Chapter 1: Measurements

Chapter 1: Measurements
The metric system
Becoming more common in our country.
The SI system
Developed in 1960, also called the
international system of units.
Set of seven standard quantities of which
all others can be derived.
Common Metric Units
Metric and SI unit is the Meter (m)
1 m = 39.4 inches
1 inch = 2.54 centimeters (cm)
Amount of space occupied by a substance
Metric unit = Liter (L), SI unit = m3
1 L = 1.06 quarts
946 milliliters = 1 quart
Common Metric Units
Metric unit is the gram (g), SI unit is the
kilogram (kg)
1 kg = 2.2 pounds
454 g = 1 pound
Metric and SI unit is the second (s)
Common Metric Units
Indicates how hot or cold an object is
Metric unit is Celsius (C) and SI unit is
Kelvin (K)
These will be discussed in more detail in
Chapter 2.
Scientific Notation
Used when numbers are really small or
really large.
General format:
A x 10n
A is a number between 1 and 10.
n is the exponent on 10 (called the power)
and is equal to the number of places that
the decimal place is moved.
Scientific Notation
0.000058 = 5.8 x 10-5
• The decimal is moved five places to the right,
hence the –5 power. Note that small numbers
(0< x <1) will always have a negative exponent.
58,000 = 5.8 x 104
• The decimal is moved four places to the left,
hence the power of 4 is used. Note that large
numbers (>1) will always have a positive
Scientific Notation
On your calculator, use the EE or EXP
key when entering these numbers.
(1.8 x 105) x (3.2 x 10-3) =
576 or 5.76 x 102
Measured & Exact Numbers
Measured numbers are those which come
from using any kind of a measuring device.
Examples are: ruler, graduated cylinder,
thermometer, scale, etc.
Include both certain digits and one uncertain digit.
Certain digits are digits that all would agree on.
Uncertain digit is the last digit that requires YOU
to make a “best guess.”
Measured & Exact Numbers
“Best Guess” depends on the smallest increments on
the measuring device.
A ruler may have markings for every 0.1cm. Your
“best guess” might be to 0.05cm or even 0.02cm.
Measured & Exact Numbers
A graduated cylinder
– depends on size.
A 10-mL has
markings every 0.2mL.
A 50-mL has
markings every 1-mL.
Measured & Exact Numbers
A Thermometer
has markings
every degree.
Measured & Exact Numbers
If a measurement results where it is exactly
lined up with a mark, then a zero at the end
may be needed.
Not everyone will agree on the “best guess,”
which means that all measured numbers have
some uncertainty.
Digital devices – the last digit in display is
Measured & Exact Numbers
An exact number is a either a counted
number or an established relationship
between two units.
Examples are:
12 computers in the room
15 apples in the bag
1 foot = 12 inches
1 pound = 16 ounces
Learning Check
Are the following numbers exact or
There are 50 pages in a book
A coin weighs 2.87 grams
There are 100 centimeters in one meter
A graduated cylinder contains 40.0mL of
Significant Figures
Goal: Report answers from calculations by
rounding the answer to the correct number of
significant figures.
Accuracy refers to measurement(s) that are
close to the true (accepted) measurement.
Precision refers to the agreement of values
obtained while repeating the same
In the lab, we desire both of these.
Significant Figures
Rules are:
A number is significant if it is:
A non-zero digit
A zero between two non-zero digits
A zero at the end of a decimal number
Any digit in the coefficient of a number written in
scientific notation
A number is not significant if it is:
A zero at the beginning of a decimal number
A zero used as the placeholder in a large number
without a decimal point.
Learning Check
How many significant digits do each of the
following measured numbers have?
2.50 x 10-8
Significant Figures in
A calculation (using measured numbers)
is only as accurate as the number that
had the least number of significance.
Calculators almost NEVER provide
answers with the proper significance.
Thus, answers will need to be rounded
(or possibly require additional zero’s).
Rounding Numbers
If the first digit to be dropped is a 4 or less, it
and all other following digits can be dropped.
If the first digit to be dropped is a 5+, then the
last retained digit is increased by one.
If the digit to be dropped is exactly 5 (nothing
after it), then round up if it makes the digit even
or down if that makes the digit even.
Learning Check
3.46, round to two s.f.’s.
54.48, round to two s.f.’s.
135.51, round to three s.f.’s.
8.74528, round to three s.f.’s.
Multiplication & Division
The final answer is rounded to have the same
number of significant digits as the
measurement with the fewest s.f.’s.
Example: 24.65 x 0.67 = 16.5155 (calc.)
Example: (2.85 x 67.4) / 4.39 = 43.756264 (calc.)
Example: (8.00) / (0.250) = 32 (calc.)
Addition & Subtraction
The final answer is rounded to have the same
number of decimal places as the measured
number with the fewest decimal places.
Example: 2.045 + 34.1 = 36.145 (calc.)
Example: 255 – 175.65 = 79.35 (calc.)
Example: 89.15 – 82.95 = 6.2 (calc.)
Mixed Calculations
Apply the rules for each type of calculation.
Don’t round intermediate answers – round
only at the very end.
Example: (23.8 + 4.25) / 67.85 = 0.413411938
Example: (17.92 – 16.82) x 0.01957 =
SI & Metric Prefixes
To increase or decrease metric units, prefixes
are used as a multiplier.
Prefixes that increase the size:
G = giga,
M = mega, 106 or 1,000,000
k = kilo, 103 or 1,000
Prefixes that decrease the size:
d = deci, 10-1 or 0.1
c = centi, 10-2 or 0.01
m = milli, 10-3 or 0.001
m = micro, 10-6 or 0.000001
Some equalities
1 m = 100 cm
1 kg = 1000 g
1 L = 1,000,000 mL
1000 m = 1 km
10 dL = 1 L
Note: Volume – a milliliter is equivalent to a
cube that measures 1 cm x 1cm x 1cm or 1
cm3 (also referred to as a cubic centimeter –
Conversion Factors
To change from one unit to another,
you will multiply by the appropriate
conversion factor.
Conversion factors are written in the
form of a fraction.
Example: 100 cm = 1 m can be written as:
100 cm
100 cm
Using Conversion Factors /
Dimensional Analysis
 Fence-post method uses number +
 When conversion factor is multiplied
correctly, then all of the units – except
those desired in the answer – will
cancel out.
Dimensional Analysis
 Ex) Convert 35 inches to centimeters
 Ex) Convert 160 pounds to kilograms
Dimensional Analysis
For problems with more than one step,
they can be done either as a series of
steps or as one continuous problem.
Note that you will only round answers
at the very END!
Dimensional Analysis
Convert 25,000 feet to km
As one continuous conversion…
Dimensional Analysis
Problems with units in both the
numerator and denominator can create
Will see some problems involving
“clinical” calculations.
Dimensional Analysis
Ex) Convert 35 miles per hour to meters
per second
Dimensional Analysis
Ex) A certain medicine requires that
250mg per kilogram of body mass is to
be given. What dose should be given
to a child that weighs 48lbs?
Dimensional Analysis
Area and volume conversions can also
be easily missed.
Ex) A concrete footing measuring 3.0
feet by 2.0 feet by 1.5 feet is poured.
What volume of concrete is needed, in
Note: 1 ft3  12 in3,
Rather: 1 ft3 = (12)3 in3 = 1728 in3 !
Dimensional Analysis
Volume of concrete = 3.0ft x 2.0ft x
1.5ft = 9.0ft3
Alternatively, EACH dimension could be
converted to cm first, then multiplied
out to yield the volume.
The density of an object is equal to its
mass divided by its volume.
You will be asked to solve for any one
of the D, m, or V’s.
Density Problems
Ex) An object has a mass of 35.0g and
occupies 5.2mL. What is its density?
Density Problems
Ex) The mass of an iron bar is 1500g.
What volume does it occupy? The
density of iron is 7.9g/mL.
Can rearrange formula or use Density
as conversion factor.

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