Laws of Arithmetic

Date: 02/12/99 at 11:37:56
From: Rose Zagaja
Subject: Laws of Arithmetic

I'm looking for information (definitions) on various laws of arithmetic 
- in particular, the distributive law, and also anything on the 
associative law, identity, and commutative equations.

Thanks.
Rose

Date: 02/22/99 at 01:02:16
From: Doctor Bonnie
Subject: Re: Laws of Arithmetic

It is really nice to hear a student ask this particular question. Math 
is just a matter of defining things, and then working with them 
according to your definitions. Following is the definition of a field, 
the real number system under regular addition and multiplication. This 
gives the definitions of all the "laws of arithmetic."

1) Take a set. Call it F (it can be numbers or other stuff).

2) Define on it "addition" according to these rules:
  a) if a and b are elements from F, then a+b is also an element in F
  b) if a, b, and c are elements of F then (a + b) + c = a + (b + c)
  c) if a and b are elements of F then a + b = b + a
  d) there is an element in F (call it z) so that for every element 
     a in F, a + z = a and z + a = a
  e) for each element a in F, there is an element b in F so that 
     a + b = z and b + a = z (where z is the element described in 2d)

3) Define on F "multiplication" according to these rules:
  a) if a and b are elements from F, then a*b is also an element in F
  b) if a, b, and c are elements of F then (a*b)*c = a*(b*c)
  c) if a and b are elements of F then a*b = b*a
  d) there is an element in F (call it n) so that for every element 
     a in F, a*n = a and n*a = a
  e) for each non-zero element a in F, there is an element b in F so 
     that a*b = n and b*a = n (where n is element described in 3d)

4) For all elements a, b and c in F, a*(b + c) = (a*b) + (a*c) and
   (b+c)*a = (b*a) + (c*a)

F is then called a field under this addition and multiplication.
  
For 2a and 3a, we say that F is closed with respect to addition and 
multiplication (respectively). For 2b and 3b, we say addition and 
multiplication are associative. For 2c and 3c addition and 
multiplication are commutative. The "z" in 2d is called the additive 
identity (and in the real numbers this is zero). The "n" in 3d is 
called the multiplicative identity (and in the real numbers this is 
one). The b in 2e is called the additive inverse of a (it is -a in the 
real numbers). The b in 3e is called the multiplicative inverse of a 
(1/a in the real numbers). In3e, 'non-zero' means not the z, as 
described in 2d. Number 4 says that multiplication distributes over 
addition (the distributive property).

I hope this answers your questions. I know it is quite detailed, but I 
think this is what you are looking for. You can find some less 
detailed information at the following pages in the Dr.Math archives:

 Associative, Distributive Properties
 http://mathforum.org/dr.math/problems/biondi.9.5.96.html   

 Basic Real Number Properties
 http://mathforum.org/dr.math/problems/tuesday7.31.97.html   

 Properties of Algebra
 http://mathforum.org/dr.math/problems/march.5.8.96.html   

Be sure to write back if you have any other questions, or if you need 
any clarification.

- Doctor Bonnie, The Math Forum
  http://mathforum.org/dr.math/