When you try to add two numbers of vastly different magnitudes, a and b, you'll have problems.īizarre effects of round off errors can be illustrated graphically. The problem isn't the magnitude of the numbers involved in the calculation, but rather their relative magnitudes. Now, the result is correct since at no time does the machine need to add two numbers whose magnitudes are vastly different. Note that this example can be used to illustrate that, to a calculator, addition is not associative. Now, when 10 ^ 15 is subtracted, the result is 0. When 10 ^ 15+7.2 is calculated, therefore, the result is exactly 10 ^ 15 (the 7 and the 2 are past the 13th digit). The TI-83, for instance, keeps 13 digits. The problem is that every digital calculator uses a finite number of digits to store numbers (thus all numbers that get stored are rational). Whenever a finite precision machine attempts to add numbers of vastly different magnitudes, accuracy can be lost. Example 1: Round Off Error and Limited Precision Most of these examples can be adapted to any calculator model, with minor variations. In a few cases, the examples also serve as a starting point for lively discussions about calculus concepts. Over the years, I’ve assembled examples whose main purpose is to develop in students a healthy skepticism for machine results. Many students who come into my AP Calculus class place far too much faith in calculator results, blindly accepting results without questioning whether they’re correct or even reasonable.
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