Significant Figures, Measurement, and
Calculations in Chemistry
Carl Hoeger, Ph.D.
University of California, San Diego
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Part 1: Measurements, Errors, and
Significant Figures
Carl Hoeger, Ph.D.
University of California, San Diego
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Measurements
•  Science is about making observations and measurements; how
do we convey to someone just HOW good/sure of our data we
are?
•  All quantitative measurements are made using some sort of
device (analog or digital). Need value, units, and error.
•  All data collection has errors associated with it:
–  Determinate Errors: aka systematic errors; have a definite
direction and magnitude and have an assignable cause.
–  Indeterminate Errors: aka random errors or noise; errors
that simply arise from uncertainties in a measurement,
sloppy techniques, anomalous physical and environmental
factors; these errors are typically untraceable.
Determinate errors can be eliminated; Indeterminate errors can not.
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Measurements (cont.)
•  Consider the measurement of a length of bar:
•  Is it 8.3, 8.32, or 8.323? Or 8.322? What are the units?
Error? Depends!
•  Calibration
•  Multiple measurements to give means and standard
deviations:
x=
∑x
i
N
; s=
∑ (x –x );
i
N –1
CV = s × 100
x
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Accuracy and Precision
Accuracy: closeness of measurement to the true value
Precision: closeness of individual measurements to one another
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Significant Figures-1
Two types of numbers exist: measured and exact numbers.
•  Measurements or “determined” values contain significant
figures and must convey to the reader the ‘certainty’ of
your measurement. This is where significant figures
(SigFigs) comes in.
•  Exact values (exact numbers) have an unlimited number
of significant figures and never detract from a
measurement or its subsequent modification or use.
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Exact Values
Exact numbers can either be cardinal (counted) or defined.
•  Cardinal or counted values represent values that, by their very
nature or how they are obtained, are indivisible without
destroying the identity of the unit.
10 dogs or 85 cars;
123 molecules; 13 silver atoms; 2500 electrons;
2 chloride ions in 1 calcium chloride.
•  Defined values are those for which the value has been exactly
defined. “1” in these expressions is EXACTLY 1
1 in = 2.54 cm; 1000 m = 1 km; 1 mg = 0.001 g; 1 lb = 16 oz
All metric to metric or English to English conversions
•  What about constants? c = 2.998 x 108 m/sec?
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Measured Values
•  Determined values have SigFigs associated with them.
•  Reported values must convey the level of precision of the
measurement to a detached third-party (i.e. an unseen
colleague).
•  A measured value is comprised of digits we know for certain
plus ONE we are uncertain of (the last digit to the right)
3
2
1.66 mL
1
Consider the graduated cylinder to the left; we
know the first two digits with certainty (you can
read them directly from the markings) but the last
“6” is estimated. Another person might read it as
a “5” or a “7”, hence it is uncertain.
By convention, the last digit is considered to be
variable by ±1 unit (unless specified otherwise)
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Significant Numbers/Digits
When given a measurement or a determined value, what digits are
significant?
•  All non-zero integers always count as sig figs;
•  There is assumed to be a ± 1 difference in the last significant
digit shown;
•  Zeros are tricky--depends on the type:
–  Captive zeros: always significant;
1001 cars; 1.7007 cm
–  Leading zeros: never significant;
0010 screws; 0.00021 L
–  Trailing zeros: only significant if a decimal point is present
210. Atoms; 213.0002 mm; 0.00056700 Angstroms
•  NOTE: for any number expressed in scientific notation, all
digits shown are significant (except leading zeros)
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Sig Figs: Calculation Considerations
Determined by the type of mathematical operation carried out
•  Multiplication or Division: least precise number determines sf in answer;
14.3 × 0.012
4
= 0.04 29 (1sf )
•  Addition or Subtraction: Answer can have no more decimal places than the
one with the fewest;
(14.3 – 0.012) + 8.0221 = 22.3101 (1dp )
•  Logs: No more digits to right of decimal than sf in original number;
log1.43 × 10 –2 = –1.844 66 = –1.845 (3 dp; note : the 1 is not a sf )
•  Antilogs: No more sf in answer than digits to right of decimal in original
number;
antilog –2.86 = 10 –2.86 = –1.380 × 10 –3 = –1.4 × 10 –3 (2 sf )
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Rounding
A number is rounded off to the desired number of significant figures by
dropping one or more digits to the right. If you are rounding to the tens,
hundreds place or higher, you must put zeroes in the lesser places (the ones
place, for example) to indicate to what place you have rounded. Use the
following guidelines when rounding off numbers:
–  When the first digit dropped is less than 5, the last digit remains
unchanged.
–  When the first digit dropped is 5 or greater, the last digit retained is
increased by 1.
–  When rounding, ignore ALL digits after the first digit you are dropping
in making your decision.
–  Always round at the end of any set of calculations.
Round each to 3 sf: i) 3203; ii) 1.768; iii) 0.01455; iv) 24.74999
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Combined Calculations
In calculations involving BOTH addition/subtraction AND
multiplication/division, significant figure considerations must
be noted before and after each calculation involving addition
or subtraction. You still round at the end. A simple rule of
thumb is as follows:
  Take note of how many sig figs the multiplication and division
steps allow;
  Take note of how many decimal places the addition and
subtraction steps allow;
  Your answer can have no more decimal places than allowed by the
addition/subtraction but no more sig figs than allowed by the
multiplication/division. Usual way to do this is to get your answer,
set the correct number of sig figs and then lessen the number of
decimal places if necessary.
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Scientific Notation
When expressing an extreme large number such as the number of atoms in a
mole, or a very small number such as the mass of a single atom, scientists use
scientific notation. The basic format of scientific notation is M x 10n, where M
is a real number between 1 and 10 and n is a whole number.
• 
• 
• 
• 
• 
• 
• 
100 = 1 (note: any number to the 0 power is 1)
101 = 10
102 = 10 * 10 = 100
103 = 10 * 10 * 10 = 1000
10-1 = 1 / 10 = 0.1
10-2 = 1 / 10 / 10 = 0.01
10-3 = 1 / 10 / 10 / 10 = 0.001
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Scientific Notation (con’t)
For practical reasons it is common to express all numbers that are larger than
1000 or smaller than 0.01 in scientific notation.
When converting a number to scientific notation, move the decimal place to
the right or to the left until what remains is a number to the left of the
decimal place between 1 and 9.
Moving the decimal to the right will result in a negative power of ten, to the
left in a positive power of ten:
Y Y Y Y Y. X X X X X
Express 3456 in scientific notation:
Express 0.0006700 in scientific notation:
3 4 5 6. = 3.456 x 103
0.0 0 0 6 70 = 6.70 x 10-4
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Power Notation: pX values
It is not always easy or convenient to use scientific notation,
especially when comparing or tabulating long lists of
numbers in scientific notation; used primarily when
discussing acidity and basicity.
•  Use pX values; by definition:
pX = – log X
X = 10 – pX
Therefore:
X ⎯⎯
→ pX
pX ⎯⎯
→ X
6.24 × 10 5 ⎯⎯
→ –5.795
11.59 ⎯⎯
→ 2.6 × 10 –12
3.1 × 10 –8 ⎯⎯
→ 7.51
–2.115 ⎯⎯
→ 1.30 × 10 2
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Units
•  Just as important as the value are the units associated with that
value, as they tell us about what we are measuring!
•  Units need to be as detailed as necessary:
–  12.7 lbs vs. 12.7 lbs of hamburger vs. 12.7 lbs of lean hamburger;
–  3 atoms vs. 3 sodium atoms vs. 3 sodium ions
•  SI or metric units almost exclusively in chemistry.
•  Fundamental units vs. Derived units:
–  Length: meter (m); Mass: kilogram (kg); Time: second (sec);
Temperature: Kelvin (K); Chemical amount: mole (mol)
–  Energy: joule (J); Pressure: atmosphere (atm); Volume: liter (L)
•  Units quite often begin with modifiers (“power prefixes”) to
relay size of quantity without having to use scientific notation:
–  5.32 x 103 J or 5.3 kJ
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Common Unit Modifiers (“Power Prefixes”)
giga - G - 109
centi - c - 10-2
mega - M - 106
milli - m - 10-3
kilo - k - 103
micro - ! - 10-6
deci - d - 10-1
nano - n - 10-9
pico - p - 10-12
fempto - f - 10-15
atto - a - 10-18
yocto - y - 10-24
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Questions? Comments?
This presentation has been brought to you through the
generous support of the University of California, San
Diego’s Instructional Improvement Program.
If you have questions, comments, or wish to use these
presentations in your University’s courses, please feel free
to send an email to Dr. Carl Hoeger at
[email protected]
This is episode “SigFig1”; if you have specific comments
or questions regarding this episode please note this in your
email
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