numerical methods in science --how many scientists does it take to change a light bulb? --scientists...

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

• Round to the nearest million

• .00088536

• Round to the nearest 100,000th

• 7432

• Round to the nearest ten

• .0653

• Round to the nearest 1000th

Examples

• 27,454,352

• Check

• .00088536

• Check

• 7432

• Check

• .0653

• Check

Examples

• 27,454,352

• Check

• .00088536

• Check

• 7432

• Check

• .0653

• Check

Round down

Round down

Round up

Round down

Examples

• 27,454,352

=27,000,000• (fill in 0’s to keep the

same size)

• .00088536=.00089

• (change the 8 to 9, do not fill in 0’s after a decimal!)

• 7432 =7430

• (fill in 0 to keep the same size)

• .0653=.065

• (do not fill in 0’s after a decimal!)

Round to the nearest:

1) 1.22 (tenth)

2) .0004528 (1000th)

3) 12,900,000 (million)

4) .00100 (10000th)

5) 3,045,000,000 (million)

6) .00003 (100th)

7) 7 (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) 1.22

2) .0004528

3) 12,900,000

4) .00100

5) 3,045,000,000

6) .00003

7) 5.30 x 10 14

Adding and subtracting

1.22 + .452 =


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


1) 1.22 - .047

2) 1290 + 100

3) .00034 + .000038

4) 5.30 - 2.30

5) 153000 - 12


1) 1.22 - .047 = 1.17

2) 1290 + 100 = 1400

3) .00034 + .000038 = .00038

4) 5.30 - 2.30 = 3.00

5) 153000 - 12 = 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?




$1 million.




$1 million.

Not $1,069,400


• 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.


1) 138422 x .047

2) 1390 ÷ 150

3) .34 x .038

4) 5.30 ÷ 23521

5) 3 x 4


1) 138422 x .047 = 6500

2) 1390 ÷ 150 = 9.3

3) .34 x .038 = .013

4) 5.30 ÷ 23521 = .000225

5) 3 x 4 = 10

A little bit of algebra

• If Density = mass/volume (It does.) then:

D=m/v ,

m=vD,

v=m/D

and


• 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



• The steps are:

2) Multiply both sides by v (the denominator)

vD=vm/v



• The steps are:

2) Multiply both sides by v (the denominator)

vD=vm/v = m

V cancels on the right



• The steps are:

3) Divide both sides by D

m = vD

D D



• The steps are:

3) Divide both sides by D

m = vD =v

D D D cancels on the right


• 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!

Conversions1) 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 = ______ m

3) .00084 km = _______ mm

4) 8.00 mm = _______nm

Metric System prefixes

• Prefix Symbol Meaning• giga G 109 (1 000 000 000)• mega M 106 (1 000 000)• kilo k 103 (1 000)• deka dk 101 (10)• deci d 10-1 (0.1)• centi c 10-2 (0.01)• milli m 10-3 (0.001)• micro m 10-6 (0.000 001)• nano n 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 meter m

• Mass kilogram kg

• Time second s

• electric current ampere A

• temperature kelvin K

• amount of substance mole mol

• luminous intensity candela 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

6

5

3

2

1

10

9

8

15

10

5

7

8

9

6

4

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

• .00089

• 7430

• .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

• 1.20 x 10 -4

• 2.17 x 10 8

• 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) 1.22

2) .0004528

3) 12,900,000

4) .00100

5) 3,045,000,000

6) .00003

7) 5

Take out of scientific notation

1) 1.82 x 10 -5

2) 4.28 x 10 4

3) 1.60 x 10 -6

4) 1.030 x 10 7

5) 7.045 x 10 -3

6) 9 x 10 0

7) 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

1

2

3

4

5

AP Chemistry Scores, Denver South High School 2004-2008

14

2

10

14 12

Extremely well qualified Well Qualified

Qualified

Possibly qualified

No recommendation

Extremely well qualified

Well Qualified

Possibly qualified

Qualified

No recommendation

Bar Graphs• Used when the categories don’t add up to any

definite total

A&P Grades, last names K-P

0

20

40

60

80

100

120

1 2 3 4 5 6 7 8 9 10

Student number

Pe

rce

nt

gra

de

Line Graphs• Used when both sets of data are numbers

Mass of Lead iodide recovered

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 2 4 6 8 10 12

Milliliters of lead solution

Mass,

in

gra

ms

numerical methods in science --how many scientists does it take to change a light bulb? --scientists...

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