Goal: To understand the mathematics that will be necessary for this course which you do not get in a math class

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Goal To understand the mathematics that will be
necessary for this course which you do not get in
a math class

• Objectives
• Learning how to use Significant Figures
• Learning how to work with Scientific Notation
• Learning about how to use Units/Directions
• Learning about the basics of Vectors

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Significant Figures

• Significant Figures are the digits that you know
  their values.
• There is no guessing you know what it is.
• If you DONT know a digit you have to put in a
  zero by default (called a place holder).
• The more significant digits you have the more
  accurate the number.

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How to determine them

• For numbers greater than one
• A) if there is no decimal place then all the
  zeros at the end are NOT significant. They are
  called place holders.
• So, 10400 meters has 3 significant figures (which
  I will hereafter call sig figs).
• B) If there IS a decimal then ALL of the digits
  are significant.
• So, 10.0203 has 6 sig figs

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Less than 1

• For numbers less than one the 0s on the left
  side are placeholders.
• So, 0.0102 only has 3 sig figs

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Some more complicated ones

• How many sig figs for each of the following s
• A) 2030.03020
• B) 0.0020030
• C) 4050700
• D) 102

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How many to use?

• Suppose I wanted to find the area of the white
  board, who would I do that?
• Find the area of the whiteboard.

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What do these distances represent?

• 0.0000000000000007 meters
• 100000000000000000000000000 meters

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Scientific Notation

• A 10B
• A is greater than or equal to 1 but less than 10.
• A can have a decimal which means that ALL of the
  digits in A are significant (the 10B is the
  placeholder)
• B is an integer (which means 0, 4, -6, but not
  2.3)

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Math in scientific notation

• For addition or subtraction you almost have to
  treat the powers of 10 as a Unit.
• That is to say that to add or subtract without
  using a calculator that is you need to have
  everything have the same powers of 10.
• Once you have that you can just add and subtract
  the s in front and leave the powers of 10.

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For Example

• 5.0 104 2.4 104 7.4 104

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Multiplication

• Suppose you have two numbers in scientific
  notation such as
• A 10B and C 10D
• Multiplied you get A C 10(BD)
• What is (3 104) (2 103)?

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Quick note about calculators

• If I give you a number of 1102 when you enter it
  into your calculator be sure to enter 1 power 2
  and not 10 power 2.
• Go ahead and try this you should get 100.
• If you do 10 power 2 you will get 1000 because
  your calculator thinks you are trying to enter
  10102

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One more note

• If you get the number in front to be more than 10
  then you have to adjust it.
• To do so take off factors of ten off of the front
  number (i.e. move the decimal)
• Each time you withdraw a factor of 10 from the
  number in front you have to deposit that factor
  of 10 into the powers of ten (by adding 1 to the
  integer for each time you move the decimal)

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Quick one I will do

• (3 104) (7 103)

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Division

• For division you divide the numbers in front and
  subtract the exponents in the denominator
• So when you divide A 10B by C 10D you get A
  / C 10(B-D)
• You try, but no calculator for now
• Find (4 105) / (2 103)

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Calculator note

• Everyone use their calculators to find the answer
  to the following problem (even if you can do this
  one in your head)
• (2 6) / (3 4)

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On the calculator

• Always put the denominator in brackets, i.e.
  ()s. Otherwise your calculator wont calculate
  the problem correctly.

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One more note

• If the number in front becomes less than 1 you
  have to adjust by adding factors of 10.
• That is you move the decimal place.
• Each time you add a factor of 10 you have to
  subtract a factor of 10 from the powers of 10 by
  decreasing the exponent by 1 for each power of 10
  you add.

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Using units/directions

• In physics values do not come usually as just a
  number.
• For example you dont go to a store to buy 5.
• You dont go to the store to buy 5 pounds.
• You may go to buy 5 pounds of apples.
• In this case pounds is a unit as is apples.

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Units include

• Physics uses the mks system which stands for
  meters (m), kilograms (kg), and seconds (s)
• Combinations of these units can make up other
  units as well.
• For example velocity is m/s direction
• Direction is also a unit (so up -down)

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Very Important

• Units can be very important.
• Unit errors in real life have had disastrous
  effects.
• A units mistake was once claimed to cause the
  crash of a Mars probe.
• Some claimed that a unit error (million and
  billion can sometimes be considered a unit) was
  rumored to wipe a TRILLION dollars of value from
  the stock market.

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Adding values

• When you add or subtract values you have to add
  values with the same unit.
• It would not make sense to add 5 apples to 5
  pears.

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Multiplying/dividing

• When multiplying or dividing you treat units like
  you would a variable in algebra.
• So, a meter times a meter is a square meter.
• A meter divided by a meter is 1 (i.e. they cancel)

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Direction

• One special type of unit is direction.
• In this class it is VERY beneficial to think of
  direction as a unit and to include it for any
  value that requires it (which I will call vector
  values).
• Examples include and are not limited to up,
  down, forwards, backwards, towards an object,
  away from an object, North, South, East, West.
• If you use graphing axis directions can be plus
  or minus x (called x hat) and plus or minus y
  (called y hat).

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Adding Direction units

• Just like with other units you can only add 2
  values if they have the same unit.
• So, you cannot add a North to a South much like
  you cant add 5 cm to 2 m UNLESS you convert one
  of the two units first.
• The conversion is straightforward often times
  because North - South
• However this also means you cannot add North to
  West as there is no conversion i.e. these
  become two separate values as we will utilize in
  this course.

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Vectors

• Vectors take advantage of the fact that each
  dimension (dimensions are separated by 90
  degrees) is independent for the most part.
• Vectors in vector form have a component in each
  dimension (which for this class will be usually 2
  dimensions).
• When you add vectors you have to add the same
  components together while separating out the
  components that are not in the same direction.
• If you want the hypotenuse of the vector, or the
  total value without sign or direction this is
  called a magnitude.

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Conclusion

• We have learned the basics of math that we will
  need to succeed in this course.
• We have learned how to use significant figures
  and what they represent.
• We have learned how to use scientific notation
  even without a calculator.
• We have learned how to use units/directions and
  how they apply to vectors.