Metric Measurements
Throughout this semester, you will make measurements using Metric units. The metric system takes a base unit
of measurement (length, volume, mass, time, etc.), and either subdivides the unit into smaller units, or
multiplies the unit to make larger units.
Commonly Used Base Units
Dimension
Unit
Length
Meter
Mass
Gram
Volume
Liter
Time
Second
Energy
Joule
Symbol
m
g
L
s
J
The metric system divides or multiplies the base unit using modifiers that are multiples of 10.
Example:
1 mm (millimeter) = 1/1000th of a meter, or 1000 mm = 1 m
1 km (kilometer) = 1000 meters, or 1 m = 1/1000th of a kilometer
The kilo prefix multiplies the base by 1000, while the milli prefix divides the base unit by 1000.
This is quite unlike measurement systems we use in everyday life, where units can be multiplied or
divided by various multiplies.
Example:
1 foot = 12 inches
1 minute = 60 seconds
1 gallon = 4 quarts
1 mile = 5280 feet
×base (Expanded)
Examples of the Relationship between the base
unit and prefixed unit
G
M
k
h
da
giga
mega
kilo
hecto
deca
×109
×106
×103
×102
×101
1Gm=1,000,000,000m (109)
1ML=1,000,000 L (106)
1kg=1000 g (103)
1hm=100m
1daL=10L
d
c
m
μ
n
p
deci
centi
milli
micro
nano
pico
×10-1
×10-2
×10-3
×10-6
×10-9
×10-12
×1,000,000,000
×1,000,000
×1,000
×100
×10
Base
×1/10
×1/100
×1/1000
×1/1,000,000
×1/1,000,000,000
×1/1,000,000,000,000
CHEM 30A, William E. Trego, Laney College
1m=10dm
1g=100cg
1m=1000mm
1L=1,000,000 μL (106)
1m=1,000,000,000nm (109)
1g=1,000,000,000,000pg (1012)
←Divides Base
Symbol prefix
Multiplies base→
Metric Prefixes
Listed below are a number of the metric modifiers, or what we might call prefixes (since when writing the
symbol for a modified unit, we write the prefix with the base). (You will be responsible for knowing the
boldfaced prefixes.)
Accuracy, Precision and Error in Measurements, and Significant Figures
There are no exact measurements. There are exact numbers. For instance, the number of students in the
classroom can be determined without uncertainty. Measurements are not exact numbers, there is always a
degree of uncertainty (some error) associated with a measurement. A scientist seeks to know the degree of
certainty in their measurements, and ensures that the degree of certainty is acceptable for their work.
Example:
Suppose we attempt to measure the bar below with the ruler. We can be certain (provided that the ruler is
calibrated properly) that the length is greater than 3.5, but less than 3.6. The 3 and 5 are certain digits. But, we
can estimate an extra digit for the measurement. We might attempt to read between the markings (interpolate),
and report the length as 3.52. A measurement is typically reported with all the certain digits, plus one additional
uncertain (estimated) digit. The uncertain digit is the last one (right most) in the measurement.
The uncertainty (or potential error) of the measurement below enters in the hundredths place (the 2).
Related to the question of certainty are the concepts of accuracy and precision. Accuracy refers to the
“closeness” of a measurement to the “true” value. Precision refers to the reproducibility of a measurement—
whether the same measurement can be attained if it is repeated. It is important to realize that a measurement
that expresses a high degree of certainty could potentially be inaccurate. Suppose that we use a balance that is
readable to 1/10th of a milligram (0.0001 g)…this balance expresses a high degree of certainty. But, if the
balance is not properly calibrated (i.e. set to a known standard) it may deliver a mass that expresses a high
degree of certainty but is inaccurate. For example, if the balance reads 0.9555 g when it should read 1.0000 g, it
is not sufficiently accurate for some experiments.
Significant Figures (Digits)
The digits in a measurement that express the certainty (plus the one uncertain digit) are referred to as the
significant digits.
When examining a measurement, we assume that all the nonzero digits are significant digits (i.e, tell us
something about the certainty of the measurement). For instance, in the length 11.234 m, all the digits are
significant.
The task of determining the significant digits becomes more complicated when dealing with zeros. Zeros
frequently are significant, but sometimes they do not express the certainty of the measurement, and instead
serve a different role.
Take for instance, the volume 10.0 mL. In this measurement, we assume that (at best) the measurement is
certain to ± 0.1 mL (to the last digit in the measurement). All the zeros in this case tell us something about the
certainty of the measurement. 10.0 mL expresses a higher degree of certainty than 10 mL (which is certain only
to ± 1 mL). 10.00 mL expresses an even higher degree of certainty.
Some zeros do not express the certainty of a measurement. These zeros are not significant.
CHEM 30A, William E. Trego, Laney College
Take for instance, the mass 0.000000125 g. The seven zeros that precede 125 are placeholders, and do not
express the certainty of the measurement—they tell us that there are no ones, not tenths, no hundredths, no
thousandths, etc.
Some zeros are ambiguous—without more information we cannot know whether these zeros are significant.
Take for instance, the distance 3000 km. Depending upon how the measurement was made, the certainty could
be ± 1 km, ± 10 km, ± 100 km, or even ± 1000 km. What we do not know is whether the zeros are expressing
the certainty of the measurement, or are just serving as placeholders.
As we can see, zeros can be confusing—we need some rules to determine whether a zero is significant.
Rules for Determining the Number of Significant Figures in a Number
1. All nonzero digits are significant. 159 g has a total of three significant figures.
2. Zeros in between nonzero digits are significant. 2005 L has four significant digits.
3. Trailing zeros (i.e. zeros that follow the last nonzero digit in a number) are significant, if the number has
a decimal point. 250.00 cm has five significant figures. All three zeros are significant.
4. Trailing zeros in a number without a decimal point are ambiguous, and are treated as not significant.
25,000 km has two significant figures. The trailing zeros are not significant in this case.
5. Leading zeros (i.e., zeros that lead up to the first nonzero digit) are never significant. 0.0011 m has only
2 significant digits. The zeros in this case are simply placeholders, and not significant in terms of the
certainty of the measurement.
Rules for Determining the Number of Significant Digits in the Results of Calculations
Multiplication/Division
When multiplying or dividing measurements, the result can have no more total significant figures than the
starting measurement with the fewest significant figures.
Example:
1.25 cm × 3.0 cm = 3.75 cm2 (before rounding)
↑
↑
3 significant digits 2 significant digits
Since the starting measurement with the fewest significant figures has 2 total, the final product is rounded to 2
significant digits. 3.75 becomes 3.8 cm2
Addition/Subtraction
When adding/subtracting measurements, the sum/difference can express no more certainty than the starting
measurement that expresses the lowest degree of certainty.
Example:
1.799 g ← certain to the thousandths place
+ 0.1 g ← certain to the tenths place (lower degree of certainty)
¯¯¯¯¯¯¯¯
1.899 g (before rounding)
Since 0.1 g expresses a lower degree of certainty than 1.799 g, the sum can express certainty only to the tenths
place. 1.899 g is rounded to 1.9 g.
CHEM 30A, William E. Trego, Laney College