Transcript Numerical methods in science

Numerical Methods in
Science
--How many scientists does it take to
change a light bulb?
--Scientists don’t change light bulbs,
that’s what engineers are for.
Rounding
• Choose where (at which digit) you want to
round.
• If the NEXT digit is 5 or more, round up;
otherwise round down
• Rounding does not change the size of the
number, just its precision.
Examples
• 27,454,352
• 7432
• Round to the nearest
million
• Round to the nearest
ten
• .00088536
• .0653
• Round to the nearest
100,000th
• Round to the nearest
1000th
Examples
• 27,454,352
• 7432
• Round to the nearest
million
• Round to the nearest
ten
• .00088536
• .0653
• Round to the nearest
100,000th
• Round to the nearest
1000th
Examples
• 27,454,352
• 7432
• Check
• Check
• .00088536
• .0653
• Check
• Check
Examples
• 27,454,352
• Check
• 7432
Round
down
• .00088536
• Check
• Check
• .0653
Round
up
• Check
Round
down
Round
down
Examples
• 27,454,352
=27,000,000
• (fill in 0’s to keep the
same size)
• 7432
• .00088536
=.00089
• (change the 8 to 9, do
not fill in 0’s after a
decimal!)
• .0653
=7430
• (fill in 0 to keep the
same size)
=.065
• (do not fill in 0’s after
a decimal!)
Round to the nearest:
1)
2)
3)
4)
5)
6)
7)
1.22
.0004528
12,900,000
.00100
3,045,000,000
.00003
7
(tenth)
(1000th)
(million)
(10000th)
(million)
(100th)
(10)
Significant figures
• All non-zero digits are significant
• Zeros
– A) Leading, not significant.
– B) Trapped (by SF)--significant
– C) Trailing, with a decimal--significant
Which digits are SF?
1)
2)
3)
4)
5)
6)
7)
1.22
.0004528
12,900,000
.00100
3,045,000,000
.00003
5.30 x 10 14
Which digits are SF?
1)
2)
3)
4)
5)
6)
7)
1.22
.0004528
12,900,000
.00100
3,045,000,000
.00003
5.30 x 10 14
Adding and subtracting
1.22 + .452 =
Adding and subtracting
1.22 + .452 = 1.67
Your calculator says “1.672”, but you
don’t know how many thousandths
there are in the first number. Round
where your knowledge ends
Adding and subtracting
1)
2)
3)
4)
5)
1.22 - .047
1290 + 100
.00034 + .000038
5.30 - 2.30
153000 - 12
Adding and subtracting
1)
2)
3)
4)
5)
1.22 - .047
1290 + 100
.00034 + .000038
5.30 - 2.30
153000 - 12
= 1.17
= 1400
= .00038
= 3.00
= 153000
Multiplying and dividing
Suppose there are 20,000 pairs of Nike Air
Pegasus running shoes in the Denver
area. Suppose each pair cost $53.47,
like mine did.
How much did those shoes cost?
Multiplying and dividing
Suppose there are 20,000 pairs of Nike Air
Pegasus running shoes in the Denver
area. Suppose each pair cost $53.47,
like mine did.
How much did those shoes cost?
$1 million.
Multiplying and dividing
Suppose there are 20,000 pairs of Nike Air
Pegasus running shoes in the Denver
area. Suppose each pair cost $53.47,
like mine did.
How much did those shoes cost?
$1 million.
Not $1,069,400
Multiplying and dividing
• Round to match the precision of the least
number of SF in your problem.
• The “20,000 pairs” is a round number,
1SF. Don’t use more than 1SF in your
answer.
Multiplying and dividing
1)
2)
3)
4)
5)
138422 x .047
1390 ÷ 150
.34 x .038
5.30 ÷ 23521
3x4
Multiplying and dividing
1)
2)
3)
4)
5)
138422 x .047
1390 ÷ 150
.34 x .038
5.30 ÷ 23521
3x4
= 6500
= 9.3
= .013
= .000225
= 10
A little bit of algebra
• If Density = mass/volume (It does.) then:
D=m/v ,
m=vD, and
v=m/D
A little bit of algebra
• You will have to be able to solve for any
variable in a formula.
• The steps are:
1) Start with your original formula.
D=m/v
A little bit of algebra
• You will have to be able to solve for any
variable in a formula.
• The steps are:
2) Multiply both sides by v (the denominator)
vD=vm/v
A little bit of algebra
• You will have to be able to solve for any
variable in a formula.
• The steps are:
2) Multiply both sides by v (the denominator)
vD=vm/v = m
V cancels
on the right
A little bit of algebra
• You will have to be able to solve for any
variable in a formula.
• The steps are:
3) Divide both sides by D
m = vD
D
D
A little bit of algebra
• You will have to be able to solve for any
variable in a formula.
• The steps are:
3) Divide both sides by D
m = vD =v
D
D
D cancels
on the right
A little bit of algebra
• So:
D=m/v
m=vD
v=m/D
In general: Solve by undoing
•
•
•
•
•
If something is added, subtract
If something is subtracted, add
If something is multiplied, divide
If something is divided, multiply
If something is raised to a power, take that
root
Practice, Practice, Practice!
Conversions
1) Start with the measurement given.
2) Multiply it by a fraction called a conversion
factor. It has three properties:
--The units you start with go on the bottom (You
want them to cancel)
--The units you want go on the top (You want to
end up with them next)
--The numbers make the top and the bottom equal
(So the fraction is equal to 1, it won't change
the value of the measurement)
3) Cancel your units, multiply the numerators, and
divide by the denominator
4) Repeat if necessary
For example:
• 74.32 mm = _______ m
For example:
• 74.32 mm = _______ m
• 74.32 mm
Start with the
measurement given.
For example:
• 74.32 mm = _______ m
• 74.32 mm x ____________ =
Multiply it by a
fraction called a
conversion factor.
For example:
• 74.32 mm = _______ m
• 74.32 mm x ____________ =
mm
--The units you start
with go on the bottom
(You want them to
cancel)
For example:
• 74.32 mm = _______ m
• 74.32 mm x ________m___ =
mm
--The units you want
go on the top (You
want to end up with
them next)
For example:
• 74.32 mm = _______ m
• 74.32 mm x __1 x 10-3 m___ =
1 mm
--The numbers make the top
and the bottom equal (So
the fraction is equal to 1, it
won't change the value of
the measurement)
For example:
• 74.32 mm = _______ m
• 74.32 mm x __1 x 10-3 m___=7.432x10-2m
1 mm
(or .07432m)
3) Cancel your units,
multiply the
numerators, and
divide by the
denominator
Convert
1) 1.26 cm = _____m
2) 5.28 m = ______ mm
3) .00084 km = _______ mm
4) 8.00 mm = _______nm
Metric System prefixes
•
•
•
•
•
•
•
•
•
•
Prefix
giga
mega
kilo
deka
deci
centi
milli
micro
nano
Symbol
G
M
k
dk
d
c
m
m
n
Meaning
109 (1 000 000 000)
106 (1 000 000)
103 (1 000)
101 (10)
10-1 (0.1)
10-2 (0.01)
10-3 (0.001)
10-6 (0.000 001)
10-9 (0.000 000 001)
SI System
•
•
•
•
•
--the International system
--used by scientists worldwide
--more consistent than the English system
--defines seven standard units
--allows combinations for derived units
• (it is no more precise or accurate than any
other system)
Measurement Unit Symbol
•
•
•
•
•
•
•
Length
Mass
Time
electric current
temperature
amount of substance
luminous intensity
meter
kilogram
second
ampere
kelvin
mole
candela
m
kg
s
A
K
mol
cd
Commonly Used Derived Units
•
•
•
•
•
•
Area
Volume
Velocity
Acceleration
Density
Dynamic viscosity
Commonly Used Derived Units
•
•
•
•
•
•
Area
=length x width (in m2)
Volume
=area x height (in m3)
Velocity
=length / time (in m/s)
Acceleration =velocity / time (in m/s2 )
Density
=mass / volume (in kg/m3)
Dynamic viscosity
(Just kidding, it’s not common)
For a chemist
• Mass: gram, kilogram, milligram
• Length: centimeter, meter, millimeter,
nanometer
• Volume: milliliter, liter, cubic meter
• Time: second, minute, hour
Making measurements
• Read the numbers
• Count the marks
• Estimate one final digit.
7
3
10
15
7
6
6
2
9
10
8
4
5
1
8
5
9
2
50
40
30
1
2
3
4
5
6
1
2
3
4
5
6
10
20
30
40
50
60
Scientific Notation
• For any real number, A, there is some a
and b, such that:
• A= a x 10b
• a is between 1 and 10
• b is a whole number
Examples
• 27,000,000
• 7430
• .00089
• .065
Examples
• 27000000
= 2.7 x 10 7
• .00089
= 8.9 x 10 -4
• 7430
= 7.43 x 10 3
• .065
= 6.5 x 10 -2
Examples
• 5.8 x 10 4
• 2.17 x 10 8
• 1.20 x 10 -4
• 5.05 x 10 -3
Examples
• 5.8 x 10 4
=58000
• 1.20 x 10 -4
=.000120
• 2.17 x 10 8
= 21,700,000
• 5.05 x 10 -3
= .00505
Put into scientific notation
1)
2)
3)
4)
5)
6)
7)
1.22
.0004528
12,900,000
.00100
3,045,000,000
.00003
5
Take out of scientific notation
1)
2)
3)
4)
5)
6)
7)
1.82 x 10 -5
4.28 x 10 4
1.60 x 10 -6
1.030 x 10 7
7.045 x 10 -3
9 x 10 0
4 x 10 1
Graphing
• A graph shows a picture of what a set of
numbers represent.
• The representation must be honest
Pie Graphs
• Used when the total of all of the numbers is some
whole value—this is for all of my AP Chemistry
students AP Chemistry Scores, Denver
South High School 2004-2008
No
recommendation
2
10
Possibly
qualified
14
Qualified
14
Extremely
well qualified
Extremely well
qualified
Well Qualified
Qualified
2
3
Possibly qualified
4
No recommendation
5
12
Well
Qualified
1
Bar Graphs
• Used when the categories don’t add up to any
definite total
A&P Grades, last names K-P
Percent grade
120
100
80
60
40
20
0
1
2
3
4
5
6
7
Student number
8
9
10
Line Graphs
• Used when both sets of data are numbers
Mass of Lead iodide recovered
0.14
Mass, in grams
0.12
0.1
0.08
0.06
0.04
0.02
0
0
2
4
6
8
Milliliters of lead solution
10
12