Irrational NumbersDate: 08/13/97 at 22:38:51 From: Julian Potvin-Bernal Subject: Irrational Numbers It's Julian's Dad typing this in, but it's truly Julian's question. First, I'll tell you how we got to the question... I had my two kids working out 1/3 of 100. Julian thought of using long division, and they ended up with 33.333... which they recognized as an irrational number. (We've been having fun with that stuff before - I've described irrational numbers as being like peanut butter stuck in between all the rational numbers.) So Julian asks: "How can three irrational numbers added together give you a rational number?" Hmmm... I was stuck for an explanation. My thinking is that an irrational number can never be exactly defined. (That's where I'm probably imprecise in my thinking...?) So how can two or more inexactly defined numbers be combined logically to give an exact number? Julian's sleeping in bed now, but he asked me to send this question to you. Thanks for your great site! (Curiosity question...Are "you" many generous people out there, or one amazing individual who answers peoples' questions? Thanks either way.) Date: 08/14/97 at 08:47:37 From: Doctor Jerry Subject: Re: Irrational Numbers Hi Julian, It has been proved that pi is not rational and is, by definition, irrational. It has been proved (by Euclid, I think) that sqrt(2) is also irrational. The numbers pi+sqrt(2), 3*sqrt(2)-pi, and 10-4*sqrt(2) are all irrational (this can be proved but at this point you may wish to take my word for it). Note that their sum is 10, a nice rational number. It's true that pi can't be written out exactly, using a terminating decimal, but most of us agree that pi is well-defined. If you can agree, for example, that the circumference of a circle with radius 1 meter is a well-defined number, then you have accepted the existence of 2*pi. We are many people, not one. One person would have to be truly amazing to handle the numbers of questions we get. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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