. .
. Inverse
Property .
. .
Introduction

There
is a
difference
between an
Inverse
Property
and an
Inverse
Operation
. Each is
explained
separately,
below.

An
Inverse
Operation
is a
procedure
in
reverse
.

People use
Inverse
Operations all
the time.
An
inverse
operation
"undoes"
an original
action .
An inverse
operation is
the opposite of
an original
procedure.

To
illustrate,
imagine looking
up a phone
number in the
phone book. To
begin, we
generally look
up a name
alphabetically.
The phone
number is
usually listed
beside the
name.

Now imagine
doing the
**exact
opposite**.
Find a name by
looking up the
phone number.
Find the phone
number, and the
name will be
listed
alongside. The
process of
beginning with
a phone number
and looking up
the name is
called an
**Inverse
Operation**.

In
this example,
all of the
numbers in the
phone book have
a name listed
beside them.
All of the
names in the
phone book have
a phone number
listed beside
them. An
inverse
operation can
be performed on
any name or
number
listed.

The
concept of an
Inverse
Operation (a
procedure in
reverse) can be
applied to many
situations:
directions on a
map (going from
point A to
point B versus
returning from
point B to
point A),
mixtures
(versus
separation),
any type of
assembly such
as building a
car, swimming
below the
surface of a
lake (versus
ascending to
the surface),
making and then
retracting a
statement,
putting on a
jacket (versus
taking it off),
converting
miles to
kilometers
(versus
converting
kilometers to
miles),
etc.

Inverse
Operations are
valuable
, and routinely
used by
everyone
because
an Inverse
Operation gets
you back to the
point where you
started
.

That is why
inverse
operations
are so
important
when dealing
with real
numbers
: **an
Inverse
Operation
reverses the
effect of an
operation**
*(an
operation is a
procedure such
as addition or
multiplication)
***and gets you
back to the
number you
started
with**.

The
inverse
operation of
addition is
subtraction
. The inverse
operation of
subtraction is
addition.

The
inverse
operation of
multiplication
is
division
. The inverse
operation of
division is
multiplication.

**Inverse
Operations are
important keys
to checking
computations in
arithmetic**.

Elementary
school students
are taught to
check their
computations
using this
principle.

Using an
Inverse
Operation to
check a
subtraction
problem:

subtract
15 from
21

check the
answer using an
inverse
operation
(addition)

The
"Inverse
Operation"
of addition
reversed the
subtraction,
and returned
the number we
started with:
the
number
21
. This means
that the
original
subtraction
problem
(21-15=6) was
completed
correctly. If
it had not been
completed
correctly, the
inverse
operation would
not have
returned the
number 21 (the
original
starting
point).

Using an
Inverse
Operation to
check a
division
problem:

divide
312 by
12

check the
answer using an
inverse
operation
(multiplication)

The
"Inverse
Operation"
of
multiplication
reversed the
division, and
returned the
number we
started with:
the
number
312
. This means
that the
original
division
problem (312
÷ 12 = 26)
was completed
correctly. If
it had not been
completed
correctly, the
inverse
operation would
not have
returned the
number 312 (the
original
starting
point).

For every
non-zero real
number, the
Inverse
Property
is another
number
(called an
Inverse
Number). An
inverse
property is
**not a
procedure**.
(The Inverse
Operation
[explained in
the preceding
section] is a
procedure.)

The entire set
of non-zero
real numbers
has the inverse
property under
addition and
multiplication
because every
element in the
set has an
inverse.

The
additive
inverse
of any
number is the
same number
with the
opposite
sign.
When a
number and its
additive
inverse are
added to one
another, the
result is
always
0
(zero) -
the identity
element for
addition.

The
additive
inverse of +15
is
-15.

When
these two
additive
inverses are
added
together:

15 +
(-15) =
0

The
additive
inverse of -66
is
+66.

When
these two
additive
inverses are
added
together:

-66 + 66
=
0

The
additive
inverse of +X
is
-X.

When
these two
additive
inverses are
added
together:

+X + (-X)
=
0

The
multiplicative
inverse
of any
number is the
reciprocal of
that
number.
When a
number and its
multiplicative
inverse are
multiplied by
one another,
the result is
always
1
(one) -
the identity
element for
multiplication.

The
multiplicative
inverse
of 15
is
.

When
these two
multiplicative
inverses are
multiplied with
each
other:

The
multiplicative
inverse
of -66
is
.

When
these two
multiplicative
inverses are
multiplied with
each
other:

The
multiplicative
inverse
of X
is
.

When
these two
multiplicative
inverses are
multiplied with
each
other:

Inverse
Properties
are important
keys
which can be
used **to
simplify
equations**.

Example 1:
Using the
Additive
Inverse
Property

Solve
for
X

X + 3 =
5

To eliminate
the +3, the
additive
inverse
property of -3
can be used . .
. because +3 -
3 =
0 (the
additive
identity
element).

Add the
additive
inverse of -3
to each side of
the
equation.

Example 2:
Using the
Multiplicative
Inverse
Property

Solve
for
X

2X =
6

To eliminate
the +2 (of the
2X), the
multiplicative
inverse
property (of
the number 2)
of ½
can be used . .
. because (2)
*
( ½
) =
1 (the
multiplicative
identity
element).

Multiply each
side of the
equation by the
multiplicative
inverse of
½
.

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Resources

Inverse
Properties

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