A. Definition of real numbers
1. definition
2. decimal representation
a. terminating decimal
b. repeating decimal
c. change fractional form to decimal form
d. change terminating decimal form to fractional form
3. real number line
a. unit distance
b. number line
c. dense set
4. classifications within
the set of real numbers
B. Identity and inverse properties
1. closure for addition
and multiplication
2. associative for addition
and multiplication
3. commutative for addition
and multiplication
4. distributive for multiplication
over addition
5. identity for addition;
definition of zero
6. identity for
multiplication; definition of one
7. inverse property for
addition; definition of opposite
8. inverse property for
multiplication; definition of reciprocal
C. Algebraic structure
1. group
2. commutative group
3. Abelian group
4. field
5.6 Essential Ideas
Even though the field properties for
real numbers are stated in terms of addition and multiplication,
they may be applied to any set with any two operations.
In particular, we state these properties for a general
set and operation:
