Precise and accurate measurements are so important that an entire
field of study, metrology, is dedicated to the science of measurement.
In the United States, the National Institute of Standards and
Technology, or NIST, is an agency of the U.S. Department of Commerce's
Technology Administration established by Congress to promote U.S.
economic growth by working with industry to develop and apply
technology, measurements, and standards.
 |
Figure 1: For nearly
100 years the
NIST laboratories have worked
with industry and government
to advance measurement science
and develop standards. |
Since measurements cross international borders, so must measurement
standards. The International Bureau of Weights and Measures determines,
conducts and monitors international metrological research. The
units of measurement this prestigious group administers are the
accepted International System of Units, or SI, which is based
on the metric system. Scientists through out the world use the
SI units for their measurements. Only the U.S. still uses traditional
units in addition to SI units. Just like there are rules for grammar
and spelling, there are specific rules for writing SI units. For
example, when writing a quantity and a unit, there is a space
between the two, such as 15 m, not 15m.
There are seven base SI units.
| Physical Quantity |
Name of Unit |
Abbreviation |
| Mass |
kilogram |
kg |
| Length |
meter |
m |
| Temperature |
Kelvin |
K |
| Amount of substance |
Mole |
mol |
| Time |
second |
s |
| Electric current |
ampere |
A |
| Luminous intensity |
candela |
cd |
Table 1: The seven base
SI units.
There are several SI units that can be expressed in terms of one
or more of the seven base SI units listed in the above table,
these are derived SI units.
| Quantity |
Definition |
Derived unit name |
| Area |
length times length |
m2 |
| Volume |
area times length |
m3 |
| Force |
mass times acceleration |
(kg x m)/s2 (Newton, N)
Newton: a unit of force, defined as that force which gives
a mass of one kilogram an acceleration of one meter per
second per second |
| Energy |
force times distance |
(kg x m2)/s2 (Joule, J)
The joule, a unit of energy, defined as the work done when
the point of application of a Newton is displaced one meter
in the direction of the force |
Table 2: Some derived
SI units.
A problem with the SI units of measurement is that very large
numbers or very small numbers must often be used to express many
measurements. When the size of the numbers become cumbersome,
then scientific notation is used.
Scientific notation makes perceiving very large and very small
numbers much easier. For example, there are about 602 200 000
000 000 000 000 000 atoms in one mole of a substance. The same
number can be expressed in scientific notation as 6.022 x 1023.
 |
| Figure 2: The definition
of a mole. If 10-3 moles is a millimole, if 10-6
moles is a micromole, then what is a "guacamole"?
|
Not only is this easier to write, but it gives us an immediate
idea of just how large this number is. The exponent, the small-size
number in the upper right-hand corner of the 10, is 23. The
exponent of 23 tells us immediately that there are 23 place-holding
digits to the right of the first digit of the coefficient. In
this example, 6.022 is the coefficient. To write very small
numbers, such as the diameter of a carbon atom, we use negative
exponents. The diameter of a carbon atom is about 0.0 000 000
001 m. In scientific notation, a carbon atom is 1.0 x 10-10
m in diameter.
When converting numbers to scientific notation follow these
simple rules:
- Determine the power of ten:
- For numbers greater than or equal to 1, add the number
of places to the right of the first digit. (Note: it
is possible to have a 0 exponent,100=1).
 |
| For example, there are 7 places to the right of
the first digit in the number, 12 000 000
and the power of ten is 107. |
- For numbers less than 1, subtract the number of places
to the left of the first nonzero digit.
 |
| For example, there are 7 places to left of the
first non-zero digit in the number, 0.00 000
012 and the power of ten is 10-7. |
- Determine the coefficient:
- Put a decimal point to the right of the first non-zero
digit and include any digits to the right that are considered
significant.
 |
For 12 000 000, put 1.2 as the coefficient.
|
 |
For 0.00 000 012, put 1.2 as the coefficient.
|
- Combine the power of ten and the coefficient into one expression.
-
 |
12 000 000 is written 1.2 x 107.
|
-
 |
0.00 000 012 is written 1.2 x 10-7.
|
Metric Prefixes
Sometimes it is desirable to express the magnitude of numbers
with words or symbols rather than scientific notation. These words
represent powers of ten and are typically applied as prefixes
to units of measurement. Some common prefixes are listed below.
|
Factor
|
Power of ten
|
Prefix
|
Symbol
|
Example
|
|
1 000 000 000 000
|
1012
|
tera
|
T
|
teraherz
|
|
1 000 000 000
|
109
|
giga
|
G
|
gigabyte
|
|
1 000 000
|
106
|
mega
|
M
|
megabyte
|
|
1 000
|
103
|
kilo
|
k
|
kilogram
|
|
100
|
102
|
hecto
|
h
|
hectometer
|
|
10
|
101
|
deka
|
da
|
dekagram
|
|
0.1
|
10-1
|
deci
|
d
|
deciliter
|
|
0.01
|
10-2
|
centi
|
c
|
centimeter
|
|
0.001
|
10-3
|
milli
|
m
|
milliliter
|
|
0.000 001
|
10-6
|
micro
|
m
|
micrometer
|
|
0.000 000 001
|
10-9
|
nano
|
n
|
nanometer
|
|
0.000 000 000 001
|
10-12
|
pico
|
p
|
picogram
|
|
0.000 000 000 000 001
|
10-15
|
femto
|
f
|
femtosecond
|
|
0.000 000 000 000 000 001
|
10-18
|
atto
|
a
|
attomole
|
Table 3: Metric Prefixes
for some Powers of Ten.
No matter which SI unit is used to report a measurement, there
is an element of uncertainty. Every measurement is limited by
the reliability of the measuring instrument and the skill of
the operator.
The metric unit selected for use is based upon the size of the
object that is being measured. Often however, the quantity used
must be converted to a different size unit. SI unit conversion
can be done by simply moving the decimal to increase or decrease
the size of the number. Then the appropriate prefix for the new
unit must be given.
 |
Figure 3: Metric
unit conversion
is simply a matter of moving the
decimal in the correct direction
and the correct number of places. |
Unit conversion is done by taking the numerical difference
between the exponents and moving the decimal in the appropriate
direction. If the new unit is smaller, then you will need more
of them, so the decimal will be moved to the right. If the new
unit is larger, you will need fewer of them, so the decimal
will be moved to the left.
Go through the following activity to practice making conversions.
This activity takes you step-by -step through the conversion
process. Once you have mastered moving the decimal the right
direction and the right number of places, then select the "Advanced
Practice" button to try problems without the step-by-step instructions.
Precision and accuracy are often used as synonyms but when referring
to uncertainty in measurement each term means something different.
It is possible for a measurement to be precise or accurate or
both or neither. If the measurement is close to repeated measurements
of the same value then it is considered precise. If the measurement
is close to the true value (or an expected value if the true value
is unknown) then it is considered accurate. Precise numbers are
often also accurate, but it is possible to repeatedly measure
the same error and come up with precise, but inaccurate numbers.
- Precision is how closely repeated measurements match
each other.
- Accuracy is how closely a measurement matches the
correct or expected value.
|
 |
|
Figure 4: A ball
player's height can be measured precisely, but still not
be accurate. |
|
|
Figure 5: The above
image shows the internals of a RAM in a computer. We shall
try to measure the length of the pink silicon rectangle
(shown in box).
|
|
Figure 6: (a)These
measurements represented by the red arrows are accurate
because they are close to the actual length of the rectangle
and they are precise because they are nearly equal to each
other.
(b)The length of these measurements are close to each other
but not to the actual length of the rectangle. They are
precise but not accurate.
(c)These measurements are neither close to the actual length
of the rectangle nor close to each other. They are neither
precise nor accurate.
|
In this activity, see how accurate and precise you can measure.
Food for thought: How would changing the resolution of
your screen affect precision and accuracy? How do the camera or
microscope used to take the picture affect precision and accuracy?
Some numbers are exact. There are exactly 100 cents in a dollar,
no more and no less. By definition there are exactly 100 centimeters
in a meter and 1000 grams in a kilogram. Usually when we measure
something however, the numbers are not exact. How close they
are to exact is a matter of how we measure them and what equipment
we use to measure them. If we measure the mass of a mouse as
75g, does the mouse weigh 75g exactly, or could it be 75.3g
or 75.34g or 75.345g?
If we could count every single atom of the mouse and add up
all the weights of the atoms, we might consider this exact but
this measurement would still depend on how exact our measurements
of the atoms were. Since there is no laboratory equipment that
can make exact measurements like this, we must quantify how
exact our measurements are by using significant figures. Significant
figures are the digits in a number that we can be certain of,
plus one additional number we estimate. If we report the mass
of the mouse to be 75.3g we are sure of the 7 and the 5 and
we estimate the 3. Our estimate is usually based on an average
of measurements and the degree of disagreement between each
of the measurements.
It is usually easy to tell how many significant figures a measurement
has. Unless a number has zeros in it the number of significant
digits is simply the number of digits.
There are four rules for numbers with zeros:
 |
1) Zeros in the middle of number are always significant.
For example, 101 m has 3 significant figures.
2) Zeros at the beginning of a number only hold
places to the right of the decimal point they are not
significant. So 0.00 101m has only three significant figures.
3) Zeros at the end of a number and after a decimal
point are significant. There would be no reason to show
them if they were not. 0.1 010 has four significant figures.
4) Zeros at the end of a number and before a
decimal point may or may not be significant. Therefore
10100 m may have 3, 4 or 5 significant figures depending
on whether or not the last zeros are part of the measurement
or only placeholders.
|
There are two more important rules when using significant figures:
 |
Whenever we multiply or divide numbers with a finite number
of significant figures the answer cannot have more significant
figures than any of the original numbers. If our mouse is
expected to gain 21% more weight in a month our calculator
says 75.3g x .21= 15.813g is the weight gain. Since .21
only has two significant digits we must report an expected
weight gain of 16 g by rounding to two significant figures. |
 |
Whenever we add or subtract numbers with finite significant
figures the answer cannot have more digits to the right
of the last significant figure than any of the original
numbers. If our mouse eats a piece of cheese weighing 2.465g
our calculator says the mouse weighs 75.3g + 2.435g = 75.735g.
We must report only three significant figures so after rounding
the weight would be 75.7g |
Practice determining significant figures in this activity.
Q. With this tool of measurement (your computer screen) what
is the finest grid you can possibly use?
A. The pixels of the monitor are the finest grid. A higher
resolution monitor would allow a more exact measurement.
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