The Numbers of Science

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The Numbers of Science

• Data Analysis

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Measurements and Units
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Units of Measure

• SI units Systeme Internationale d Unites
• standard units of measurement to be understood by
  all scientists
• Base Units defined unit of measurement that is
  based on an object or event in the physical world
• there are 7 base units
• some familiar quantities are time, length, mass,
  and temp

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• Time
• second (s)
• Many chemical reactions take place in less than a
  second so scientist often add prefixes, based on
  multiples of ten, to the base units.
• ex. Millisecond
• Length
• meter (m)
• A meter is the distance that light travels though
  a vacuum in 1/299 792 458 of a second.
• What is a vacuum?
• Close in length to a yard.
• Prefixes also applyex. millimeter

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• Mass
• mass is a measurement of matter
• kilogram (kg)
• about 2.2 pounds
• Masses measured in most laboratories are much
  smaller than a kilogram, so scientists use grams
  (g) or milligrams (mg).
• How many grams are in a kilogram?
• 1000
• How many milligrams are in a gram?
• 1000

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Derived Units

• Not all quantities are measured in base units
• A unit that is defined by a combination of base
  units is called a derived unit.
• Volume and Density are measured in derived units.

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• Volume
• The space occupied by an object
• Unit cm3 mL
• Liters are used to measure the amount of liquid
  in a container (about the same volume as a quart)
• Prefixes also appliedex. milliliter

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• Density
• The ratio that compares the mass of an object to
  its volume is called density.
• Units are g/cm3
• You can calculate density by the following
  equation
• Density mass/volume
• Ex What is the density of a sample of
  aluminum that has a mass of 13.5 g and a volume
  of 5.0 cm3?
• Density 13.5g/5.0cm3
• 2.7g/cm3

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Temperature

• A measurement of how hot or cold an object is
  relative to other objects
• The kelvin (K) scale
• water freezes at 273K
• water boils at 373K
• We also use the Celsius (C) scale
• water freezes at 0oC
• water boils at 100oC

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To Convert Celsius to Kelvin

• Add 273!!
• ex -39oC 273 234 K

To Convert Kelvin to Celsius

• Subtract 273!!
• ex 234K- 273 -39C

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Dimensional Analysis
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Dimensional Analysis

• Conversion factor
• A numerical factor used to multiply or divide a
  quantity when converting from one system of units
  to another.
• Conversion factors are always equal to 1
• Dimensional analysis
• A fancy way of saying converting units by using
  conversion factors

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Dimensional analysis often uses conversion factors

• Suppose you want to know how many meters are in
  48 km.
• You have to choose a conversion factor that
  relates kilometers to meters.
• You know that for every 1 kilometer there is 1000
  meters.
• What will your conversion factor be?
• 1000m/1km
• Now that you know your conversion factor, you
  can multiply it by your knownBUT you want to
  make sure you set it up so that kilometers
  cancels out. How would you do this?

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• 48km x 1000m
• 
  1km
• 48,000 m
• TIP Put the units you already have on the bottom
  of the conversion factor and the units you want
  on top.

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Scientific Notation
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Scientific Notation

• Numbers that are extremely large can be difficult
  to deal withsooo
• Scientists convert these numbers into scientific
  notation
• Scientific notation expresses numbers as a
  multiple of two factors
• A number between 1 and 10 (only 1 digit to the
  left of the decimal!)
• Ten raised to a power

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• For example
• A protons mass 0.0000000000000000000000000017262
  kg
• If you put it in scientific notation, the mass of
  a proton is expressed as 1.62762 x 10-27 kg
• Remember
• When numbers larger than 1 are expressed in
  scientific notation, the power of ten is positive
• When numbers smaller than 1 are expressed in
  scientific notation, the power of ten is negative

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• Try these
• Convert 1,392,000 to scientific notation.
• 1.392 x 106
• Convert 0.000,000,028 to scientific notation.
• 2.8 x 10-8

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Adding and Subtracting using Scientific Notation

• Make sure the exponents are the same!!
• 7.35 x 102 2.43 x 102 9.78 x 102
• If the exponents are not the same, you have to
  make them the same!!
• Tip if you increase the exponent, you decrease
  the decimal ----- if you decrease the exponent,
  you increase the decimal
• Example
• Tokyo pop 2.70 x 107
• Mexico City pop 15.6 x 106 1.56 x 107
• Sao Paolo pop 0.165 x 108 1.65 x 107
• NOW you can add them together and carry thru
  the exponent
• Total 5.91 x 107

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Multiplying and Dividing using Scientific Notation

• Multiplication
• Multiply decimals and ADD exponents
• Ex (1.2 x 106) x (3.0 x 104) 3.6 x 1010
  6 4 10
• Ex (1.2 x 106) x (3.0 x 10-4) 3.6 x 102
  6 (-4) 2
• Division
• Divide decimals and SUBTRACT exponents
• Ex (5.0 x 108) (2.5 x 104) 2.0 x 104
  8 4 4
• Ex (5.0 x 108) (2.5 x 10-4) 2.0 x 1012
  8 (-4) 12

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Accuracy vs. Precision

• Significant Figures

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Accuracy and Precision

• Accuracy How close measurements are to the
  actual value
• Precision How close measurements are to each
  other

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Percent Error

• An error is the difference between an
  experimental value and an accepted value
• Percent error
• Percent error accepted - experimental x
  100
• accepted value
• A tolerance is a very narrow range of error

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• Example
• The accepted density for copper is 8.96g/mL.
  Calculate the percent error for each of these
  measurements.
• 8.86g/mL
• 8.92g/mL
• 9.00g/mL
• 8.98g/mL

• (8.96 8.86)/8.96 x 100 1.12
• (8.96 8.92)/8.96 x 100 0.45
• (9.00 8.96)/8.96 x 100 0.45
• (8.98-8.96)/8.96 x 100 0.22

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Significant Figures

• Significant figures include all known
  digits plus one estimated digit
• Rules
• Non-zero numbers are always significant
• Zeros between non-zero numbers are always
  significant (trapped zeros)
• All final zeros to the right of the decimal place
  are significant (trailing zeros) (but trailing
  zeros dont count if there is no decimal in the
  number)
• Zeros that act as place holders are not
  significant (convert to SN to remove placeholder
  zeros) (leading zeros)
• Counting numbers and defined constants have an
  infinite number of sig figs

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Rounding numbers

• An answer should have no more significant figures
  than the data with the fewest significant figures
• Example
• Density of a given object m 22.44g
  1.5802817g/cm3
• 
  V 14.2cm3
• How should the answer be rounded?
• 1.58 g/cm3

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Addition Subtraction

• How do you add or subtract numbers that contain
  decimal point?
• The easiest way (which you learned in third
  grade) is to line up the decimal points then
  perform the math
• Then round according to the previous rule,
  rounding to the least numbers after the decimal!
  (ex 5.25 10.3 15.55 ? 15.6)

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Multiplication Division

• When you multiply or divide, your answer must
  have the same number of significant figures as
  the measurement with the fewest significant
  figuresjust like adding or subtracting!
• Ex 38736km
• 4784km
• 8.096989967 ? 8.097

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Representing Data

• A goal of many experiments is to discover whether
  a pattern exists in a certain situationwhen data
  are listed in a table the patterns may not be
  obvious
• Soooo, scientists often use graphs, which are
  visual displays of data
• X-axis ? independent variable
• Y-axis ? dependent variable