Calculators and Irrational Numbers
Date: 05/02/2001 at 16:19:28 From: Kathy King Subject: Calculator calculation of irrationals I have explained and illustrated the idea of irrational and rational numbers to my freshman algebra students. We discussed how the decimal representation of a number is a clue to deciding about rational or irrational. My students want to conclude that the square root of 11 is rational because the decimal values returned by any calculator, when reentered and squared, equals 11. I didn't believe this until I tried it for many square roots. At 10, the decimal value returned by any calculator, when squared, equals 10, etc. How is this done? I want to be able to explain to my students that this is NOT really true, but they are having trouble believing me. Help! I need to explain again about perfect square roots and rational numbers and make it clear that decimal values for square roots should be ignored when determining rationality, but I need to be able to explain the calculator problem. Thank you for your expertise. Kathy King Mathematics teacher Tremont High School Tremont, IL
Date: 05/03/2001 at 13:43:14 From: Doctor Peterson Subject: Re: Calculator calculation of irrationals Hi, Kathy. This could lead to some useful discussion of the difference between calculators and math. You can start by pointing out how silly their claim is that the number they see on the calculator is the exact root. You mentioned that it works on ANY calculator. That implies that if you square the root you find on an 8-digit calculator, you get the original number back; and the same happens if you use a 32-digit calculator like the one on my computer. But they give different answers! How can they both be the right answer? You can get a clue to what's happening if you find the root on an 8-digit calculator, and rather than squaring it again on the same calculator, copy the number into the 32-digit calculator and square it there. You'll find that it doesn't really work out so well. What's happening is just that the errors the calculator makes in doing the square root and keeping only 8 digits (or 32) don't make any difference in the 8 (or 32) digits you see after you square it again. If you did my suggested experiment, you should find that the first 8 digits of the answer are right (or it may look like 10.999999..., which the calculator would round up); but eventually there will be an error. The first calculator's limited display just isn't big enough to show the error it introduced; and the second also has an error that it can't display. Let's be a bit more precise about this. Suppose you take the actual square root of 11, and introduce an error of about 0.00000005 by rounding to 8 digits. When you square it again, you are doing this: (sqrt(11) + 0.00000005)^2 which expands to 11 + 0.0000001 sqrt(11) + 0.000000000000001 or 11 + 0.00000033 + 0.000000000000001 The last term is invisible on the calculator; the middle term seems as if it should be visible. But I've left out an important fact that can be fun to explore: the calculator really works with a little more precision internally than what it displays, so it didn't really round the answer to only 8 digits. If it keeps an extra digit internally, then the error after squaring will be just past the end of the display, as you've seen. I'm curious what you meant when you said, "the decimal representation of a number is a clue to deciding about rational or irrational." I hope that your position in the discussion was that it is NOT a clue. Knowing the first 2 billion digits of a number tells me nothing about whether the decimal may terminate at the next digit and be rational, or continue forever as an irrational number. Irrationality is meaningful only with reference to an exact number, not an approximation, because rationals and irrationals are intermingled densely on the number line: between any two rationals there is an irrational, and between any two irrationals there is a rational. And that's why calculators are irrelevant to any discussion of irrationality, and why students who have come to think that numbers are what they see on a calculator can have a hard time believing in irrationals. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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