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• It is useful to have an idea of the physical meaning of numbers.
  • For example, there are 86,400 seconds in a day. If you had to recount the 443,912 votes cast in Broward County, FL in November 2002, how long would it take you if you could parse votes at one per second?
  • You want to rent storage space. How big exactly is 27 cubic feet? Is the space big enough to store your 30 year old Carmengia automobile?
• Scientific notation saves writing. Positive exponents (first two examples) are numbers greater than one, whereas negative exponents are numbers between zero and one.
  • 5.382x10¹² = 5,382,000,000,000 - twelve digits to the right of the five.
  • 5.382x10¹¹ = 538,200,000,000 - eleven digits to the right of the five.
  • 5.382x10^-6 = 0.000005382 - the number is less than one, write the five at the sixth place to the right the decimal.
• It is VERY helpful to know your prefixes and what they mean:
  
  • tera - trillion - 10¹² = 1,000,000,000,000
  • giga - billion - 10⁹ = 1,000,000,000
  • mega - million - 10⁶ = 1,000,000
  • kilo - thousand - 10³ = 1,000
  • centi - hundredth - 10^-2 = 1/100
  • milli - thousandth - 10^-3 = 1/1,000
  • micro - millionth - 10^-6 = 1/1,000,000
  • nano - billionth - 10^-9 = 1/1,000,000,000
  • pico - trillionth - 10^-12 = 1/1,000,000,000,000
• Significant figures determine how precise a measurement is. 
• If you have a stick that is known to be one meter long with no demarcations, is this a good tool to measure fractions of a meter? You measure a distance to be 10 meter sticks long, and then some. Is it meaningful to add the quantity 3cm to this distance? Why or why not?
• Rules of significant figures
• Leading zeroes are not significant
• 0.0017 has two significant figures
• Trailing zeroes are not significant if they are needed to hold a place
• 1700 has two significant figures
• 0.001700 has four significant figures - there is implied precision in writing the extra two digits
• Zeroes within a number are significant
• 0.0107 has three significant figures
• 1205 has four significant figures
• If you wish to ensure all digits of a whole number are significant, add a decimal point:
• 127,000. has six significant figures whereas 127,000 only has three
• Things you should know about units
• Everyone but US uses the metric system. It's easier. Would you rather answer: how many teaspoons are in a gallon or how many milliliters are in a liter?
  
• Life is easier if you work with like units. For example, if you are working with distances, decide whether to use kilometers, miles, feet, meters, etc...Pick a unit and stick with it! If you are working with time, choose whether it's better to use minutes, seconds, hours, days, etc...
  • As you learn new things in this class, you will need to learn their units. Almost every quality we talk about will have a unit. Learn it up front, like you would vocabulary in your English class, and you will be much happier.
  • Converting units (an example)
    • Convert 3.1 miles into kilometers.
      First Know the conversion factor. These are in your book in the back cover. In this case 1 mi = 1.609 km.
      Second Determine what the final unit you need is. In this case, we want kilometers. Thus we want to write our conversion factor as a ratio with kilometers on top:
      

      1.609 km

      1 mi

      
      Third Set up a table and do the math
      

      3.1 mi	x	1.609 km	=	4.9879 mi*km
      		1 mi		1 mi

      
      Finally divide out the common units (in this case mi) from the top and bottom answers on the right [so mi*km/mi = km], and divide the top number 4.9879 by the bottom number 1 [4.9879/1 = 4.9879] and put the two together to get the answer: 4.9879 km.
• Dimensional Analysis is the use of units to validate a formula or result. For example:
  • Take the formula to calculate the hypoteneuse of a triangle:
    c = \sqrt{a^2 + b^2}
    The units of a, b, and c should all be distance related (say meters). When we add m² together with m² we should get m². The square root of m² is m as we would expect for the units of c.
  • Consider also the formula for distance travelled at a constant acceleration in time t.
    x = {1 \over 2} a t^2
    Distance, x, has units of m. Acceleration, a, has units of m/s². Time has units of seconds. Replacing each algebraic symbol with its corresponding units, we have
    [m] = [constant] [{m \over {s^2}}] [s^2]
    The units cancel just as algebraic symbols do. Thus we are left with
    [m] = [constant] {{[m]*[s^2]} \over {[s^2]}} = [constant]*[m] => [m]