Understanding Positive and Negative Numbers: Proving Properties and Applications

[C0nfused]

Hi everybody,
How do we define positive and negative numbers? Also, how do we prove that (-a)(-b)=ab and also that if a<b then ac<bc for c>0 and ac>bc for c<0 ?
Thanks

 

[HallsofIvy]

What level explanation do you want?

One way to define the integers is to define an equivalence relation on the set of pairs of natural numbers: (a,b)~ (c,d) if and only if a+ d= b+ c. It's easy to show that that is an equivalence relation and so separates all such pairs in "equivalence classes". The set of integers IS the set of equivalence classes (with appropriately defined operations). There is exactly one equivalence class that consists of all pairs (a,a): that is, (a,b) with a= b. That turns out to be the additive identity and we call it "0". You can show that there is a one-to-one correspondence between equivalence classes [(a,b)] such that a> b and we associate that with the natural number n where a= b+n (all pairs in that class having the same n). We can show that [(a,b)]+ [(b,a)]= [(a,a)]= 0. If a> b so that [(a,b)] is associated with n, we call [(b,a)] "-n". The set of negative numbers.

The proof of the properties you mention are more tedious than anything else.

Here's a link to a paper I wrote on basic number systems:
http://academic.gallaudet.edu/courses/MAT/MAT000Ivew.nsf/ID/918f9bc4dda7eb1c8525688700561c74/$file/NUMBERS.pdf

 

[C0nfused]

Thanks for your answer. I am not very familiar yet with these things, but I will check your paper and your answer and if i have any problems I will post them. Impressive work by the way!