Note:
Rounding a
number
makes it
much more
convenient
to use the
number.

However,
rounding a
number
always
introduces
an error -
called the
rounding
error
.

The
question
is: Can
you afford
to ignore
the error?
Is the
rounding
error
small,
relative
to the
value of
the
rounded
number?

Often
the answer
to that
question
is
"yes".
For
example,
how many
people
live in
your
country?
The answer
is usually
rounded to
millions.
It makes
no sense
to list an
exact
number,
down to
the last
individual
person.
The
rounding
error is
small,
relative
to
hundreds
of
millions
of
people.

Rounding
errors
accumulate
. Do not
use
rounded
numbers in
calculations
because
the
rounding
errors
compound
and
accumulate.
Round the
final
result,
not the
numbers
used in
intermediate
calculations.

(Have
you ever
had a
bank, or a
grocery
store,
round
their
intermediate
calculations
in such a
way that
you lose
money?)

To
*guard
against*
the
inaccuracies
introduced
by
*accumulated
rounding
errors*,
guard
digits
should be
used. This
is the
opposite
of
rounding.
Guard
digits are
extra
digits
which are
included
in
calculations
to
compensate
for
possible
rounding
errors.
For
example,
if
calculations
are to be
carried
out to the
hundredths
place, use
the
thousands
place or
beyond in
your
intermediate
calculations.

Example
: Multiply
the
following
three
numbers
together:
2.301,
1.046,
and
3.111
. Round
the final
result to
the
nearest
hundredth.

Round
Numbers
before
multiplication
:

(2.30)(1.05)(3.11)
= 7.51

DO
NOT
Round
Numbers
before
multiplication
:

(2.301)(1.046)(3.111)
= 7.49

Final
Result
: The
**calculated
results
are
different**
due to the
extra
three
rounding
errors
introduced
when the
numbers
were
rounded
before
being
multiplied.