How Can the Product of Two Radicals Be Finite?Date: 06/27/2006 at 15:54:47 From: Keleigh Subject: (no subject) How is it possible that sqrt(5) * sqrt(20) equals exactly 10? sqrt(5) equals an infinite number (~2.236067977...) sqrt(20) equals an infinite number (~4.472135955...) sqrt(5) * sqrt(20) equals 10 I can't comprehend this. Is it possible that two infinite numbers eventually have the correct combination to make 10? Or is this just a man-made decision that the answer is approximately 10? Please help! Date: 06/28/2006 at 09:29:36 From: Doctor Ian Subject: Re: (no subject) Hi Keleigh, The first thing to do is stop using the word "infinite" to describe what's going on here. That by itself might go a long way towards clearing up your confusion. A number itself, e.g., ___ \/ 2 isn't infinite at all--it's somewhere between 1.4 and 1.5, because 1.4^2 = 1.96 ___ (\/ 2 )^2 = 2 1.5^2 = 2.25 And it's somewhere between 1.41 and 1.42, because 1.41^2 = 1.9881 ___ (\/ 2 )^2 = 2 1.42^2 = 2.0164 And we can keep narrowing this down as much as we want, limited only by our patience. But clearly, a number between 1.41 and 1.42 can't be said to be "infinite" in any meaningful sense of that word, because we can hardly expect to find an infinite number bounded by two finite numbers. What we _can_ say is this: The number ___ \/ 2 can't be expressed exactly as a fraction whose denominator is a power of 10. That is, there are no integers k and n such that ___ k \/ 2 = ---- 10^n As I said, we can generate a series of approximations to this, 14 141 1414 14142 141421 1414213 --, ---, ----, -----, ------, -------, ... 10 100 1000 10000 100000 1000000 but none of them will give us the exact value. Does this mean the number doesn't exist? Let's construct an isosceles right triangle, where the legs have length 1: A . . . . . . ? 1 . . . . . . . . C. .. . . . . . B 1 What is the length of the hypotenuse? We can agree that there is some number that describes the length, right? The Pythagorean Theorem tells us that if we call that number x, it will be true that x^2 = 1^2 + 1^2 x^2 = 2 So there is _some_ number that, when multiplied by itself, gives a product of 2. The question is, what do we call it? Its exact name is just ___ x = \/ 2 When we try to translate that into a different representation, which we find convenient for dealing with numbers in lots of situations, we find that it doesn't work very well. But this is a mismatch between the number and the representation. It says nothing about the number itself. In the cases where they "combine perfectly", note that it always reduces to a root being multiplied by itself, e.g., ___ ____ \/ 5 * \/ 20 ___ _______ = \/ 5 * \/ 5 * 4 ___ ___ ___ = \/ 5 * \/ 5 * \/ 4 ___ ___ = \/ 5 * \/ 5 * 2 Now, at this point, if we realize that ___ \/ 5 is just a name we give to the number--whatever it is!--that, when multiplied by itself, gives a product of 5 then there should be no mystery that we end up with an integer for a result: ___ ___ = (\/ 5 * \/ 5 ) * 2 = 5 * 2 = 10 So there's really no "combination" going on. It's more like "rejoining". Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 06/28/2006 at 21:13:53 From: Keleigh Subject: Thank you ((no subject)) Thank you for taking the time to answer my question so thoroughly. I think I need to eliminate "infinite" from my vocabulary so I don't get confused again! You provided several explanations and scenarios and you MADE it make sense. I really appreciate it and it definitely helped! You're great :) |
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