**
MATH 240: Properties of Real Numbers
**

This is a list of some of the properties of the set of real numbers
that we need in order to work with vectors and matrices.
Actually, we can work with matrices whose entries come from any set
that satisfies these properties,
such as the set of all rational numbers
or the set of all complex numbers.

*Closure:*
For all real numbers a,b,
the sum a + b and the product a ^{.} b
are real numbers.

*Associative laws:*
For all real numbers a,b,c,
a + (b + c) = (a + b) + c and
a ^{.} (b ^{.} c) = (a ^{.} b) ^{.} c.

*Commutative laws:*
For all real numbers a,b,

a + b = b + a and a ^{.} b = b ^{.} a.

*Distributive laws:*
For all real numbers a,b,c,
a ^{.} (b + c) = a ^{.} b + a ^{.} c and
(a + b) ^{.} c = a ^{.} c + b ^{.} c.

*Identity elements:*
There are real numbers 0 and 1
such that for all real numbers a,

a + 0 = a and 0 + a = a, and

a ^{.} 1 = a and 1 ^{.} a = a.

*Inverse elements:*
For each real number a, the equations
a + x = 0 and x + a = 0

have a solution x in the set of real numbers, called the
*additive inverse*
of a, denoted by -a.

For each nonzero real number a, the equations
a ^{.} x = 1 and x ^{.} a = 1

have a solution x in the set of real numbers, called the
*multiplicative inverse*
of a, denoted by a^{-1}.

Here are some additional properties of real numbers a,b,c,
which can be proved from the properties listed above.

- If a + c = b + c, then a = b.
- If a
^{.} c = b ^{.} c and c is nonzero, then a = b.
- a
^{.} 0 = 0
- -(-a) = a
- (-a)
^{.} (-b) = a ^{.} b