Transcript Introduction to Significant Figures & Scientific Notation

Introduction to Significant
Figures
&
Scientific Notation
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 that 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 + 83
= 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  -476
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