Numerical Methods in Science

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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 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 7432 Check .0653

Examples 27,454,352 Check .00088536 7432 Check .0653 Round down
Round up

Examples 27,454,352 =27,000,000 (fill in 0’s to keep the same size)
=.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: 1.22 (tenth) .0004528 (1000th)
12,900,000 (million) (10000th) 3,045,000,000 (million) (100th) 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? 1.22 12,900,000 .00100 3,045,000,000 .00003 5.30 x 10 14

Which digits are SF? 1.22 12,900,000 .00100 3,045,000,000 .00003 5.30 x 10 14

=

= 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

= 1.17 = 1400 = = 3.00 =

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
x .047 1390 ÷ 150 .34 x .038 5.30 ÷ 23521 3 x 4

Multiplying and dividing
x = 6500 1390 ÷ = 9.3 .34 x = .013 5.30 ÷ = 3 x 4 = 10

A little bit of algebra D=m/v , m=vD, v=m/D and
If Density = mass/volume (It does.) then: D=m/v , m=vD, v=m/D and

A little bit of algebra D=m/v
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 vD=vm/v = m
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 m = vD D D
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 m = vD =v D D
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!

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.

Multiply it by a fraction called a conversion factor.
For example: 74.32 mm = _______ m 74.32 mm x ____________ = Multiply it by a fraction called a conversion factor.

--The units you start with go on the bottom (You want them to cancel)
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)

--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 ________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 m) 3) Cancel your units, multiply the numerators, and divide by the denominator

Convert 1.26 cm = _____m 5.28 m = ______ mm .00084 km = _______ mm
8.00 mm = _______nm

Metric System prefixes
Prefix Symbol Meaning giga G 109 ( ) mega M 106 ( ) 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 ( ) nano n 10-9 ( )

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 3 10 15 7 6 50 6 2 9 10 8 4 40 5 1 8 5 9 2 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 = 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 = 2.17 x 10 8 = 21,700,000 5.05 x 10 -3 =

Put into scientific notation
1.22 12,900,000 .00100 3,045,000,000 .00003 5

Take out of scientific notation
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

Extremely well qualified
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 No recommendation 2 Extremely well qualified Well Qualified Qualified Possibly qualified No recommendation 10 14 Possibly qualified Extremely well qualified 14 12 Well Qualified Qualified

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

Line Graphs Used when both sets of data are numbers

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