# Math and Measurement by ewghwehws

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```									Math and Measurement

Unit 2
Numbers…which ones are important?
What is 13/7?

Is it 1.8571428?

Or…is it 1.86? Or 1.9? Or 2?

Where do we round?
Significant Digits
Rules for Counting Significant
Figures - Details
• Exact numbers have an infinite
number of significant figures.

1 inch = 2.54 cm, exactly
Rules for Counting Significant
Figures - Details
• Nonzero integers always count as
significant figures.

3456 has
4 sig figs.
Rules for Counting Significant
Figures - Details
•   Zeros
- Leading zeros do not count as

significant figures.

• 0.0486 has
3 sig figs.
Rules for Counting Significant
Figures - Details
•   Zeros
- Captive zeros always count as
significant figures.

• 16.07 has
4 sig figs.
Rules for Counting Significant
Figures - Details
•   Zeros
Trailing zeros are significant only if
the number contains a decimal
point.

9.300 has
4 sig figs.
Sig Fig Practice #1
How many significant figures in each of the following?

1.0070 m         5 sig figs
17.10 kg        4 sig figs

100,890 L         5 sig figs

3.29 x 103 s         3 sig figs
0.0054 cm          2 sig figs
3,200,000          2 sig figs
Rules for Significant Figures in
Mathematical Operations

•    Multiplication and Division: # sig figs in
the result equals the number in the least
precise measurement used in the
calculation.

6.38 x 2.0 =
12.76  13 (2 sig figs)
Sig Fig Practice #2
3.24 m x 7.0 m         22.68 m2              23 m2
100.0 g ÷ 23.7 cm3     4.219409283 g/cm3   4.22 g/cm3
0.02 cm x 2.371 cm     0.04742 cm2          0.05 cm2
710 m ÷ 3.0 s          236.6666667 m/s      240 m/s
1818.2 lb x 3.23 ft    5872.786 lb·ft      5870 lb·ft
1.030 g ÷ 2.87 mL      2.9561 g/mL         2.96 g/mL
Rules for Significant Figures in
Mathematical Operations
• Addition and Subtraction: The number of
decimal places in the result equals the
number of decimal places in the least
precise measurement.

6.8 + 11.934 =
18.734  18.7 (3 sig figs)
Sig Fig Practice #3
3.24 m + 7.0 m           10.24 m         10.2 m
100.0 g - 23.73 g         76.27 g        76.3 g
0.02 cm + 2.371 cm        2.391 cm       2.39 cm
713.1 L - 3.872 L         709.228 L      709.2 L
1818.2 lb + 3.37 lb       1821.57 lb     1821.6 lb
2.030 mL - 1.870 mL       0.16 mL        0.160 mL
Practice Question
Questions 1-2 refer to the following sets of
numbers.
A.1.023 g
B.0.0030 mL
C.40,500 m

1.Is a number containing three significant
figures
2.Is a measure of mass
Scientific Notation

In science, we deal with some very LARGE
numbers:
1 mole = 602000000000000000000000

In science, we deal with some very SMALL
numbers:
Mass of an electron =
0.000000000000000000000000000000091 kg
Imagine the difficulty of calculating the
mass of 1 mole of electrons!

0.000000000000000000000000000000091 kg
x 602000000000000000000000
???????????????????????????????????
Scientific Notation:
A method of representing very large or very small
numbers in the form:
M x 10n

 M is a number between 1 and 10
 n is an integer
2 500 000 000.
9 8 7 6 5 4 3 2 1
Step #1: Insert an understood decimal point
Step #2: Decide where the decimal must end
up so that one number is to its left
Step #3: Count how many places you bounce
the decimal point
Step #4: Re-write in the form M x 10n
2.5 x   10 9

The exponent is the
number of places we
moved the decimal.
0.0000579
1 2 3 4 5

Step #2: Decide where the decimal must end
up so that one number is to its left
Step #3: Count how many places you bounce
the decimal point
Step #4: Re-write in the form M x 10n
5.79 x           10 -5

The exponent is negative
because the number we
started with was less than 1.

When you move the decimal to the left, the exponent is
positive.
When you move the decimal to the right, the exponent is
negative
You Try!
Convert the following
notation
1. 21.9                      1. 2.19 X 101
2. 6022                      2. 6.022 X 103
3. 0.12011                   3. 1.2011 X 10-1
Convert the following into
expanded form             4. 1800
4. 1.8 X 103                 5. .000081
5. 8.1 X 10-5                6. 720
6. 7.2 X 102
Multiplying and Dividing in
Scientific Notation
1. Multiply or divide the      Example #1
“M” values
2. If multiplying, add the     (1.35 x 104) x (2.35 x 105)
exponents
3. If dividing, subtract the
exponents.
4. If necessary, move
exponent to get
numbers back into           (2.6 x 104) / (4.6 x 103)
scientific notation.
You try
1. (6 X 103) x (4 X 10-3)       Answers:

1. 2.4 x 101
2. (3 x 104) x (4.5 x 105)      2. 1.35 x 1010
3. 5 x 10-3
3. ( 4.5 x 10-5) / (9 x 10-3)
Nature of Measurement
Measurement – quantitative observation
consisting of 2 parts
Part 1 – number
Part 2 – unit
Examples:
20 grams
6.63 x 10-34 Joule seconds
Accurate or Precise?
Accurate
measurements are
close to the actual or
accepted value.

Precise
measurements are
close to one another.

More than one
measurement must be
taken to determine if
the measurements are
precise.
The Fundamental SI Units
(le Système International, SI)
SI Prefixes
Prefix     Abbr.         Meaning       Exponent

Giga        G        1,000,000,000       109
Mega        M          1,000,000         106
Kilo      K             1,000          103
Hecto       h              100           102
Deca       D               10           101
Base unit                     1
deci       d             1/10           10-1
Centi      c            1/100
10-2
Milli      m            1/1000
10-3
Micro                1/1,000,000
10-6
Nano        n      1/ 1,000,000,000
10-9
Converting Among SI Units
Convert 2.6 grams to   Convert 5.25 decigrams to
milligrams             micrograms
You Try!

1. 3.4 liters to milliliters   1. 3,400 mL
2. 7,899 milligrams to         2. 7.899 g
grams                       3. .0277 cg
3. 277 kilograms to            4. 2,000,000 ug
centigrams
4. 2 meters to
micrometers
Derived SI Units
• Produced by multiplying or dividing standard
units.
5m
For Example:
Area = (Length)(Width)              Width

2.5 m   Length

Area =     (2.5 m)(5 m)   = 12.5 m2
Density
The ratio of mass to volume, or mass divided by
volume.
mass                  m
Density =                     D=
volume                 V
Density
• A measure of how closely matter is packed
into a volume.
• Unique for each compound.
– Density of water is 1.00 g/mL at 25˚C.
– Increasing temperature decreases the density, so
densities are given with temperatures.
• An intensive property.
• Substances that are less dense float in
substances more dense.
Density Problems
A sample of aluminum metal has a mass of 8.4 g. The
volume of the sample is 3.1 cm3. Calculate the density
of aluminum.

Given:   m = 8.4 g            8.4 g             Don’t
V = 3.1 cm3   D=                       forget
3.1 cm3             units!
Unknown: D = ?
Equation:                                           Box
D = 2.7   g/cm3
D=       v
Don’t forget units!!!
You try!
An unknown liquid is discovered at a crime
scene. A volume of 2.3 mL has a mass of 4.1
grams, what the liquid’s density?
Density Problems
Diamond has a density of 3.26 g/cm3. What is the mass
of a diamond that has a volume of 0.350 cm3?
Density Problems
A sample of metal is found to have a mass of
4.56 g and a density of 1.98 g/mL. What is the
volume of this metal?
Density Problem (No calculator)
The typical battery in a car is filled with a
solution of sulfuric acid, which is
approximately 39.9% sulfuric acid. If the
density of this solution is 1.3 g/mL, determine
the number of grams of acid present in 500.0
mL of battery solution.
1. What is the
volume of 5
grams of this
substance?

2. What is the
approximate
density of the
substance?
Converting Temperatures
C = 5/9 (F-32)

F=

K = C + 273

C=
You Try!
temperatures
1. 293 K to Celsius        1. 20 K
2. Room temperature to     2. About 24 C
3. Internal body
temperature to Kelvin
Steps to complete these problems

Step 1: Read the problem CAREFULLY.
Step 2: Determine the unit for the answer
Step 3: Write down all values given in the problem
and retrieve any needed conversion factors
Step 4: Set up the problem (watch carefully as teacher
does this step)
Step 5: Calculate—Multiply by numbers on the top
and divide by those on the bottom
Practice #1
The record long jump is 349.5 inches. Convert
this to meters. There are 2.54 cm in an inch.
Practice #2
A car is traveling 55.0 miles per hour. Convert
this to meters per second. One mile is equal to
1.61 km.
Practice #3
How many mg are there is a 5.00 grain aspirin
tablet?
1 grain = 0.00229 oz.
There are 454 grams/lb. There are 16 oz./lb
Practice #4
Convert 24 km/h to m/s (write out all steps
before using calculator).
Practice #5
In 1980, the US produced 18.4 billion (18.4 X
109) pounds of phosphoric acid to be used in
the manufacture of fertilizer. The average cost
of the acid is \$318/ton. (1 ton = 2000 lbs).
What was the total value of the phosphoric
acid produced?

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