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### Laws of Arithmetic

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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
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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

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/
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