Graphics Reference
In-Depth Information
Appendix 1
Properties of the real numbers
Algebraic properties of a field of numbers
-
chapter 1 onwards
If
a
,
b
, and
c
are numbers, then
a
b
is a number;
a
·
b
is a number;
a
b
b
a
;
a
·
b
b
·
a
;
a
(
b
c
)
(
a
b
)
c
;
a
· (
b
·
c
)
(
a
·
b
) ·
c
;
there is a number 0 such that
there is a number 1
0, such that
a
0
a
, for all
a
;
a
· 1
a
, for all
a
0;
for each
a
there is a number
a
for each
a
0, there is a number 1/
a
such that
a
(
a
)
0;
such that
a
· (1/
a
)
1;
a
(
b
) is usually written
a
b
;
a
· (1/
b
) is usually written
a
/
b
;
a
· (
b
c
)
a
·
b
a
·
c
.
From these algebraic properties it follows that
(
a
)
a
, 1/(1/
a
)
a
,
a
· 0
0, (
a
)· (
b
)
a
·
b
and also
a
·
b
0 only when
a
0or
b
0.
While these algebraic properties hold for the rational numbers, Q,
the real numbers, R, the complex numbers, C, and somefinitesystems
such as arithmetic modulo 2, Z
, they do not hold
universally for the number system on a pocket calculator. Let
a
b
where
a
is as large a number as your calculator will show, then if
c
is as
small a number as you can key in, the machine will calculate
a
, or modulo 3, Z
(
b
c
) as 0, and (
a
b
)
c
as
c
.
Order properties of a field of numbers
-
chapter 2 onwards
The trichotomy law
If
a
is a number, then
either a
0,
or a
is
positive
or
a
is positive, and only one of these is true. When
a
is
positive,
a
is said to be negative.
