Decimals for measurement
All measurements of real objects are approximate. The number of digits
in your answer indicates the accuracy with which the measurement was made.
Understanding rounding and significant figures is important for everyday
life and work. The major obstacle to learning these topics is not understanding
the decimal notation system.
Rounding in everyday life
a) Rounding up
If you needed to buy 123 cm of fabric, you would find that, at most shops,
they cut fabric in multiples of 20 cm. So you cannot buy 123 cm of fabric:
a decision needs to be made between 120 cm and 140 cm. Usually, people
buy the larger amount and have a small amount of wastage rather than risk
not having enough. In this case, we have rounded up to the nearest
20 cm. The first row of the following table shows the length required,
and the second row shows the length purchased, after rounding up to
the nearest 20 cm.
120cm |
122cm |
124cm |
126cm |
128cm |
130cm |
135cm |
137cm |
140cm |
120cm |
140cm |
140cm |
140cm |
140cm |
140cm |
140cm |
140cm |
140cm |
b) Rounding down
If you buy groceries at the supermarket in Australia, the cost of the
good may be $24.63 but it is not possible to pay this amount of money,
due to the abolition of one and two cent coins. Rounding can be done in
various ways; the one which seems 'fairest' to the customer is when the
amount is rounded down to the nearest 5c. The first row of the
following table shows the total amount of the bill, and the second row
shows the amount to be paid, after rounding down to the nearest 5c.
$4.00
|
$4.01
|
$4.02
|
$4.03
|
$4.04
|
$4.05
|
$4.06
|
$4.07
|
$4.08
|
$4.09
|
$4.10
|
$4.00
|
$4.00
|
$4.00
|
$4.00
|
$4.00
|
$4.05
|
$4.05
|
$4.05
|
$4.05
|
$4.05
|
$4.10
|
c) Rounding to the nearest 5c
If you visit a small shop it is more likely that your total will be adjusted
by rounding to the nearest 5c. In this case the shop-owner is not
prepared to make a loss on (almost) every sale of a few cents as these
will accumulate in the long term to considerable amounts. The following
table shows the same first row (the total of the bill) as above, but the
amounts to be paid are different as this time the amounts are rounded
to the nearest 5c.
$4.00
|
$4.01
|
$4.02
|
$4.03
|
$4.04
|
$4.05
|
$4.06
|
$4.07
|
$4.08
|
$4.09
|
$4.10
|
$4.00
|
$4.00
|
$4.00
|
$4.05
|
$4.05
|
$4.05
|
$4.05
|
$4.05
|
$4.10
|
$4.10
|
$4.10
|
Summary table: Rounding to the nearest $100
|
Amount
|
$434
|
$454
|
$464
|
$484
|
$500
|
$524
|
$550
|
$574
|
Rounding up
|
$500
|
$500
|
$500
|
$500
|
$500
|
$600
|
$600
|
$600
|
Rounding down
|
$400
|
$400
|
$400
|
$400
|
$500
|
$500
|
$500
|
$500
|
Rounding to nearest
|
$400
|
$500
|
$500
|
$500
|
$500
|
$500
|
***
|
$600
|
*** Logically, because $550 is exactly halfway between $500 and $600,
$550 rounded to the nearest $100 could be either $500 or $600. By convention,
we say it is $600.
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Procedures for rounding numbers
Here are some examples of rounding to the nearest hundred, ten and tenth.
a) Rounding to the nearest hundred
Consider rounding 936 to the nearest hundred. If we draw a numberline
in hundreds,(e.g. 8 hundred, 9 hundred, 10 hundred, 11 hundred) then 936
is closer to 9 hundred than any other hundred. In fact, any number between
850 and 950 will round to 9 hundred (see lower diagram). How to treat
the numbers which are exactly equal distance from the hundred markers
(e.g. 850 and 950) needs to be decided. Usually, these are rounded up,
so 850 is rounded to 9 hundred, and 950 is rounded to 10 hundred (i.e.
one thousand).
While this process can be taught as an algorithm, (e.g. identify the
digit after the relevant place value column; if it is 5 or above, then
round up, otherwise round down) it is important to understand the process
involved.
b) Rounding to the nearest ten
If we repeat the above ideas for rounding 936 to the nearest ten, we
need a numberline with marks on the tens, e.g. 92 tens, 93 tens, 94 tens
and 95 tens. Then 936 will be seen to nearest to 94 tens (usually called
940). The first diagram shows this, while the second shows that any number
from 935 to 945 is rounded to 940 (when rounding to the nearest ten).
Remember that the numbers which are exactly in the middle need special
consideration; we usually round these up, but this has just been agreed
by people (there is no logical reason).
Instead of rounding to the nearest ten, we could try some other
rounding. To round 936 up to the nearest ten also gives 940,
and to round 936 down to the nearest ten gives 930.
c) Rounding to the nearest tenth
This time we need a numberline with tenths marked on it. For example,
27.53 can be thought of as 2753 hundredths, or 275 tenths and 3 hundredths
(see number expanders). The
numberline marked in tenths (273 tenths, 274 tenths, 275 tenths, 276 tenths)
is usually marked as below. Then 275.3 tenths is closer to 275 tenths
than 276 tenths, so 27.53 is closer to 27.5 than 27.6. As the second diagram
shows, any number between 27.45 and 27.55 is rounded to 27.5 when rounding
to the nearest tenth.
Bank interest
Interest of 3% for 1 year on $12000 is $360 per year, or $6.923076923
per week, or $0.986301369 per day. If the bank calculated the interest
daily, rounding to the nearest cent, they would pay 99 cents a day or
$361.35 over the year. If, on the other hand, they round down to
the nearest cent, they would pay 98 cents daily or $357.70 over a year,
saving themselves $2.30. Rounding is certainly an important issue when
money is concerned!
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Significant Figures -
When 12.00 is not the same as 12.0 or 12
How measurements are made
Consider the pencil shown in the diagram. We can try to measure its length
using a measuring device placed below the pencil. The length that we decide
upon depends on the divisions on the measuring device. For example, if
our measuring device only has cm marked on it, the pencil can only be
judged to be close to 4 cm, or maybe nearly 4 1/2 cm but not 4.0 cm or
40 mm as these imply we have made finer judgements than is possible with
this measuring device.
By measuring the same pencil with the second measuring device, which
is marked in tenths of centimetres (i.e. millimetres), we can make a more
accurate decision of 4.3 cm or 43 mm. Again, we could not write 4.30cm
or 43.0 mm as this implies we have made judgements about tenths of millimetres.
Under a microscope, using a measuring device marked in tenths of millimetres,
we can further judge that the length is 4.32cm.
So the measuring device that we use to measure an object affects the
judgement we make and therefore the way we record the length. The following
table summarises these ideas.
Length recorded as
|
Judged to nearest
|
Means it is between
|
4m
|
m
|
3.5m and 4.5m
|
4.7m
|
tenth of m
|
4.65m and 4.75m
|
4.0m
|
tenth of m
|
3.95m and 4.05m
|
7.39m
|
cm
|
7.385m and 7.395m
|
7.00m
|
cm
|
6.995m and 7.005m
|
So, is 12 equal to 12.00 and 12.00?
There are two answers to this simple question:
- As exact mathematical numbers, the answer is YES!
- As measurements, the answer is NO!
A measurement of 12 indicates a real size of between 11.5 and 12.5
A measurement of 12.0 indicates a real size of between 11.95 and 12.05
A measurement of 12.00 indicates a real size of between 11.995 and 12.005
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