With these
three points in
mind,
the
question
is:
How can
we use real
numbers in
practical
calculations?
What rules
apply?
. .
. Identity
Property .
.
.
The
concept of
individual
identity
can be
applied in
a variety
of
interesting
ways.
People
working in
different
fields
of study
have
developed
their own
specialized
systems to
identify
and
catalog
what is
important
to them.
Examples
of these
differences
can be
seen in
how
individual
identity
is used in
biological
identification
(DNA),
music,
psychology,
archaeology,
architecture,
etc.
However,
the
examples
used
here
will
focus
exclusively
on
numbers
.
Identity:
-
(an
identity
is
not the
same as
the
identity
property)
an
example
is
shown
below
A personal
driver's
license
number is
one form
of
identity
.
Your
driver's
license
number
allows you
to be
recognized
as a
separate
driver who
is
different
from all
other
drivers.
Your
driver's
license
number is
a
distinct
and unique
number
which is
yours
alone.
However,
your
diver's
license
number is
also part
of a large
group of
license
numbers.
It is only
one number
within a
large
group of
numbers
(called a
set
).
Identity
Property:
-
(the
identity
property
is
not the
same as an
identity)
an
example
is
shown
below
An
identity
property
only
applies to
a group of
numbers as
a
whole
(a
set
). It does
not apply
to
individual
numbers
within the
set.
The
identity
property
can be
illustrated
using the
"clock"
analogy as
follows:
The clock
(shown
above)
cannot
display
all real
numbers.
The clock
can only
display
hours,
minutes,
and
seconds up
to and
including
12
hours.
The
set
of numbers
used by
the clock
is:
all times
up to 12
hours.
Identity
The
time
displayed
by the
clock has
an
identity
. It is
2:15. 2:15
is
distinct
and
unique
from all
the other
times
which
could be
displayed
by the
clock (am
and pm are
not shown
the a 12
hour clock
used in
this
example).
Changes to
Identity
The time
on the
clock will
change.
One hour
later, the
time on
the clock
will say
3:15. 3:15
has a
completely
different
identity
than
2:15.
Identity
Property
(The
Identity
Property
is a
property
which
applies to
the entire
set of
numbers
which can
be
displayed
by the
clock. The
identity
property
does not
apply to
individual
numbers by
themselves.)
Of all the
times
(numbers)
that can
be
displayed
by the
clock,
the number
12 hours
has a
unique
quality.
If 12
hours is
added to
any time
shown on
the clock,
the clock
continues
to show
the
original
time. (The
original
time does
not loose
its
identity.)
The time
currently
displayed
on the
clock is
2:15.
However,
if 12
hours is
added to
2:15, the
clock will
continue
to show
2:15.
The number
12
hours
is
called
an
identity
element
(also
called a
neutral
element).
Every
number on
the clock
remains
unchanged
whenever
12 hours
is added
to it.
Because
the number
12 hours
has this
unique
quality,
the
set
of
numbers
used by
the clock
possesses
an
Identity
Property
of
addition
.
Identity
Properties
for the
set of all
Real
Numbers
An
additive
identity
is a
number
that can
be added
to any
number
without
changing
the value
of that
other
number.
The
additive
identity
for the
set
of all
real
numbers
is
0
(zero).
The number
0 can be
added to
any real
number
without
changing
its
value.
Example
1:
(
positive
integers
)
2
+
0
=
2
Example
2:
(
negative
numbers
)
-2
+
0
=
-2
Example
3:
(
fractions
)
¾
+
0
=
¾
Example
4:
(
decimals
)
1.1
+
0
=
1.1
Example
5:
(
algebraic
notation
)
a
+
0
=
a
A
multiplicative
identity
is a
number
that
can
be
multiplied
by
any
number
without
changing
the
value
of
that
other
number.
The
multiplicative
identity
for
the
set
of
all
real
numbers
is
1
(one).
Any
real
number
can
be
multiplied
by
the
number
1
without
changing
its
value.
Example
1:
(
positive
integers
)
2
*
1
=
2
Example
2:
(
negative
numbers
)
-2
*
1
=
-2
Example
3:
(
fractions
)
¾
*
1
=
¾
Example
4:
(
decimals
)
1.1
*
1
=
1.1
Example
5:
(
algebraic
notation
)
a
*
1
=
a
The
Multiplicative
Identity
of
1
is the
key
to
adding
fractions
with
different
denominators:
Example
1:
add the
following
two
fractions
+
(1)
Using
the
Multiplicative
Identity
of
1
,
multiply
each
fraction
by
1
.
Note:
Multiplying
a fraction
by 1 does
not change
the value
of the
fraction:
multiplied
by 1 is
still
multiplied
by 1 is
still
(2)
Replace
the
Multiplicative
Identity
of
1
,
with
a
fraction
Select
the
fraction
as
follows:
-
The
fraction
must
equal
1.
The
numerator
and
denominator
of
the
fraction
must
be
the
same.
Examples
of
the
type
of
fractions
which
can
be
used
are:
-
-
Choose
fractions
which
will
produce
common
denominators
when
multiplied
with
the
original
fractions.
For
this
example
those
fractions
can
be:
-
(3)
Complete
the
Multiplication
of
the
two
groups
of
fractions.
Note
that
after
the
multiplication
is
complete,
the
denominators
of
both
of
the
remaining
fractions
are
the
same:
Note:
The
denominators
of
both
of
the
remaining
fractions
are
the
same.
(4)
Add
the
Fractions
. The
two
remaining
fractions
can
now
be
added
since
their
denominators
are
the
same:
(5)
The
Final
Answer
:
Example
2:
add the
following
two
fractions
+
(1)
Using
the
Multiplicative
Identity
of
1
,
multiply
each
fraction
by
1
.
Note:
Multiplying
a fraction
by 1 does
not change
the value
of the
fraction:
multiplied
by 1 is
still
multiplied
by 1 is
still
(2)
Replace
the
Multiplicative
Identity
of
1
,
with
a
fraction
Select
the
fraction
as
follows:
-
The
fraction
must
equal
1.
The
numerator
and
denominator
of
the
fraction
must
be
the
same.
Examples
of
the
type
of
fractions
which
can
be
used
are:
-
-
Choose
fractions
which
will
produce
common
denominators
when
multiplied
with
the
original
fractions.
For
this
example
those
fractions
can
be:
-
(3)
Complete
the
Multiplication
of
the
two
groups
of
fractions.
Note
that
after
the
multiplication
is
complete,
the
denominators
of
both
of
the
remaining
fractions
are
the
same:
Note:
The
denominators
of
both
of
the
remaining
fractions
are
the
same.
(4)
Add
the
Fractions
. The
two
remaining
fractions
can
now
be
added
since
their
denominators
are
the
same:
(5)
The
Final
Answer
:
Free
Resources
"Identity"
Properties
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