Real Numbers Closed under DivisionDate: 09/24/2002 at 19:41:38 From: Dave Subject: Are real numbers closed under division? Real numbers are not closed under division because we can't divide by 0. Is this correct? Date: 09/24/2002 at 22:53:23 From: Doctor Mike Subject: Re: Are real numbers closed under division? Hi Dave, You have a good point. It's good to keep alert and thinking about what you hear in math classes. What you may have missed (or perhaps it was not actually said in so many words) is that the division X/0 is officially "not defined," so technically, division of real numbers IS closed with respect to division. And the reason division was carefully defined the way it was is so the non-closure would not happen. By the way, usually the main operations for real numbers are multiplication and addition. The operations of subtraction and division are by-products of the other two operations. The way this usually happens for subtraction is that -a is used to represent the additive inverse of a, such that a + (-a) = 0. The way this usually happens for division is that 1/a is used to represent the multiplicative inverse of a, such that a * (1/a) = 1. HOWEVER, and this is important, zero does not have a multiplicative inverse, so 1/0 isn't defined at all. Then subtraction is defined as a - b = a + (-b) and division is defined as a / b = a * (1/b) if b is not zero. - Doctor Mike, The Math Forum http://mathforum.org/dr.math/ |
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