Closure &
Density
Properties
Closure
Property

The term
"closure"
is used in many
diverse fields.
Generally,
these
definitions are
not compatible
with the
precise
definition used
in
mathematics.
Outside
the field of
mathematics,
closure can
mean many
different
things. For
example, it can
mean something
is enclosed
(such as a
chair is
enclosed in a
room), or a
crime has been
solved (we have
"closure").
Visual Closure
means that you
mentally fill
in gaps in the
incomplete
images you see.
These kinds of
concepts should
not be carried
over to
mathematics.
When the
term
"closure"
is used in
mathematics, it
applies
to sets and
mathematical
operations
. The sets can
include
ordinary
numbers,
vectors, to
matrices,
algebra, and
other elements.
The operations
can include any
operation
(addition,
multiplication,
square root,
etc.).
However .
. . here, our
concern is only
with the
closure
property as it
applies to real
numbers
.
The
Closure
Property
states
that when
you perform an
operation (such
as addition,
multiplication,
etc.) on any
two numbers in
a set, the
result of the
computation is
another number
in the same
set
.
As an
example,
consider the
set of all blue
squares,
highlighted on
a yellow
background,
below:
"Blue
Squares"
The
set of
blue squares is
closed
under
addition
because when
any blue
squares are
added, the
result is more
blue squares.
You have not
moved outside
the set of all
blue
squares.
However,
the
set of blue
squares is
NOT
closed under
multiplication
. If you
multiply some
blue squares by
½ the
result
sometimes
includes
½ a blue
square. The
½ blue
square does not
belong to the
set of all blue
squares.
Real
numbers are
closed with
respect to
addition and
multiplication
. Real numbers
are not closed
with respect to
division (a
real number
cannot be
divided by
0).
Example 1:
Adding
two real
numbers
produces
another real
number
The number
"21"
is a real
number.
Example 2:
Multiplying
two real
numbers
produces
another real
number
The number
"312"
is a real
number.
Why is
this important
to
know?
It is important
because
equations which
only involve
addition and
multiplication
have a solution
which is also a
real number 
you know that
in advance.
For example,
you know for
certain that
you can add the
costs of all
the items in a
shopping basket
and get a
"real
number"
answer. This is
due to the
property of
closure.
If this were
not the case,
you could never
be sure that
addition would
quite work.
Density
Property

The
Density
Property
states that
there is
always a
another real
number between
any two real
numbers
: there is
a limitless
supply of real
numbers.
This idea
is illustrated
by the number
lines shown
below .
No matter what
two numbers are
chosen, there
are always more
numbers in
between the
two.