A Beautiful Mind Incorrect Maths?Date: 04/22/2002 at 00:33:45 From: Kerry Cue Subject: A Beautiful Mind incorrect Maths? Dear Dr Maths, I am a journalist working for the Herald Sun, Melbourne, Australia. This is Australia's largest circulation newspaper. But I was a maths teacher for 10 years. Problem: In all publicity shots for "A Beautiful Mind" there is a picture of Russell Crowe looking through a window covered in Maths. Usually on his forehead is the statement 0 less than pi less than one You will find this statement (or Domain definition) on the official Web page, http://www.abeautifulmind.com/main.html . I have published in the paper that this is wrong. A reader disagrees. Can this statement ever be true? Kerry Cue Date: 04/22/2002 at 11:02:54 From: Doctor Ian Subject: Re: A Beautiful Mind incorrect Maths? Hi Kerry, Note that sometimes 'pi' is the ratio of the circumference of a circle to its diameter; and sometimes 'pi' is just a variable that stands for something else, usually something beginning with 'p': perimeter, probability, price, population, and so on. (Why would you do this? Mathematicians often use Greek letters for variables; also, if you have more than one quantity whose name begins with 'p', you can only use 'p' to represent one of them. Then you have to look around for other alternatives: the Greek pi, the Hebrew peh, the Russian equivalent, and so on.) Without knowing the context in which the equation 0 < pi < 1 - - appears, there is no way to say whether it's correct or not, for the same reason that you can't say whether 0 < x < 1 - - is correct or not, without knowing a lot more about the context in which the equation appears. (Note that just above the equation is another equation, in which pi is used as a subscript on another variable, which suggests that he's using pi as a variable, and not in its 'usual' sense.) For what it's worth, even when pi _does_ represent the ratio of the circumference of a particular circle to its diameter, that ratio can depend on the shape of the space in which the circle appears. The following item from the Dr. Math archives Why is Pi a Constant? http://mathforum.org/library/drmath/view/57828.html shows how that ratio for a particular circle can be equal to 2. If you move the circle farther towards the pole opposite the pole from which the radius is being measured, the circumference continues to shrink while the radius continues to grow. Thus, you can make the ratio as close to zero as you desire. So there are at least two ways in which the 'domain definition' 0 < pi < 1 - - could be true: (1) It defines the range over which some variable whose name is 'pi' is defined; or (2) it defines a range of latitudes on a sphere for which the ratio of circumference to diameter for a circle centered at the north pole is between 0 and 1. But there are certainly lots of other ways in which it could be true as well. The moral of the story is that it's very difficult to write an equation that can't be true under _some_ circumstances - even if those circumstances make the equation useless. The same is true in English. A sentence like President Bush is taller than President Lincoln. is 'obviously' false... unless it's uttered, for example, by someone who is collecting figurines of American Presidents, and noting a fact that happens to be true of two objects in his collection. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 04/22/2002 at 19:54:14 From: Kerry Cue Subject: A Beautiful Mind incorrect Maths? Dear Dr Maths, Many thanks for doing the hard work overnight while I slept soundly in my bed in Australia. The answer that pi can be used as a variable solves the problem. As if there weren't enough letters in the Greek alphabet. Why couldn't they keep pi sacred? Is nothing sacred anymore? I disagree about the circle and pole analogy. The radius of any line of latitude should be measured from the core of the earth. I hope you had a Happy Pi Day on 3.14. Thanks again. Kerry Cue Date: 04/22/2002 at 20:08:26 From: Doctor Ian Subject: Re: A Beautiful Mind incorrect Maths? Hi Kerry, I'm glad I was able to help. The point of the circle and pole analogy is that if you're a two-dimensional being on the surface of a sphere, there _is_ no 'core' from which to measure anything. There is only the surface; so it is only possible to measure along the surface. You look at it and think it's 'wrong' because you're living in three dimensions; but the people in the analogy are not. The trick is to imagine what might happen if you started trying to measure circles in a similarly curved space, in which you can only sense three dimensions out of four. If the idea still seems 'wrong' to you, may I suggest that you read the delightful little book _Flatland_? You can find it online at http://www.math.brown.edu/~banchoff/gc/Flatland/ It's partly math, and partly political satire. It will definitely stretch your mind a little. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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