Commutative Property and DivisionDate: 02/21/2002 at 11:12:51 From: Danielle Subject: Commutative Property *(Division)* I am trying to prove that the commutative property following certain rules will work in all circumstances for division. The rules are: - no using Zero - the first number has to stay the same - you have to use more than two numbers (ex. 26/4/2 = 26/2/4) Can you find a way to disprove this? Has anyone ever discovered this? Why does it work? Thanks- Danielle Date: 02/21/2002 at 11:33:25 From: Doctor Peterson Subject: Re: Commutative Property *(Division)* Hi, Danielle. Well, it's not a genuine commutative property, but I can tell you why it works. It's actually sort of interesting. What's happening is that repeated division is the same as a multiplication and a division: (26/2)/4 = (26/2) * (1/4) = 26/(2*4) Since the 2 and 4 (the two divisors) are really being multiplied together, you can commute them: (26/4)/2 = (26/4) * (1/2) = 26/(4*2) You can do the same thing to prove this in general: (a/b)/c = (a/b)(1/c) = a/(bc) = a/(cb) = (a/c)(1/b) = (a/c)/b A true commutative property of division would have to say that 4/2 = 2/4 which of course is not true. And, when you see your equation expressed with the parentheses in place, you can see that you are never really operating on the 2 and the 4; no property of an operation can be applied across a parenthesis. What you've done is sort of like a magic trick, hiding what's really going on. I would not use this as a "property," but would do exactly what I did above when I need to work with such a repeated division. I also prefer to use parentheses to make it clear what I mean, since repeated division is easy to misinterpret. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 02/21/2002 at 12:11:35 From: Danielle Subject: Commutative Property *(Division)* Has anyone discovered that before? So it can't it or can it be proved wrong with those rules? Thanks so much for the help - Danielle Date: 02/21/2002 at 12:26:48 From: Doctor Peterson Subject: Re: Commutative Property *(Division)* Hi, Danielle. It's been discovered many times before, I'm sure; it just isn't a "property" that really deserves a name, so no one pays much attention to it. But I showed you how to prove that it is always true for any a, b, and c, as long as b and c are non-zero. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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