Rounding To Decimal Places by pmm93834

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									Rounding

   •   Introduction
   •   Rounding To Decimal Places
   •   Rounding To Whole Numbers, Nearest 10, 100 And 1000
   •   Some Issues With Rounding
   •   Significant Figures
   •   Rounding To 1 Significant Figure
   •   Rounding To 3 Significant Figures
   •   Limits Of Accuracy, Upper And Lower Bounds
   •   Maximum And Minimum Values


Introduction

Rounding is the process whereby we approximate an answer to
some specified level of accuracy.


Rounding To Decimal Places

If I use an ordinary 30cm ruler it would be hard to get our
answer to more than 1/2 mm of accuracy. It is more likely that we
would only get our answers to 1 mm (i.e. 0.1 cm) of accuracy.
For instance, I could measure a length to be 3.6 cm, but I would
be unlikely to measure a length so that it was 3.63 or 3.63767
cm! I'd have to have very good eyesight (and a very good ruler)!

However, when we work with numbers, especially division, we
often get answers that have many decimal places, particularly
when we use a calculator. Let's look at an example. I have a
35cm length of wood. I want to divide it into 7 pieces. Easy, just
do the sum 35 ÷ 7 = 5cm. No problemo!

Now what about a 30cm piece of wood split into 7 equal pieces.
Well, do the sum 30 ÷ 7 = 4.285714286 on a calculator. How am
I supposed to measure to that degree of accuracy! Even an
industrial saw with laser control couldn't cut that accurately.

What we need to do is approximate the answer so that it is
reasonable. For a length using a ruler we have already seen that
an answer with just one decimal position or place will often do.

But before we do the sum let's just look closer at our numbers
and the decimals: Look at this diagram.
This diagram represents the number 1.1111. Each piece is 10
times smaller than the one to the left of it.

It is easy to see that if you were to lose the last 1, you are not
really losing that much. So we could say that 1.1111 to three
decimal places was 1.111. Now if I look at the last bit now, it's
still pretty small compared with the bits to the left of it. So I
could ignore that piece as well and get 1.1111 to be 1.11 to two
decimal places. Again I could lose the second decimal part
without making too much difference and so 1.1111 would become
1.1 to one decimal place. I could even lose that bit, since it is so
small in comparison with the whole number that 1.1111 will
become 1 to the nearest whole number.

We must always remember though that we have lost some of the
original number and sometimes even though these parts are very
small in comparison with the rounded number, they can be
important. For instance if you imagine that the above number
represented tonnes of gold, you might not want to lose even the
smallest amount of gold!

When we lose or leave off parts of the number in this way we say
that we have truncated the number.

Consider though the following number as a diagram:




If we now truncate the number to one decimal place we will get
1.1. But look at the diagram. If we leave off the 0.09 part we are
losing a quite substantial chunk of the number. If you think about
it you should see that really the number is almost 1.2 really.
There is only another 0.01 needed to make it 1.2 So really we
should make this number equal to 1.2 when rounded to one
decimal place.

What about this diagram:




It seems clear that we could ignore the 0.03 and get 1.1 to one
decimal place.

Consider this one:




I think you will agree that this second decimal is quite large and
so the number should be closer to 1.2 when rounded to one
decimal place rather than 1.1.

But what about this situation:
We now have a bit of a dilemma. Should we lose the last bit and
make it closer to 1.1 or should we think that it makes the number
closer to 1.2?

We could choose either. In fact mathematicians have agreed that
it should make the number 1.2 rather than 1.1.
Another way of looking at the situation is to think of a dial or
meter with numbers on the scale:




Obviously 1.17 is closer to 1.2 than to 1.1




In this example, 1.14 is closer to 1.1
Again for this example, mathematicians have agreed that it is
closer to 1.2 than to 1.1. You might not agree, but we all have to
accept it.




Note: One of the difficulties that some have with these ideas is
that they don't really understand the decimal system. When they
see a number such as 1.17 they read it a "one point seventeen"
instead as "one point one seven". If you read it as "one point
seventeen" then you might think that this number is bigger than
1.2 "one point two". Actually you'd be wrong. Look at the
diagram here to see why.




Here we can see that 1.2 is actually larger than 1.17. The
problem is that with decimals as you go to the right the numbers
are representing ever smaller things. So we should never say
"one point seventeen", but rather "one point one seven". The
only exception to this is with money where we would say "one
pound, seventeen pence" for £1.17. We would not however, see
£1.2 (except on a calculator display), but we would add on the
extra 0 to make it £1.20 and say it as "one pound, twenty
(pence)". It's clear that this is now larger than £1.17!

The other thing to note about rounding is that we don't need to
even think about any of the other, smaller decimal places. For
instance 1.149999999 is still 1.1 when rounded to one decimal
place. 1.159999999 is still 1.2 when rounded to 1 decimal place.
If you are not sure why, draw yourself some diagrams and you
will see that the extra decimals never have as much influence as
the one to the left of each one.

One tricky area is when we get a number such as 1.99999. What
is this rounded to 1 decimal place? Obviously when truncated the
number is simply 1.9. But since the second decimal place is
greater than 5 we need to round up. Well the next number to
round up to is not 1.99, because that has two decimal places, so
it can only be 2.0.

Now that we have some understanding of what is happening let
us work through some examples to see how we can do rounding
quickly in practice.


Look carefully through the following       examples.   Each   one
introduces a slightly different point.


Example 1: Round 23.834 to one decimal place.

First truncate the number to 1 d.p. That is 23.8

Now look at the digit in the second decimal place. It's a 3. Since
this is smaller than 5 we don't need to round up.

So our answer is 23.8 (to 1 d.p.) - Note how we must tell the
reader of our answer that we have rounded the answer.


Example 2: Round 34.7823 to one decimal place.

First truncate the number to 1 d.p. That is 34.7

Now look at the digit in the second decimal place. It's an 8. This
is lager than 5 so we need to round up. The next number after
34.7 is 34.8.

So our answer is 34.8 (to 1 d.p.)
Example 3: Round 287.653209 to one decimal place.

Truncate to 1 d.p. i.e. 287.6

The second decimal is a 5. We use the agreed rule and round up.
The answer is 287.7 (to 1.d.p)


Example 4: Round 29.98087 to one decimal place.

Truncate first, i.e. 29.9 The second decimal is an 8 so round up.
The next number after 29.9 is 30.0

So the answer is 30.0 (to 1 d.p.)


Example 5: Round the number 2.9876 to two decimal places.

This time we truncate to 2 decimal places i.e. 2.98

We now look at the third decimal which is a 7, so we need to
round up.
So the answer is 2.99 (to 2 d.p.)



Example 6: Round 0.0034 to two decimal places.

Truncate to 2 decimals. We get 0.00. Look at the third decimal,
it's a 3 so we don't round up.

So the answer is 0.00 (to 2 d.p.)

However consider the implications of this. If you divide by 0 you
will cause the computer or calculator to come back with an error.
This is because the answer is infinity. No computer or calculator
can handle that. However the answer is not infinity. It might be
large, but it isn't that large! Suppose your computer was
controlling a nuclear power station and you decided to round the
answers without thinking about divisions by 0. If you were lucky
your computer might just crash. If you were unlucky the whole
power station might crash!


Example 7: Round 9,999.999999 to three decimal places.

Truncate first, we get 9,999.999. Now look at the fourth decimal,
it's a 9 so round up. The next number is 10,000.000

So the answer is 10,000.000 (to 3 d.p.)
Rounding To Whole Numbers, Nearest 10, 100 And
1000.

So far we have just looked at rounding to a certain number of
decimal places. But we can also round to the nearest whole
number, the nearest 10, the nearest 100, and nearest 1000 etc.
We already have the basic ideas, so it will probably be easier to
look at examples and see how it works in practice.

But first let's make sure you understand how numbers work in
terms of their position and relative value.




Does it look familiar - it should, for it is the same diagram as for
decimals, but with different numbers above.
The number represented is 11,111

Rounded   to   the   nearest   10 this number would be 11,110
Rounded   to   the   nearest   100 this number would be 11,100
Rounded   to   the   nearest   1000 this number would be 11,000
Rounded   to   the   nearest   10,000 this number would be 10,000

Let's look at some examples to see how this rounding works:

Example 1: Round 78.765 to the nearest whole number.

A whole number doesn't have any decimals in it, so truncate the
number to 78. Now we look at the first decimal place to see if it
should be rounded. Since it is a 7 it should be rounded up.

So the answer is 79 (to nearest whole number).


Example 2: Round 768.9 to nearest 10

Truncate everything less than the tens column to get 760. Note
we ignore any decimals. look at the units column to see if we
should be rounding. The units column is an 8, so we should round
up to the next number.

So the answer is 770 (to the nearest 10).
Example 3: Round 657,987 to the nearest 1000

We set everything in the units, tens and hundreds column to 0,
so we get 657,000. Now we look in the hundreds column to see if
we should round up and since it is a 9 we definitely should.

So the answer is 658,000 (to the nearest 1000).


Example 4: Round 9989 to the nearest 100.

We set the tens and units column to zero to get 9900. Then look
at the tens column to see if we need to round. It's an 8 so we
need to round.

So the answer is 10,000 (to the nearest 100). This because going
one hundred up from 9900 gets us to 10,000


Example 5: Round 372 to the nearest 1000.

We set the hundreds, tens and units column to 0, so we get 0 !.
Now we look at the hundreds column to see if we should round. It
is a 3 so we shouldn't.

So the answer is 0 (to the nearest 1000). This is obviously
strange, and possibly dangerous if we are going to divide by this
number!


Some Issues With Rounding

We have already noted that when rounding we can get a zero,
which in some situations can lead to unforeseen effects.

It should be noted though that rounding too early in a calculation
can also result in quite drastic errors. As a rule answers should
only be rounded at the end of all calculations. Consider the
following example:

Question: Given a circle of circumference 234m work out its
area.

Answer: Firstly what value should we use for π (pi)? The true
number is 3.1415926535897... Often in examinations you are
told to use the number 3.14 which is π rounded to 2 decimal
places. However on a calculator you might press the key for π
and on my calculator this uses the value 3.141592654 which is π
rounded to 9 decimal places. Will it make a difference. Well in the
first answer I shall use the most accurate values possible at all
times and in the second solution I shall keep rounding values to 2
decimal places.
Solution 1:




Solution 2:




Notice the difference between the answers: 4359.29 - 4357.34 =
1.95
This might not seem like much, but it shows how answers can
become quite inaccurate very easily. Think if there were even
more steps involved.

Moral: round only at the end of all your calculations.


Significant Figures

Another form of rounding is where we round to a certain number
of significant figures.

Take any number and the digit at the left hand side is the most
significant, or most important part of it. The digit at the right
hand is the least significant digit.

e.g. Consider 3,456 The three is the most significant digit since it
represents 3000. The 6 is the least significant because it only
represents 6 units.

e.g. Consider 0.003456 The 3 is again the most significant digit
since it represents 0.003 or 3 thousandths. Whereas the 6 is the
least significant digit because it represents only 0.000006 or 6
millionths.

The usual forms of rounding to significant figures are rounding to
one significant figure or rounding to three significant figures.


Rounding To One Significant Figure

In this we just keep the most significant figure and set all other
digits to zero.

Example 1: Round 345098 to 1 significant figure.

Firstly keep the most significant digit i.e. 300000. Now check the
second significant digit to see if we should round up. Its a 4 so
we don't round up.

So the answer is 300,000 (to 1 s.f.)


Example 2: Round 0.065709 to one significant figure.

Firstly keep the most significant figure and ignore all the other
decimals. i.e. 0.06. Now check the second significant figure,
which is a 5 so we round up.

So the answer is 0.07 (to 1 s.f.)

Example 3: Round 96.987 to 1 sig. fig.
Firstly keep the most significant digit i.e. 90. Check the second
significant figure, which is a 6 so we round up.

The answer is 100 (to 1 s.f.)


Example 4: Round 0.009609999 to one sig. fig.

Keep the most significant digit, i.e. 0.009. The second significant
digit is a 6 so round up.

The answer is 0.01 ( to 1 s.f.)


Rounding To Three Significant Figures

Here we keep the three most significant figures, using the fourth
to decide whether to round up or not.

Example 1: Round 65487 to 3 sig. figs.

Keep the three most significant digits. i.e. 65400. The fourth
significant digit is an 8 so round up.

The answer is 65,500 (to 3 s.f.)

Example 2: Round 0.098734 to 3 sig. figs.

We get 0.0987, the fourth digit is a 3 so don't round up.

The answer is 0.0987 (to 3 s.f.)


Example 3: Round 0.00999898 to 3 sig figs.

We get 0.00999, the fourth digit is an 8 so round up.

The answer is 0.01 (to 3 s.f.)


The great advantage of rounding to significant figures is that it
keeps the relative size of the number clear, without it becoming a
zero.




Limits Of Accuracy And Upper And Lower Bounds
If we are give a number, say 25 cm and we are told that this
value has been rounded to the nearest cm, then it’s true value
could actually be a little bit less than 25cm, or a little bit more
than 25cm. How can we know what the lower bound is (i.e. the
smallest value) and what the upper bound is (i.e. the largest
value)?

Obviously, the lower bound cannot be 24 cm. Because 24 cm to
the nearest cm is 24 cm.

What about 24.1 cm. Well if we round this we get 24 cm.

What about 24.5 cm? Well this rounded is 25cm. Can we get
smaller than this. No. 24.4999 is still only 24 cm to the nearest
cm.

So our lower bound is 24.5 cm.

What about the maximum value it could be. 26 cm is too large,
25.6 cm would round to 26cm. What about 25.4 cm. Well this
would round to 25cm.

So, now we look at 25.5 cm. If we rounded this we would get
26cm. So our true value must be less than this. So 25.5 cm is the
upper bound of possible values.

Clearly the actual value cannot equal 25.5 cm, but we cannot
easily write a number that is just less than this, so we allow 25.5
to be the maximum possible value.
This does cause a little confusion with people, because 25.5 cm
would actually round to 26cm, so how can it be the maximum
value.

Well, if you tried to write the biggest number that is just less
than 25.5, you will find you can’t do it.

25.49 can be made a little larger by using 24.99, which can be a
bit larger as 24.999 and so on.

But, you can’t say 24.999 recurring, because if you go to an
infinite number of decimal places, all being 9, you are actually
really writing 25.5 anyway.

(This is because 0.9999 recurring actually equals 1!)

It is easy to think of the lower and upper bounds as being part of
a class interval or inequality.

24.5
Notice, that, since we have rounded to the nearest cm, then our
lower bound is the rounded value (25cm) minus ½ of 1cm and
the upper bound is 25cm + ½ of 1cm.
Consider another example. The length of a football pitch is 110 m
to the nearest 5m. What is the smallest possible value and the
largest possible value for the length of the pitch?

Well if the pitch has been rounded to the nearest 5m, then it
could be ½ of 5m on either side of 110.

So we have 110 – 2.5m or 110 + 2.5m
i.e. the pitch could vary from 107.5 m to 112.5 m

It’s very simple really. Just find half of the thing you are rounding
to.

1.34 is rounded to 2 d.p. so it could vary from 1.34 – 0.005 to
1.34 + 0.005, i.e. from 1.335 to 1.345


If the number has been rounded to significant figures then you
have a little more thinking to do.

The number 1300 has been rounded to 2 significant figures. What
is the smallest and largest values it could have?

Well, being rounded to 2 s.f. implies for this number that it is
rounded to the nearest 100, so we need 50 on each side. So it
could vary from 1250 to 1350.


Maximum And Minimum Values

We often use these upper and lower bounds to work out the
minimum and maximum values of an expression.

The usual situation is if there is a division involved, such as when
calculating speed or density.


Suppose we have A =
                        x
                        y

To obtain a maximum value for A, I need a maximum value of x,
but a minimum value for y!

To obtain a minimum value for A, I need a minimum value of x,
but a maximum value for y.

								
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