrives at. What do you think of Leibniz' response? How would you answer Arnauld?
  4. Wallis gives a demonstration that "the ratio of a positive number to a negative number may be supposed greater than infinity." Give the general term f(n) of the sequence he uses, and sketch the graph: {(n,f(n))| n  an integer}. How would you answer him? Illustrate graphically Euler's response, and comment upon whether it satisfies you.
  5. Leibniz and Euler each find 1/2 as the sum of the series
    1-1+1-1+ ¼
    Discuss whether their respective arguments are persuasive (that is, psychologically convincing) and valid (that is, mathematically convincing).
  6. Jean-Charles Callet presented to Lagrange a paradoxical argument purporting to show, for any given fraction m/n,  0 < m < n, that the series in question 5 has sum m/n. See ref.9 for the details, and (before reading the answer of Lagrange and Poisson) decide what your own response is. Then, after working through the probability argument (for the case m=3, n=5) of Lagrange and Poisson, given in ref.9, show how to obtain the sum 2/3 for the case m=2, n=3.
    Are you satisfied that the paradox has been resolved?
  7. What do you think of Euler's argument that -1 must be greater than infinity? What makes him believe that his reasoning is more rigorous than Wallis'?
  8. Euler says he is faced with an "insoluble contradiction" at finding "two distinct series, each with sum -1." His series are:
    1+2+4+8+¼,         1+1+2+3+¼ .
    He says that he cannot possibly equate them. How would you answer him?
    Construct two distinct convergent series with the same sum. Do you think Euler would have been bothered by this?
  9. Euler says: "I do sometimes entertain doubts as to how freely we may assign values to variables .... possibly there exists some peculiar limitation of domain."
    What is the modern definition of convergent and divergent series? What do you know about the convergence of the power series
    a0+a1x+a2x2+a3x3+ ¼
    for various values of x?
  10. What is the modern status of divergent series as legitimate and useful players on the mathematical stage? (See refs. 12, 17, 24.)
  11. What made great mathematicians like Euler so bold (we could say rash) in manipulating series - especially geometric series, and so reluctant to recognise a distinction between convergent and divergent series? In view of the permanent and undoubted prestige of Euler as a mathematician, what lessons for teaching can we draw from his struggles with series?
  12. Try to extend the idea of logarithms to negative numbers; and to complex numbers. What is your reaction? Can you guess at the experience of the pioneers of the eighteenth century when they tried to do the same? (See Boyer, ref.9, pages 422, 428, 438 of the 1991 paperback edition.)
    What do you make of this argument (which, according to Boyer, "puzzled the best mathematicians of the earlier part of the eighteenth century"):
    log(-1)2=log(+1)2 Þ 2log(-1)=2log(+1)Þ log(-1)=log(+1)=0Þ 1=e0=-1.
    [In 1747, Euler wrote to d'Alembert and expounded correctly the status of logarithms of negative numbers; in particular, ln(-1)=pi.]


Gavin Hitchcock , "Alien Encounters - Questions and Exercises for Act 1," Convergence (May 2011)