### Exact Numbers

```Numbers in Science
Chapter 2
Measurement

What is measurement?
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A measurement has 2 Parts – the Number and the Unit
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Quantitative Observation
Based on a comparison to an accepted scale.
Number Tells Comparison
Unit Tells Scale
There are two common unit scales

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English
Metric
The Unit
The measurement System units

English (US)

Metric (rest of the world)

Length – inches/feet
Distance – mile
Volume – gallon/quart
Mass- pound

Length – meter
Distance – kilometer
Volume – liter
Mass - gram
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Related Units in the Metric System
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All units in the metric system are related to the
fundamental unit by a power of 10
The power of 10 is indicated by a prefix
The prefixes are always the same, regardless of the
fundamental unit
Fundamental Unit 100
Fundamental SI Units

Established in 1960 by an international agreement to
Name of Unit
Abbreviation
standardize science units
These
metric system
Mass units are in theKilogram
kg
Physical Quantity

Length
Meter
m
Time
Second
s
Temperature
Kelvin
K
Energy
Joules
J
Pressure
Pascal
Pa
Volume
Cubic meters
m3
Length…..
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SI unit = meter (m)
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About 3½ inches longer than a yard
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1 meter = distance between marks on standard metal rod in a Paris vault
or distance covered by a certain number of wavelengths of a special color
of light
Commonly use centimeters (cm)
1 inch (English Units) = 2.54 cm (exactly)
Figure 2.1: Comparison of English and
metric units for length on a ruler.
Volume

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Measure of the amount of three-dimensional space
occupied by a substance
SI unit = cubic meter (m3)
Commonly measure solid volume in cubic centimeters
(cm3)
Commonly measure liquid or gas volume
in milliliters (mL)
◦
◦
1 L is slightly larger than 1 quart
1 mL = 1 cm3
Mass

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Measure of the amount of matter present in an object
SI unit = kilogram (kg)
Commonly measure mass in grams (g) or milligrams
(mg)

1 kg = 2.2046 pounds, 1 lbs.. = 453.59 g
Temperature Scales
Any idea what the three most common temperature
scales are?
 Fahrenheit Scale, °F
◦

Celsius Scale, °C
◦
◦

Water’s freezing point = 32°F, boiling point = 212°F
Temperature unit larger than the Fahrenheit
Water’s freezing point = 0°C, boiling point = 100°C
Kelvin Scale, K (SI unit)
◦
◦
Temperature unit same size as Celsius
Water’s freezing point = 273 K, boiling point = 373 K
Thermometers based on the three temperature scales
in (a) ice water and (b) boiling water.
The number
Scientific Notation
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Technique Used to Express Very Large or Very Small
Numbers
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135,000,000,000,000,000,000 meters
0.00000000000465 liters
Based on Powers of 10
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What is power of 10 Big?
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0,10, 100, 1000, 10,000
100, 101, 102, 103, 104
What is the power of 10 Small?
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0.1, 0.01, 0.001, 0.0001
10-1, 10-2, 10-3, 10-4
Writing Numbers in Scientific
Notation
1. Locate the Decimal Point : 1,438.
2. Move the decimal point to the right of the
non-zero digit in the largest place
- The new number is now between 1 and 10
- 1.438
3. Now, multiply this number by a power of 10
(10n), where n is the number of places you
moved the decimal point
- In our case, we moved 3 spaces, so n = 3 (103)
The final step for the number……
4. Determine the sign on the exponent n
If the decimal point was moved left, n is +
If the decimal point was moved right, n is –
If the decimal point was not moved, n is 0
- We moved left, so 3 is positive
- 1.438 x 103
Writing Numbers in Standard Form
We reverse the process and go from a number in
scientific notation to standard form…..
1
Determine the sign of n of 10n
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2
Determine the value of the exponent of 10
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3
If n is + the decimal point will move to the right
If n is – the decimal point will move to the left
Tells the number of places to move the decimal point
Move the decimal point and rewrite the number
Try it for these numbers: 2.687 x 106 and 9.8 x 10-2
Let’s Practice…..
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Change these numbers to Scientific Notation:
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1,340,000,000,000
697, 000
0.00000000000912
1.34 x 1012
6.97 x 105
9.12 x 10-12
Change these numbers to Standard Form:
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3.76 x 10-5
8.2 x 108
1.0 x 101
0.0000376
820,000,000
10
Are you sure about that number?
Uncertainty in Measured Numbers
cm
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A measurement always has some amount of uncertainty, you
always seem to be guessing what the smallest division is…
To indicate the uncertainty of a single measurement scientists
use a system called significant figures
The last digit written in a measurement is the number that is
considered to be uncertain
Rules, Rules, Rules….
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We follow guidelines (i.e. rules) to determine what numbers
are significant
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Nonzero integers are always significant
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2753
89.659
.281
Zeros
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Captive zeros are always significant (zero sandwich)
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1001.4
55.0702
4780.012
Significant Figures – Tricky Zeros
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Zeros
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Leading zeros never count as significant figures
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0.00048
0.0037009
0.0000000802
Trailing zeros are significant if the number has a decimal point
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22,000
63,850.
0.00630100
2.70900
100,000
Significant Figures
Scientific Notation
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All numbers before the “x” are significant. Don’t worry about
any other rules.
7.0 x 10-4 g has 2 significant figures
2.010 x 108 m has 4 significant figures
How many significant figures are in these numbers?
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102,340
1.0 x 107
1,908,021.0
0.01796
1,200.00
0.000002
92,017
0.1192
8.01010 x 1014
Have a little fun remembering sig figs
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Exact Numbers
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Exact Numbers are numbers known with certainty
Unlimited number of significant figures
They are either
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counting numbers
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number of sides on a square
or defined
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100 cm = 1 m, 12 in = 1 ft, 1 in = 2.54 cm
1 kg = 1000 g, 1 LB = 16 oz
1000 mL = 1 L; 1 gal = 4 qts.
1 minute = 60 seconds
Calculations with Significant Figures
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Exact numbers do not affect the number of significant
Answers to calculations must be rounded to the
proper number of significant figures
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For addition and subtraction, the last digit to the right
is the uncertain digit.
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round at the end of the calculation
Use the least number of decimal places
For multiplication, count the number of sig figs in each
number in the calculation, then go with the smallest
number of sig figs
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Use the least number of significant figures
Rules for Rounding Off
If the digit to be removed
•
•
is less than 5, the preceding digit stays the same
 Round 87.482 to 4 sig figs.
is equal to or greater than 5, the preceding digit is
increased by 1
 Round 0.00649710 to 3 sig figs.
In a series of calculations, carry the extra digits to the
final result and then round off
Don’t forget to add place-holding zeros if necessary to keep
value the same!!
Round 80,150,000 to 3 sig figs.
Examples of Sig Figs in Math
1)
5.18 x 0.0208
2)
21 + 13.8 + 130.36
3)
116.8 – 0.33
proper number of
significant digits!!!
Solutions:
1)
0.107744 round to proper # sig fig
1)
2)
165.47
1)
3)
5.18 has 3 sig figs, 0.0208 has 3 sig figs so answer is 0.108
Limiting number of sig figs in addition is the smallest number
of decimal places = 12 (no decimals) answer is 165
116.47
1)
Same rule as above so answer is 116.5
Moving unit to unit: Conversion
Exact Numbers
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Exact Numbers are numbers known with certainty
They are either
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counting numbers
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number of sides on a square
or defined
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100 cm = 1 m, 12 in = 1 ft, 1 in = 2.54 cm
1 kg = 1000 g, 1 LB = 16 oz
1000 mL = 1 L; 1 gal = 4 qts.
1 minute = 60 seconds
The Metric System
Fundamental Unit 100
Movement in the Metric system
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In the metric system, it is easy it is to convert numbers to
different units.
 Let’s convert 113 cm to meters
Figure out what you have to begin with and where you need
to go..
 How many cm in 1 meter?
 100 cm in 1 meter
Set up the math sentence, and check that the units cancel
properly.
 113 cm [1 m/100 cm] = 1.13 m
Let’s Practice converting metric units
 250 mL to Liters
 0.250 mL
 475 cg to kg
 47,500,000 or
 4.75 x 107
 1.75 kg to grams
 1,750 grams
 328 mm to dm
 3.28 dm
 88 µL to mL
 0.088 mL
 0.00075 nL to µL
 0.75 µL
Converting Between Metric and
non-Metric (English) units
Converting non-Metric Units
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Many problems involve using equivalence statements
to convert one unit of measurement to another
Conversion factors are relationships between two
units
Conversion factors are generated from equivalence
statements
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e.g. 1 inch = 2.54 cm can give
2.54 cm
1in
or
1in
2.54 cm
Converting non-Metric Units

Arrange conversion factor so starting unit is on the bottom of
the conversion factor
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You may string conversion factors together for problems that
involve more than one conversion factor.
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Convert kilometers to miles
Convert kilometers to inches
Find the relationship(s) between the starting and final units.
Write an equivalence statement and a conversion factor for
each relationship.
Arrange the conversion factor(s) to cancel starting unit and
result in goal unit.
Practice
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Convert 1.89 km to miles
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Convert 5.6 lbs to grams
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Find equivalence statement 1mile = 1.609 km
1.89 km (1 mile/1.609 km)
1.17 miles
Find equivalence statement 454 grams = 1 lb
5.6 lbs(454 grams/1 lb)
2500 grams
Convert 2.3 L to pints
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Find equivalence statements: 1L = 1.06 qts, 1 qt = 2 pints
2.3 L(1.06 qts/1L)(2 pints/1 qt)
4.9 pints
Temperature Conversions
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To find Celsius from Fahrenheit
 oC
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To find Fahrenheit from Celsius
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oF
= 1.8(oC) +32
Celsius to Kelvin
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= (oF -32)/1.8
K = oC + 273
Kelvin to Celsius
 oC
= K – 273
Temperature Conversion Examples
1)
180°C to Kelvin
1)
2)
2)
23°C to Fahrenheit
1)
2)
3)
3)
Use the conversion factor: F = (1.80)C + 32
F = (1.80)23 + 32
F=73.4 or 73°F
87°F to Celsius
1)
2)
3)
4)
To convert Celsius to Kelvin add 273
180+ 273 = 453 K
Use the conversion factor C=5/9(F-32)
C = 5/9(87-32)
C = 30.5555555… or 31°C
694 K to Celsius
1)
2)
To convert K to C, subtract 273
694-273= 421°C
Measurements
and
Calculations
Density
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Density is a physical property of matter representing the mass
per unit volume
For equal volumes, denser object has larger mass
For equal masses, denser object has small volume
Solids = g/cm3
Mass
Liquids = g/mL
Density 
Gases = g/L
Volume
Volume of a solid can be determined by water displacement
Density : solids > liquids >>> gases
In a heterogeneous mixture, denser object sinks
Using Density in Calculations
Mass
Density 
Volume
Mass
Volume 
Density
Mass  Density  Volume
Density Example Problems

What is the density of a metal with a mass of
11.76 g whose volume occupies 6.30 cm3?
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What volume of ethanol (density = 0.785 g/mL)
has a mass of 2.04 lbs?
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What is the mass (in mg) of a gas that has a
density of 0.0125 g/L in a 500. mL container?
How could you find your density?
Volume by displacement
To determine the volume to insert into the
density equation, you must find out the difference
between the initial volume and the final volume.
 A student attempting to find the density of
copper records a mass of 75.2 g. When the
copper is inserted into a graduated cylinder, the
volume of the cylinder increases from 50.0 mL to
58.5 mL. What is the density of the copper in
g/mL?
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A student masses a piece of unusually shaped metal
and determines the mass to be 187.7 grams. After
placing the metal in a graduated cylinder, the water
level rose from 50.0 mL to 60.2 mL. What is the
density of the metal?
A piece of lead (density = 11.34 g/cm3) has a mass of
162.4 g. If a student places the piece of lead in a
graduated cylinder, what is the final volume of the
graduated cylinder if the initial volume is 10.0 mL?
Percent Error

Percent error – absolute value of the error divided by
the accepted value, multiplied by 100%.
% error = measured value – accepted value x 100%
accepted value
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Accepted value – correct value based on reliable
sources.
Experimental (measured) value – value physically
measured in the lab.
Percent Error Example
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In the lab, you determined the density of ethanol to be
1.04 g/mL. The accepted density of ethanol is 0.785 g/mL.
What is the percent error?
The accepted value for the density of lead is 11.34 g/cm3.
When you experimentally determined the density of a
sample of lead, you found that a 85.2 gram sample of lead
displaced 7.35 mL of water. What is the percent error in
this experiment?
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Joe measured the boiling point of hexane to be 66.9 °C.
If the actual boiling point of hexane is 69 °C , what is the
percent error?
A student calculated the volume of a cube to be 68.98
cm3. If the true volume is 71.08 cm3, what is the student’s
percent error?
Tom used the density of copper and the volume of water
displaced to measure the mass of a copper pipe to be
145.67 g. When he actually weighed the sample, he found
a mass of 146.82 g. What was his percent error?
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