Practical Applications of Negative NumbersDate: 10/13/2002 at 01:07:55 From: Michael Weinberg Subject: Practical application of negative numbers I am preparing a unit on operations with negative numbers for a class of very bright and accelerated fifth graders. I would like them to have some of the theory and background on negative numbers other than the obvious temperature and checkbook examples. What is the history of negatives? Why were they invented, considering that you can't have -2 of a given object? Can they be used to count anything? What have they been used for? I would like to go deep, so any guidance you have would be most appreciated. Thanks. Date: 10/13/2002 at 23:14:59 From: Doctor Peterson Subject: Re: Practical application of negative numbers Hi, Michael. This site, listed in our FAQ under Math History, should be your first source for questions on history: MacTutor Math History Archive http://www-history.mcs.st-and.ac.uk/ In particular, you will find some relevant material under the history of zero: http://www-history.mcs.st-and.ac.uk/HistTopics/Zero.html Check the link on Brahmagupta for some further details on the earliest use. The Chinese also used negative numbers very early, writing them in black and positive numbers in red. Negative numbers were not taken seriously, though, until the time of Cardan and Stifel, whom you can look up there; and it was not until some time later that negative numbers were given "equal rights" with positives, so that one did not need to know ahead of time whether a value was positive or negative. We have some discussion of this topic here (check out the links): Negative Number History http://mathforum.org/library/drmath/view/52593.html To answer your specific questions, I would say that negative numbers were invented not to count anything (unless you think of "counting" a debt, which is how Brahmagupta described the concept), but in order to make it much easier to work with equations. The best example I can think of, however, is beyond your students, unless you just show them the problems without going into the solutions. This is the solution of quadratic equations. Until the use of negative numbers, each kind of equation had to be treated separately: x^2 + ax = b x^2 - ax = b and so on. Once it was found that negative and positive numbers could be handled in the same way, the same methods could be used for all quadratic equations, since all could be expressed in the same general form by allowing variables to be have either sign: x^2 + ax + b = 0 (Note that if a and b are positive, this has no positive solutions, so they would not even have considered it.) In my mind the most significant use of negative numbers is in coordinate systems. We can locate points going both left and right, up and down from an origin, which would be impossible using only positive numbers. We would have to find an origin for which all points of interest were on the same side. So allowing negative numbers frees us up to describe things that happen anywhere in space, such as orbits of planets or graphs of equations. (Similarly, by allowing negative temperatures, we don't have to use a scale that starts at the lowest temperature we can observe; that's how the Fahrenheit scale originated, as a way to avoid negative numbers.) So really the number line is the central concept in working with negative numbers. Without them, we can't name all the points on the line, but only "half" of them. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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