Transcript Introduction to Significant Figures & Scientific Notation

Introduction to Significant
Figures
&
Scientific Notation
Scientific Method
Logical approach to solving problems by
observing
collecting data
formulating a hypotheses
testing Hypotheses
Formulating theories
Significant Figures
• Scientist use significant figures to
determine how precise a measurement
is
• Significant digits in a measurement include
all of the known digits plus one estimated digit
For example…
• Look at the ruler below
• Each line is 0.1cm
• You can read that the arrow is on 13.3 cm
• However, using significant figures, you must
estimate the next digit
• That would give you 13.30 cm
Let’s try this one
• Look at the ruler below
• What can you read before you
estimate?
• 12.8 cm
• Now estimate the next digit…
• 12.85 cm
The same rules apply with all
instruments
• The same rules apply
• Read to the last digit that you know
• Estimate the final digit
Let’s try graduated cylinders
• Look at the graduated cylinder below
•
•
•
•
What can you read with confidence?
56 ml
Now estimate the last digit
56.0 ml
One more graduated cylinder
• Look at the cylinder below…
• What is the measurement?
• 53.5 ml
Rules for Significant figures
Rule #1
• All non zero digits are ALWAYS
significant
• How many significant digits are in the
following numbers?
•274
•3 Significant Figures
•25.632
•5 Significant Digits
•8.987
•4 Significant Figures
Rule #2
• All zeros between significant digits are
ALWAYS significant
• How many significant digits are in the
following numbers?
504
3 Significant Figures
60002
5 Significant Digits
9.077
4 Significant Figures
Rule #3
• All FINAL zeros to the right of the
decimal ARE significant
• How many significant digits are in the
following numbers?
32.0
3 Significant Figures
19.000
5 Significant Digits
105.0020
7 Significant Figures
Rule #4
• All zeros that act as place holders are
NOT significant
• Another way to say this is: zeros are
only significant if they are between
significant digits OR are the very final
thing at the end of a decimal
For example
How many significant digits are in the following numbers?
0.0002
6.02 x 1023
100.000
150000
800
1 Significant Digit
3 Significant Digits
6 Significant Digits
2 Significant Digits
1 Significant Digit
Rule #5
• All counting numbers and constants
have an infinite number of significant
digits
• For example:
1 hour = 60 minutes
12 inches = 1 foot
24 hours = 1 day
How many significant digits
are in the following numbers?
0.0073
100.020
2500
7.90 x 10-3
670.0
0.00001
18.84
2 Significant Digits
6 Significant Digits
2 Significant Digits
3 Significant Digits
4 Significant Digits
1 Significant Digit
4 Significant Digits
Rules Rounding Significant
Digits
Rule #1
• If the digit to the immediate right of the last
significant digit is less than 5, do not round up
the last significant digit.
• For example, let’s say you have the number
43.82 and you want 3 significant digits
• The last number that you want is the 8 –
43.82
• The number to the right of the 8 is a 2
• Therefore, you would not round up & the
number would be 43.8
Rounding Rule #2
• If the digit to the immediate right of the last
significant digit is greater that a 5, you round
up the last significant figure
• Let’s say you have the number 234.87 and
you want 4 significant digits
• 234.87 – The last number you want is the 8
and the number to the right is a 7
• Therefore, you would round up & get 234.9
Rounding Rule #3
• If the number to the immediate right of the
last significant is a 5, and that 5 is followed by
a non zero digit, round up
• 78.657 (you want 3 significant digits)
• The number you want is the 6
• The 6 is followed by a 5 and the 5 is followed
by a non zero number
• Therefore, you round up
• 78.7
Rounding Rule #4
• If the number to the immediate right of the
last significant is a 5, and that 5 is followed by
a zero, you look at the last significant digit
and make it even.
• 2.5350 (want 3 significant digits)
• The number to the right of the digit you want
is a 5 followed by a 0
• Therefore you want the final digit to be even
• 2.54
Say you have this number
• 2.5250
(want 3 significant digits)
• The number to the right of the digit you
want is a 5 followed by a 0
• Therefore you want the final digit to be
even and it already is
• 2.52
Let’s try these examples…
200.99
(want 3 SF)
201
18.22
(want 2 SF)
18
135.50
(want 3 SF)
136
0.00299
(want 1 SF)
0.003
98.59
(want 2 SF)
99
Scientific Notation
• Scientific notation is used to express
very large or very small numbers
• It consists of a number between 1 & 10
followed by x 10 to an exponent
• The exponent can be determined by the
number of decimal places you have to
move to get only 1 number in front of
the decimal
Large Numbers
• If the number you start with is greater than 1,
the exponent will be positive
• Write the number 39923 in scientific notation
• First move the decimal until 1 number is in
front – 3.9923
• Now at x 10 – 3.9923 x 10
• Now count the number of decimal places that
you moved (4)
• Since the number you started with was
greater than 1, the exponent will be positive
• 3.9923 x 10 4
Small Numbers
• If the number you start with is less than 1, the
exponent will be negative
• Write the number 0.0052 in scientific notation
• First move the decimal until 1 number is in
front – 5.2
• Now at x 10 – 5.2 x 10
• Now count the number of decimal places that
you moved (3)
• Since the number you started with was less
than 1, the exponent will be negative
• 5.2 x 10 -3
Scientific Notation Examples
Place the following numbers in scientific notation:
99.343
9.9343 x 101
4000.1
4.0001 x 103
0.000375
3.75 x 10-4
0.0234
2.34 x 10-2
94577.1
9.45771 x 104
Going from Scientific Notation
to Ordinary Notation
• You start with the number and move the
decimal the same number of spaces as
the exponent.
• If the exponent is positive, the number
will be greater than 1
• If the exponent is negative, the number
will be less than 1
Going to Ordinary Notation
Examples
Place the following numbers in ordinary notation:
3 x 106
6.26x 109
5 x 10-4
8.45 x 10-7
2.25 x 103
3000000
6260000000
0.0005
0.000000845
2250
Significant Digits
Calculations
Significant Digits in
Calculations
• Now you know how to determine the
number of significant digits in a number
• How do you decide what to do when
adding, subtracting, multiplying, or
dividing?
Rules for Addition and
Subtraction
• When you add or subtract measurements,
your answer must have the same number of
decimal places as the one with the fewest
• For example:
20.4
1.322
= 104.722
Addition & Subtraction
Continued
• Because you are adding, you need to look at the
number of decimal places
20.4 + 1.322 + 83 = 104.722
(1)
(3)
(0)
• Since you are adding, your answer must have the
same number of decimal places as the one with the
fewest
• The fewest number of decimal places is 0
• Therefore, you answer must be rounded to have 0
decimal places
• Your answer becomes
• 105
Addition & Subtraction
Problems
1.23056 + 67.809 =
69.03956  69.040
23.67 – 500 =
- 476.33  -500
40.08 + 32.064 =
72.1440  72.14
22.9898 + 35.453 =
58.4428  58.443
95.00 – 75.00 =
20  20.00
Rules for Multiplication & Division
• When you multiply and divide numbers
you look at the TOTAL number of
significant digits NOT just decimal
places
• For example:
67.50 x 2.54
= 171.45
Multiplication & Division
• Because you are multiplying, you need to look at the
total number of significant digits not just decimal
places
67.50 x 2.54 = 171.45
(4)
(3)
• Since you are multiplying, your answer must have the
same number of significant digits as the one with the
fewest
• The fewest number of significant digits is 3
• Therefore, you answer must be rounded to have 3
significant digits
• Your answer becomes
• 171
Multiplication & Division
Problems
890.15 x 12.3 =
10948.845  1.09 x 104
88.132 / 22.500 =
3.916977  3.9170
(48.12)(2.95) =
141.954  142
58.30 / 16.48 =
3.5376  3.538
307.15 / 10.08 =
30.47123  30.47
More Significant Digit
Problems
18.36 g / 14.20 cm3
= 1.293 g/cm3
105.40 °C –23.20 °C
= 82.20 °C
324.5 mi / 5.5 hr
= 59 mi / hr
21.8 °C + 204.2 °C
= 226.0 °C
460 m / 5 sec
= 90 or 9 x 101 m/sec
1.
How many significant digits are in each of the following?
a. 12.5 b. .00230 c. .01000 d. 100.025 e. 100.000
3
3
4
6
6
2.
Round each of the following to three sig figs.
a. 125.365 b. .3002536458 c. 455.5
d. 278.96 e. 9.96543
.300
456
279
9.97
125
3.
Change each of the following
to scientific notation.
360.4
a. 100.00
b. .0020
c. 1000000 d. .02500
e. 70
2.0 x 10-3
1.0000 x 102
1 x 106
2.500 x 10-2 7 x 101
4.
Add the following using sig figs.
a. 110.1
250.326
360.4
b. 78.59681
10.
89
5. Subtract each of the following using sig figs
.
a. 125.63
b. 56.056
25.364
25.4
100.27
30.7
6. Multiply each of the following using sig figs.
a. 200.00 x 30.0
b. 25.11 x 5.0
6.00 x 103
130