Real numbers
have unique
properties
which make them
particularly useful
in everyday life.
First,
Real
numbers are an
ordered set of
numbers. This
means real numbers
are sequential. The
numerical value of
every real number
fits between the
numerical values two
other real
numbers.
Everyone is
familiar with this
idea since all
measurements
(weight, the
purchasing power of
money, the speed of
a car, etc.) depend
upon the fact that
some numbers have a
higher value than
other numbers. Ten
is greater than
five, and five is
greater than four .
. . and so on. This
is an important math
property.
Second,
we
never run out of
real numbers.
The quantity of real
numbers available is
not fixed. There are
an infinite number
of values available.
The availability of
numbers expands
without end. Real
numbers are not
simply a finite
"row of
separate
points" on a
number line. There
is always another
real number whose
value falls between
any two real numbers
(this is called the
"density"
property).
Third,
when
real numbers are
added or multiplied,
the result is always
another real
number (this is
called the
"closure"
property). [This is
not the case with
all arithmetic
operations. For
example, the square
root of a -1 yields
an imaginary
number.]
With these three
points in mind,
the question
is: ,
How can we use
real numbers in
practical
calculations?
What rules
apply?
- How
should
numbers be
added,
subtracted,
multiplied,
and
divided?
What
latitude
do we
have?
-
Does
it matter
what we do
first?
second?
third? . .
.
-
Can
we add
a series
of numbers
together
in
any
order?
Will the
final
answer be
the same
regardless
of the
order we
choose?
-
Can
we
multiply
a series
of numbers
together
in
any
order?
Will the
final
answer be
the same
regardless
of the
order we
choose?
The
following properties of
real numbers answer
these types of
questions.
The property
characteristics
which follow show
how much latitude
you have to change
the mechanics of
calculations which
use real numbers
without changing the
results.
-
Associative
Property
-
Commutative
Property
-
Distributive
Property
-
Identity
Property
-
Inverse
Property
In addition, the
following three math
properties of
equivalence
determine when one
algebraic quantity
can be substituted
for another without
changing the
original value.
-
Reflexive
Property
-
Symmetric
Property
-
Transitive
Property
Properties of
Equivalence
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Equivalence
- Reflexive,
Symmetric, and
Transitive
Properties
-
What
guarantees
that a
relationship is
an
Equivalence
Relationship
?
The following
three
properties
are
true for every
equivalence
relationship:
•
Reflexive
Property
•
Symmetric
Property
•
Transitive
Property
All
three
properties
should be
considered
together.
For example,
all three
properties must
be true for the
equal (=),
congruent
(
) ,
or
similarity
(
) sign
to be
valid in a
mathematical
relationship
such as a
function
or an
equation.
Reflexive
Property:
(a=a)
a
is
equivalent
to
a
("a"
equals
"a")
Examples
of
reflexivity:
2
=
2
x
=
x
(numbers
are listed in
exactly the
same
order)
5+4
=
5+4
X+2
=
X+2
The property of
reflexivity may
seem
ridiculously
obvious.
However, it is
used
extensively in
proofs.
For example,
the
property of
reflexivity can
be used to
prove that the
following two
triangles are
congruent.
Both
triangles have
a common side:
AC
Symmetric
Property:
(if a=b,
then
b=a)
The Symmetric
and Commutative
Property are
similar.
a =
b
is
equivalent
to
b =
a
Examples
of the
Symmetric
Property:
(numbers
are listed in
reverse order -
a
reflection)
5+4
=
4+5
X+2
=
2+X
This
property is
often used to
teach
elementary
school children
how numbers
work.
For
example:
Young children
are taught that
the numbers can
be rewritten as
follows:
Young children
are taught that
the numbers can
be rewritten as
follows:
Transitive
Property:
(if a=b
and b=c, then
a=c)
The Transitive
Property
illustrates how
logic and
deductive
reasoning are
used in
mathematics.
The Transitive
Property shows
how to draw
conclusions
from the
information
available.
The logic
provided by the
Transitive
Property can be
summarized as
follows: if one
item is equal
to a second
item, and the
second item is
equal to a
third item,
then the first
item is also
equal to the
third item.
For example, if
"Kevin is
the doctor for
my family"
and "the
doctor for my
family is my
cousin",
then the
transitive
property shows
that
"Kevin is
my
cousin".
An
example of how
this principle
can be applied
to mathematics
is:
.5
=
1/2
1/2
=
2/4
Therefore,
.5
=
2/4
Free
Resources
Reflexive,
Symmetric,
and
Transitive
Properties
Videos
Flash
Cards/Slides
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