Identity and Inverse Properties for Zero
Date: 01/07/2004 at 10:51:00 From: Sean Subject: Properties of identity and inverse not true for subtraction If we subtract 0 from a number and get the same number, doesn't that make 0 an identity for subtraction? Also, can't a number be its own inverse for subtraction? I have a text book which doesn't explain this--it simply gives the answer. I feel subtraction should have the identity properties in the case of all number systems which include 0.
Date: 01/07/2004 at 12:15:19 From: Doctor Peterson Subject: Re: Properties of identity and inverse not true for subtraction Hi, Sean. If your text defines these properties carefully, the definitions should be something like this: an element e is called an identity for the operation o if e o x = x and x o e = x for all x in the set an element y is called the inverse of x if x o y = e and y o x = e Note the two-sidedness of the definitions. Let's apply this definition of the identity to addition: If 0 is an identity element, then 0 + x = x and x + 0 = x for all x; this is clearly true Now try applying the definition of the identity to subtraction: If 0 is an identity element, then 0 - x = x and x - 0 = x for all x; the first is false Since there is no true (double-sided) identity, we can't define an inverse. The lack of associativity also makes it very difficult to apply the identity in the case of subtraction. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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