Using Scientific
Measurements
We have talked about measured quantities:
Quantities that can be measured directly.
Examples – Length, Mass, Volume…
What about quantities that can not be measured
directly, or, derived quantities:
Quantities found by the mathematical
manipulation of one or more measured
quantities.
Examples – Area, Volume, Density…
How can Volume be both measured and derived?
Density:
• Derived quantity from mass and volume
• Physical property of a substance
• Is temperature dependent
m
d=
V
Use the picture below to calculate the
density of a piece of chewing gum that has a
mass of 25.3 grams.
Sample Problem:
What is the density of aluminum if a 3.0
cm by 3.0 cm by 2.5 cm block has a mass of 58.7
grams?
Notice, the volume can be calculated
from a geometric formula and the mass can be
measured directly. But what about a wad of
chewing gum. How do we finds its volume to
find its density?
Accuracy vs. Precision
The goal of all measurements is to be
both accurate and precise!
• Accuracy refers to the
proximity of a measurement to
the true value of a quantity.
• Precision refers to the
proximity of several
measurements to each other.
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Remember, each measured quantity must
have an uncertain digit.
The uncertain digit relays the accuracy of
a measurement.
How then does the accuracy of a set of
measurements become relayed in a calcualted,
or derived, quantity?
A derived quantity can not be more accurate
than the least accurate measurement!
Determining Significant Figures
1. All nonzero digits are significant.
2. Zeroes between two significant figures
are themselves significant.
3. Zeroes at the beginning of a number
are never significant.
4. Zeroes at the end of a number are
significant if a decimal point is written
in the number.
5. Exact numbers are infinitely significant.
Significant Figures
• When addition or subtraction is
performed, answers are rounded to the
least significant decimal place.
• When multiplication or division is
performed, answers are rounded to the
number of digits that corresponds to the
least number of significant figures in any
of the numbers used in the calculation.
Significant Figures
• The term significant figures refers to
digits in a number that were
measured.
• When rounding calculated numbers,
we pay attention to significant
figures so we do not overstate the
accuracy of our answers.
1. How many significant
figures are in each of the
following measurements?
24 mL
2 significant figures
3001 g
4 significant figures
0.0320 m3
3 significant figures
6.4 x 104 molecules 2 significant figures
560 kg
2 significant figures
Significant Figures
Addition or Subtraction
The answer cannot have more digits to the right of
the decimal
point than any of the original numbers.
89.332
+1.1
90.432
3.70
-2.9133
0.7867
one significant figure after decimal point
round off to 90.4
two significant figures after decimal point
round off to 0.79
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Significant Figures
Multiplication or Division
The number of significant figures in the result is set
by the original number that has the smallest number
of significant figures
4.51 x 3.6666 = 16.536366 = 16.5
3 sig figs
round to
3 sig figs
6.8 ÷ 112.04 = 0.0606926 = 0.061
2 sig figs
round to
2 sig figs
Rules for Rounding
• If the number to be retained is followed by a
number larger than 5, round up.
• If the number to be retained is followed by a
number less than 5, drop.
• If the number following the number to be
retained is 5 followed by any non-zero number,
round up.
• If the last number to be retained is a 5 and is
followed by a zero, then always make it even.
Perform each of the following, rounding the
answer to the correct number of sig. fig’s.:
1. 298 g – 43.7 g
2. 4.218 cm x 6.6 cm
3. 4.23 m2 ÷ 18.941 m
4. 0.0653 g + 0.08538 g + 0.07654 g + 0.0432 g
5. 50 L x 5.23 L
6. 85.621 s – 5.5 s
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