Measurement is
assigning a real number to some physical quantity
such as length, width, height, volume, mass,
brightness, etc. A physical apparatus
suitable for the measurement to be made is used to
make the measurement.
Measurement
involves error, so it is appropriate to discuss
accuracy and precision at some point.
Examples:
A yard stick measures length in yards.
A meter stick measures length in meters.
A ruler measures length in feet.
A thermometer measures temperature in degrees F or
C.
Using these tools
we can measure length and width to find area, and if
we measure height, we can then find volume.
With yardsticks
and rulers we deal heavily with fractions and
conversions between these fractions. With
meter sticks we deal with decimal points, moving
them left or right as we convert, no fractions
involved.
Below is part of
a ruler, exaggerated in size for clarity:
The
top scale is in inches (in), that means the major
marks, labeled with numbers, represent 1 inch
in length. We use the double quote " to mean
inches, so each major mark represents 1".
Counting all the marks between 0" and 1' we find 16
altogether that are equally spaced between 0" and
1'. So, each mark represents 1/16".
Continuing, taking 2 of these marks at a time
we see there are 8 such pairs between 0" and 1".
As you can see this mark represents 2 of the 1/16"
marks and there are 8 of them, so each of these
8 marks represents 1/8". Likewise, the
3rd larger mark groups 2 of these 1/8" marks and 4
of the 1/16" marks and there are 4 of these groups,
so this mark must represent 1/4". Finally, the
next larger mark represents 2 of these 1/4" marks, 4
of the 1/8" marks and 8 of the 1/16" marks, and 2 of
these marks represents 1", so one mark would
represent 1/2".
Moving the
other way, each mark in order from larger to
smaller represents 1/2 of the previous mark.
Starting with 1", then next largest mark is 1/2 of
1" or 1/2". Now moving to the next mark,
that mark is 1/2 of the 1/2" mark, so it represents
1/4". Continuing, the next mark is 1/2
of 1/4" so it represents 1/8". And finally the
smallest mark is 1/2 of 1/8" which is 1/16".
We say that
this scale is calibrated to the nearest 1/16".
This means that it's precision is 1/16". But, since
we cannot measure to 1/32" or smaller, the best we
can say is a measurement made with this ruler is
accurate to
1/16" ± 1/32".
The error
is expected to be 1/2 above or below the smallest
unit of measurement with any instrument. One
cannot speak of an exact measurement, only of an
approximation with an error bound.
Now, let's
investigate the metric side of that ruler.
Each major
mark represents 1cm. There are 10
smallest marks in equally spaced each of these
marks. So each of these marks would represent 1/10
of 1cm. 1/10 of 1cm is 1mm. Notice there
are marks that group 5 of these smallest marks,
these next larger marks would represent 1/2 of 1cm
which is 0.5cm.
In the
English system we use powers of 2 to measure
lengths; In the metric system we use powers of 10.
The metric system is easier since powers of ten
require only movement of the decimal left or right.
The English system requires a multiplication or
division by some power of 2 then a conversion to a
decimal.
Example, say we
measured a length of 1 and 5/16" and wanted to
express this in feet. We know there are 12" in
a foot. So we need to divide 1 5/16" by 12".
Let's see, 1 5/16" is 21/16"; now divide this by 12
and we get (21/(16*12)) = 21 / 192 ft = (I need my
calculator here...) = 0.0625 ft.
Hmm, looking at
the ruler above, 1 5/16" looks to be approximately
32mm. 1mm is 1/1000 of a meter, that is 1mm =
0.001m. If we have 32 of these mm then that
must be 0.032m. And we're done. Let's
look at this a bit closer.
1m = 10dm
(dm = decimeter, dec --> 10)
1m = 100cm
(cm = centimeter, cent --> 100)
1m = 1000mm (mm =
millimeter, mil --> 1000)
We rarely speak of
decimeters. So let's focus on cm and mm.
So
we measured 32 mm. We want this in
meters.
32.0mm a mm is smaller
than 1m by 3 decimal places
look, 1m =
1000 mm, divide by 1000, and we get 1/1000 m = 1 mm.
Dividing
any number by 1000 involves shifting the decimal to
the left by 3, or 3 powers of 10 (1000 = 10 *
10 * 10, 3 tens, 3 decimals)
So.
3.20 (1 shift) 0.320 (2
shifts) 0.0320 (3 shifts)
0.0320 m
Let's try
another. What is this measurement in cm?
32.0 mm = 3.20 cm
Ok, since 1m = 100cm and 1m = 1000mm a meter
is a meter so
100cm = 1000mm, dividing by 100 we get 1cm =
10mm (which we saw in the ruler above)
So, 1mm = 1/10 cm, and we have 32 mm, dividing by 10
moves the decimal to the left by 1.
32.0 mm = 3.20 cm.
One last
example. Say we have 1.32 m.
How may cm is this?
1.32 m = 132 cm
1m = 100cm.
we multiply by 100 that is, shift the decimal to the
right by 2.
1.32m = 13.2 dm (1 shift) = 132 cm (2 shifts)
Converting between the English
system and the metric system requires a calculator.
For length we have
2.54 cm = 1"
(this should be committed to memory)
all other length conversions
can then be done.
For
example:
12in * 2.54 cm/in = 30.48cm
(0.3048 m, 3048 mm)
1 yard = 3 ft, so 1 yard = 3 * 30.48cm = 91.44cm
(0.9144m)
(so a yard stick is slightly smaller than a meter
stick. )
A 100 yard football field would be 91.44m.
1 mile is 5280ft/mi * (0.3048 m / ft) = 1609.3m
We can 1km = 1000m, so we have 1609.3/1000 =
1.6093km
1 mile = 1.6093 km.
If you look at the speedometer 60 MPH is just
under 100 KmPH.
Other metric
scales follow this moving decimal pattern where the
units are all powers of ten and all use the same
prefixes.