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Decimal notation is just
a shorthand way of expressing
certain fractions, namely
those fractions with denominators
that are powers of 10. For
example, consider the number
2.345
Because of the place-values
of the decimal digits, this
really means
Because all the denominators
are powers of 10, it is very
easy to add these fractions
by finding a common denominator.
In this example, the common
denominator is 1000, and we
get
This suggests a general rule
for converting a decimal number
to its fraction form:
- Put
all the digits over the
denominator that corresponds
to the last decimal place
value.
In the number 2.345, the
last decimal place value is
the thousandths place, so
we put the digits 2345 over
the denominator 1000.
Of course we would usually
want to reduce the resulting
fraction to its simplest form.
In this case
The only time this method
does not work is for repeating
fractions. We know that 1/3 = 0.3333333...
but how could we go from 0.3333333...
back to 1/3? There is no last
decimal place because the
decimals repeat forever. Fortunately,
there is a simple trick for
this:
·
Put
the repeating digit over a
denominator of 9.
So we see that in the case
of 0.3333333...., the repeating
digit is 3, and we make the
fraction 3/9, which reduces
to 1/3.
If there is a group of more
than one digit that repeats,
- Put
the repeating group of digits
over as many 9s as there
are digits.
For example, in the fraction
we see that the group of
digits 15 repeats, so we
put 15 over a denominator
of 99 to get
o
One
warning: This only works for
the repeating fraction part
of a number. If you have a
number like 2.33333..., you
should just work with the
decimal part and rejoin it
with the whole part after
you have converted it to a
fraction.
o
Irrational
numbers like p or  have non-repeating
decimals, and so they cannot
be written as fractions. You
can, however, round them off
at some point and produce
an approximate fraction for
them.
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If you have learned
enough algebra to follow
these steps, it is not
difficult to see why
this works. Lets start
with the simple example
0.3333333... Since we
already know that the
answer is 1/3, we can
concentrate on the procedure,
not the answer.
Let
x = 0.3333333...
Multiply both sides
by 10 to get
10x = 3.33333...
Notice that the decimal
part (.33333
) is still
the same. In fact, it
is the very thing that
we called x in
the first place, so
we can say that
10x=
3 + x
Now solve for x:
This method will also
work for repeating fractions
that contain a group
of repeating digits,
but you have to multiply
by a higher power of
10 in order to make
the decimal portion
stay the same as it
was before. For example,
suppose we had 0.345345345....
Let
x = 0.345345345....
Multiply both sides
by 1000 to get
1000x = 345.345345345....
Notice that the decimal
part is still the same.
In fact, it is still
the thing that we called
x in the first
place, so we can say
that
1000x = 345 + x
Now solve for x:
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We know the decimal equivalents
for some common fractions
without having to think about
it: 1/2 = 0.5, 3/4 = 0.75,
etc. But how do we arrive
at these numbers? Remember
that the fraction bar means
the same thing as division.
- To
convert a fraction to a
decimal, do the division.
For example,
You can do the division with
a calculator or by hand with
long division.
- Look
only one digit to the right
- 5s
or higher round up (there
is some dispute about this
rule, but it is good enough
for most purposes)
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Why round fives up?
The number 3.5 is exactly
halfway between 3.0
and 4.0, so it makes
just as much sense to
round it down as it
does to round it up.
Most of the time there
is no harm in using
the always round fives
up rule. This is the
rule that the United
States Internal Revenue
Service advises you
to use on your taxes,
and who is going to
argue with them?
Sometimes, though,
it can cause problems.
Suppose you are adding
a very large number
of values that have
all been rounded by
this rule. The sum that
you get will be a little
bit bigger than it ought
to be. This can be a
very serious problem
in computer programs.
When thousands or even
millions of additions
are being performed,
the accumulated roundoff
error can be quite large.
One way of dealing
with this problem is
the even-odd rule.
This rule says that:
- If
the five is the last
significant digit
and the round-off
digit (the one to
the left of the 5)
is odd, round up.
- If
the five is the last
significant digit
and the round-off
digit is even, dont
round up.
Actually, you could
reverse even
and odd in this
rule. All that matters
is that about half the
time you will be rounding
up on a 5, and half
the time down.
The
reason it matters that
the five is the last
significant digit is
because if there are
any other non-zero digits
past the five then you
must round up,
because the part that
you are chopping off
is more than 50% of
the roundoff place-value.
For example, suppose
you want to round 3.351
to the nearest tenth.
The decimal part represents
the fraction 351/1000,
which is 1/1000 closer
to 400/1000 than it
is to 300/100. Therefore
you would always round
this up to 3.4.
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Example:
Round 11.3826 to the nearest
hundredth.
Solution:
The hundredths place is where
the 8 is. We look one digit
to the right and see a 2,
so we do not round up, leaving
us with 11.38.
Example:
Round 11.3826 to the nearest
thousandth.
Solution:
The thousandths place is where
the 2 is. We look one place
to the right and see a 6,
so we round the 2 up, getting
11.383.
Trouble
with 9s.
If the digit you are rounding up is a 9, then rounding it
up will make it a 10, which
is too big for one place.
What happens is that the extra
1 gets added to the place
to the left.
Example:
Round 3.49721 to the nearest
hundredth.
Solution:
The hundredth place has a
9 in it. One step to the
right is a 7, so we have
to round up. This makes the
9 into a 10, but we really
cant write the new number
as 3.4(10). Instead, the extra
1 moves one place to the
left and is added to the 4,
giving us 3.50.
- It
can happen that the place
to the left contains another
9, in which case the extra
1 will cause it
to become a 10, which
pushes the 1 still further
on to the next place to
the left.
Example:
Round 75.69996217 to the nearest
ten-thousandth
Solution:
The ten-thousandths place
is the last 9 in the number,
and the place to its right
is a 6, which means we round
up. This makes the 9 into
a 10, like this:
75.699(10)
But
of course we cannot put a
10 in that place, so the
1 moves to the left and
gets added to the 9 there,
making it into a 10:
75.69(10)0
This
leaves us with the same problem,
a 10 in one decimal place,
so the extra 1 moves one
more step to the left, turning
that 9 into a 10:
75.6(10)00
Well,
we still have the same problem,
so we move the 1 yet another
step to the left, where it
adds on to the 6, finally
leaving us with an acceptable
answer:
75.7000
In general,
·
The extra 1
migrates to the left until
it finds a resting-place.
This means that the 1 moves
to the left until it can be
added to a digit less than
9, or until it falls off
the end as a new digit out
in front.
Example:
Round 999.96 to the nearest
tenth.
The
digit to the right of the
tenths place is a 6, so
we have to round up. But when
we round up the 9 it becomes
a 10, forcing the one to
be added to the left. Unfortunately,
we find another 9 there
and the process is repeated
for each of the 9s until
we reach the leading 9,
which becomes a 10 resulting
in 1000.0
Although calculators have
made it much easier to do
arithmetic with decimal numbers,
it is nice to know that you
can still do it without a
calculator.
To
add or subtract decimal numbers,
you use the familiar column
method that you learned back
in grade school. To use this
method, the place values of
the two numbers must be lined
up. This means that the decimal
points must be lined up, and
you can fill in with zeros
if one number has more decimal
places than the other.
Example:
5.46 + 11.2
Becomes:
To
multiply two decimal numbers,
you can use the column method
just as you would with whole
numbers. You ignore the decimal
points as you carry out the
multiplication, and then you
put the decimal point in the
result at the correct place.
The product will have the
number of decimal places as
the total number of decimal
places in the factors. In
the following example, the
first factor has 2 decimal
places and the second factor
has 1 decimal place, so the
product must have 3 decimal
places:
You can
divide decimal numbers using
the familiar (?) technique
of long division. This can
be awkward, though, because
it is hard to guess at products
of decimals (long division,
you may recall, is basically
a guess-and-check technique).
It can be made easier by multiplying
both the dividend and divisor
by 10s to make the divisor
a whole number. This will
not change the result of the
division, because division
is the same thing as fractions,
and multiplying both the numerator
and denominator of a fraction
by the same number will not
change the value of the fraction.
For example, consider
This
is the same as the fraction
 , which
is equivalent to the fraction
 , obtained
by multiplying the numerator
and denominator by 10. Thus
 ,
which
can be attacked with long
division:
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