Multiplying negative numbers

Jul 31, 2011

First of all have to accept following rules for combining numbers

1. a + b = b + a

2. ab = ba

3. a(b + c) = ab + ac

4. a + 0 = a

5. a0 = 0

6. a1 = a

7. a + -a = 0

***********

1.1 = 1 [6]

1(1 + -1) = 1.0 [7] = 0 [5]

but 1(1 + -1) = 1.1 + -1.1 [3] =1 -1 = 0 [7]

OK

*****

8a. -1.1 = -1

Now consider incorrect 8b. -1.-1 = = -1

-1.0 = 0 [5] (*)

-1 + 1 = 0 [7]

-1.(-1 + 1 ) combing above two results

-1.(-1 + 1) = -1.-1 + -1.1 [3] = -1 + -1 [8a, 8b] = -2 (contradiction with *)

***************

Some of the rules above maybe redundant in that some rules can be derived by combining others – not had time to check that.

So … defining -1.-1 = -1 leads to inconsistent results in calculations and that is a bad thing.

Of course need to check completeness. I think 13 rules/axioms are needed to deifne arithmetic.

what about the “rules” to combine terms? or are they “implicit” in the rules and at its base we can consider arithmetic to be symbol manipulation without regard to its “meaning”?

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