Thread: What is zero
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matt grime
Feb5-04, 03:06 PM
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I think I'll start a new thread for this, but to answer some specific questions.

1. 'omthe' is a typo that should read 'other'

2. No 0 is defined in C already, you just can't divide by it - check the defintion of a field, F is a field if it is an abelian group, with identity 0 under operation +, and F, omitting 0, is also an abelian group under the operation *.

3. In some structure ( ring usually) we say x divides y (is a divisor of) if there some other z with x*z=y. So in the ring of integers mod 8, 2 divides 0 in a non-trivial way (obviously x.0=0 is a trivial statement), that is what we mean by zero-divisors (the non-trivial is implicit).

4. When I say it is not a good place to do arithmetic, I mean things like finding roots of ax+b=0 is not as easy as it ought to be, because usually we would say x= -b/a. However, when non-trivial zero divisors exist this isn't true, as we can no longer divide by a. I mean the multiplicative inverse for 2 does not exist in mod 8 arithmetic.

To convince yourself of what's going on, lets do mod 3 arithemetic, what is 1/2? It is by definition the thing that when multiplied by two gives 1, agreed? So we are seeking a y such that 2y=1 (mod 3). By inspection 2*2=4=1 mod 3, so 1/2 = 2! Really we ought not to write 1/2 as it is too suggestive, but instead write 2^{-1}

In cases where x*y=0 for non-zer x and y we cannot say 0/x =y and vice versa - or at least whilst you may write it, it is not valid as a mathematical statement. To see why, consider mod 16 arithmetic - 4*4=0 and 4*8=0, so you cannot define 0/4 - there are two possibitlities.

LOok out for a new posting.