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.
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
answers 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
The Seven
Fundamental
Properties
of Real
Numbers
- click
description
Commutative
Property
-
Addition and
Multiplication
(Note that the
Commutative
Property
only
applies to
addition and
multiplication.
It does not
apply to other
operations in
arithmetic.)
Commutative
Property
of
Addition:
Any
two
addends
can
be
added
in either
order
without
changing
the
results.
This
applies
to all
real
numbers
(including:
fractions,
decimals,
and
negative
numbers).
Example
1:
5 +
9 =
9 +
5
(
positive
integers
)
Example
2:
[-5] +
9 =
9 +
[-5]
(
negative
integers
)
Example
3:
¾ +
½
=
½ +
¾
(
fractions
)
Example
4:
1.1 +
0.0055
=
0.0055 +
1.1
(
decimals
)
Example
5:
a +
b =
b +
a
(
algebraic
notation
)
Commutative
Property
of
Multiplication:
Any
two
factors
can
be
multiplied
in either
order
without
changing
the
results.
This
applies
to all
real
numbers
(including:
fractions,
decimals,
and
negative
numbers).
Example
1:
5 *
9 =
9 *
5
(
positive
integers
)
Example
2:
[-5] *
9 =
9 *
[-5]
(
negative
integers
)
Example
3:
¾ *
½
=
½ *
¾
(
fractions
)
Example
4:
1.1 *
0.0055
=
0.0055 *
1.1
(
decimals
)
Example
5:
a *
b =
b *
a
(
algebraic
notation
)
