Transcript Scientific Notation

Significant Figures and Scientific
Notation
Significant Figures
► When using our calculators we must determine the correct
answer; our calculators are mindless drones and don’t know
the correct answer.
► There are 2 different types of numbers
– Exact
– Measured
► Measured number = they are measured with a measuring
device so these numbers have ERROR.
► When you use your calculator your answer can only be as
accurate as your worst measurement
Chapter Two
2
Exact Numbers
An exact number is obtained when you count objects
or use a defined relationship.
Counting objects are always exact
2 soccer balls
4 pizzas
Exact relationships, predefined values, not measured
1 foot = 12 inches
1 meter = 100 cm
For instance is 1 foot = 12.000000000001 inches? No
1 ft is EXACTLY 12 inches.
3
Learning Check
Classify each of the following as an exact or a
measured number.
1 yard = 3 feet
The diameter of a red blood cell is 6 x 10-4 cm.
There are 6 hats on the shelf.
Gold melts at 1064°C.
4
Solution
Classify each of the following as an exact (1) or a
measured(2) number.
This is a defined relationship.
A measuring tool is used to determine length.
The number of hats is obtained by counting.
A measuring tool is required.
5
Measurement and Significant Figures
► Every experimental
measurement has a
degree of uncertainty.
► The volume at right is
certain in the 10’s place,
10mL<V<20mL
► The 1’s digit is also
certain, 17mL<V<18mL
► A best guess is needed
for the tenths place.
Chapter Two
6
What is the Length?
1
2
3
►We can see the markings between 1.6-1.7cm
►We must guess between .6 & .7
►We record 1.67 cm as our measurement
7
4 cm
Learning Check
What is the length of the wooden stick?
A. 4.5 cm
B. 4.54 cm
C. 4.547 cm
Below are two measurements of the mass of the
same object. The same quantity is being described
at two different levels of precision or certainty.
Chapter Two
9
Note the 4 rules
When reading a measured value, all nonzero digits
should be counted as significant.
There is a set of rules for determining if a zero in a
measurement is significant or not.
► RULE 1. Zeros in the middle of a number are like any
other digit; they are always significant. Thus, 94.072
g has five significant figures.
► RULE 2. Zeros at the beginning of a number are not
significant; they act only to locate the decimal point.
Thus, 0.0834 cm has three significant figures, and
0.029 07 mL has four.
Chapter Two
10
► RULE 3. Zeros at the end of a number and after
the decimal point are significant. It is assumed
that these zeros would not be shown unless they
were significant. 138.200 m has six significant
figures. If the value were known to only four
significant figures, we would write 138.2 m.
► RULE 4. Zeros at the end of a number and before
an implied decimal point may or may not be
significant. We cannot tell whether they are part
of the measurement or whether they act only to
locate the unwritten but implied decimal point.
Chapter Two
11
Practice
45.8736
6
•All digits count
.000239
3
•Leading 0’s don’t
.00023900 5
•Trailing 0’s do
48000.
5
•0’s count in decimal form
48000
2
•0’s don’t count w/o decimal
3.982106 4
1.00040
6
•All digits count
•0’s between digits count as well
as trailing in decimal form
Examples of Rounding
For example you want a 4 Sig Fig number
0 is dropped, it is <5
4965.03
4965
780,582
780,600 8 is dropped, it is >5; Note you
must include the 0’s
1999.5
2000.
5 is dropped it is = 5; note you
need a 4 Sig Fig
Practice Rule #2 Rounding
Make the following into a 3 Sig Fig number
1.5587
1.56
.0037421
.00374
1367
1370
128,522
129,000
1.6683 106
1.67 106
Your Final number
must be of the same
value as the number
you started with,
129,000 and not 129
RULE 1. In carrying out a multiplication or division,
the answer cannot have more significant figures than
either of the original numbers.
Chapter Two
15
Multiplication and division
32.27  1.54 = 49.6958
49.7
3.68  .07925 = 46.4353312
46.4
1.750  .0342000 = 0.05985
.05985
3.2650106  4.858 = 1.586137  107
1.586 107
6.0221023  1.66110-24 = 1.000000
1.000
Chapter Two
►RULE 2. In carrying out an addition or
subtraction, the answer cannot have more
significant digits BEFORE or AFTER the DECIMAL
point than either of the original numbers.
Chapter Two
17
Addition/Subtraction
25.5
+34.270
59.770
59.8
32.72
- 0.0049
32.7151
32.72
320
+ 12.5
332.5
330
Addition and Subtraction
.56
__ + .153
___ = .713
__ .71
82000 + 5.32 = 82005.32
82000
10.0 - 9.8742 = .12580
.1
10 – 9.8742 = .12580
0
Look for the
last important
digit
Mixed Order of Operation
8.52 + 4.1586  18.73 + 153.2 =
= 8.52 + 77.89 + 153.2 = 239.61 =
239.6
(8.52 + 4.1586)  (18.73 + 153.2) =
= 12.68  171.9 = 2179.692 =
Chapter Two
2180.
How wide is our universe?
210,000,000,000,000,000,000,000 miles
(22 zeros)
This number is written in decimal
notation. When numbers get this large,
it is easier to write them in scientific
notation.
Scientific Notation
A number is expressed in scientific
notation when it is in the form
a x 10n
where a is between 1 and 10
and n is an integer
Write the width of the universe in
scientific notation.
210,000,000,000,000,000,000,000 miles
Where is the decimal point now?
After the last zero.
Where would you put the decimal to make
this number be between 1 and 10?
Between the 2 and the 1
2.10,000,000,000,000,000,000,000.
How many decimal places did you move the
decimal?
23
When the original number is more than 1,
the exponent is positive.
The answer in scientific notation is
2.1 x 1023
Express 0.0000000902 in
scientific notation.
Where would the decimal go to make the
number be between 1 and 10?
9.02
The decimal was moved how many places?
8
When the original number is less than 1, the
exponent is negative.
9.02 x 10-8
Write 28750.9 in scientific notation.
A.
B.
C.
D.
2.87509 x 10-5
2.87509 x 10-4
2.87509 x 104
2.87509 x 105
Express 1.8 x 10-4 in decimal notation.
0.00018
Express 4.58 x 106 in decimal notation.
4,580,000
On the calculator, scientific notation is
done with the
button.
4.58 x 106 is typed 4.58
6
Use a calculator to evaluate:
4.5 x 10-5
1.6 x 10-2
Type 4.5
-5
1.6
-2
You must include parentheses if you don’t use those
buttons!!
(4.5 x 10
-5)
(1.6 x 10
-2)
0.0028125
Write in scientific notation.
2.8 x 10-3
Use a calculator to evaluate:
7.2 x 10-9
1.2 x 102
On the calculator, the answer is:
6.E -11
The answer in scientific notation is
6.0 x 10 -11
The answer in decimal notation is
0.000000000060
Write (2.8 x 103)(5.1 x 10-7) in
scientific notation.
A.
B.
C.
D.
14.28 x 10-4
1.4 x 10-3
14.28 x 1010
1.428 x 10-3
Write in PROPER scientific notation.
(Notice the number is not between 1 and 10)
234.6 x 109
2.346 x 1011
0.0642 x 104
6.42 x 10 2
Write 531.42 x 105 in scientific
notation.
A. .53142 x 102
B. 5.3142 x 103
C. 53.142 x 104
D. 531.42 x 105
E. 53.142 x 106
F. 5.3142 x 107
G. .53142 x 108