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hpr1502 :: Practical Math - Units - Distances and Area, Part 2

Charles in NJ continues Practical Math with an episode on units of distance and area

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Part of the series: Practical Math

Goal for the series: Embracing units, and carrying them along as you go, can help you work with confidence in using maths in your life.

HPR Episode: Using and Converting Between Units of Distance

Intro: Last time, we talked in general terms about units, numbers and
how they might be useful in practice.  In this episode, we address some
specific measurement units that apply to distance and area, and how we
might convert from one system to another to better understand both.

Entire point of this episode is this: Carry units in calculations on 
  distances and areas, and you'll have more success in using them in 
  your life.  

Segment 1:  Review of Distance and Area conversions in the English system

1. Links from last time

   Table of Units:

   To see why the story is tremendously more complicated than my account

   Fun article on the mile.
   High school student theme on the furlong. by silverAlex2000

   Brief dictionary article on the mile, referenced by Dr. Math 
      Referred by

   Resource: StackExchange Physics and Maths sections ("mile" question)

2. Converting between units
   a. Units of distance usually defined as multiples of each other
      - 1 mile = 5,280 feet            - 1 hand = 4 inches
      - 1 foot = 12 inches             - 1 yard = 36 inches
      Skipping ahead to look at the metric system, we now have:
      - 1 inch = 2.54 centimeters (exact). Regularized in recent years.

   b. This works because there's consensus on Zero distance, so we don't
         have to adjust for differing origins, as we do with the 
         non-absolute temperature scales like Fahrenheit and Celsius. 
      - We'll get to temperature, non-absolute scales in a later show.

   c. For absolute scales, we can convert from one unit to another using
         a "conversion factor".  That is, we can convert a measurement 
         expressed in one unit to its equivalent in another unit by 
         multiplying or dividing by some number to stretch or compress 
         the original unit to match the target unit.
      - Example: I know that 1 foot is 12 inches, so how many inches are
          there in 10 feet?  How feet are there in 660 inches?
      - It is clear that a factor of 12 ought to be involved, but how do
          I know when to multiply or divide by 12 in the conversion?
      - Wait!  I'm serious.  When you see this problem for the first 
          time, you have to think this through to get it right.  
        * Without a system in place, you always have to think about it.
      - Answers in naive setup:
          (i) 10 feet = (12 * 10) inches = 120 inches
         (ii) 660 inches = (660 / 12) feet = 55 feet

3. Having a system.  Or units conversion as "multiplying by One"
   a. In each of the solutions I wrote down above, I start with an 
         equation that looks like this: X inches = Y feet.

   b. Inches are not feet, and this way of writing down the calculation 
        does NOT help you figure you how the conversions should work, or
        whether you should multiply or divide to get the right answers.

   c. Here's a system for creating conversion factors that tell you what
        to do at each step in the units conversion process.  It is based
        on the very obvious fact that when I multiply any number by '1',
        its value remains unchanged.
      - Start with one of the identities we wrote down at the beginning.
	    In this case, let's use:  12 inches = 1 foot
      - If I divide equals by equals, the results are equal.
	    So I can write:
	                                   12 inches      1 foot
	  12 inches = 1 foot implies that  ---------  =  --------- = 1
		                                 1 foot      12 inches
      -	Get the first term by dividing my original identity by (1 foot).
      - Get the second term by dividing my original identity by (12 in).

   d. To make a conversion from feet to inches, I use:
                           12 in    10 ft 
      10 ft * 1 = 10 ft * ------- = ------ * 12 in = 10 * 12 in = 120 in
                           1 ft      1 ft
      - Note: In the fraction (10 ft) / (1 ft), the units "cancel out", 
                which leaves a unitless number.
      - Suppose we start with the other form for the conversion factor:
                               1 ft    10 square feet 
      10 feet * 1 = 10 feet * ------ = -------------- = ???
                              12 in       12 inches
      - See?  When I use the form where the units don't cancel each 
          other, I get a resulting equation that is still correct.  It 
          just doesn't make much sense to me as a reader.
      - This is what you get when you "divide by 12" to convert feet to 
          inches, but the difference is that you KNOW something's wrong.  
      - You do not have to even look at the numbers to know that this 
         could not possibly be the right number of inches in 10 feet.

Brilliant Insight #2: When you use unit conversion factors, you help 
  your cause by carrying along both sets of units in the form of a 
  fraction as you go through your calculation.  
  - If the units on the right-hand side of your final equation don't 
    match the units you want (after everything else cancels out), your 
    numerical answer is almost certainly WRONG.
  - The implication here?  To convert units of distance, you need to 
       multiply or divide by a 
       conversion factor = (X New_Units) / (Y Old_Units).  
    When you do this, write the conversion factor in its full fractional
       form, and carry out all of the multiplications and cancellations.
  - If you do the conversion this way, and the units match, you only 
       have to check your arithmetic to be sure you've got it right.

  - If the units you want do not match those on the right side of the 
       equal sign, you are solving the wrong problem.  The equation may 
       be correct, but it is not expressed in the units you wanted.

 6. Let's use the system to solve the second example:
                               1 ft       660 in * 1 ft 
      660 in * 1 =  660 in *  -------  = --------------- = 55 feet
                               12 in        12 in
      Why?  The "inches" units cancel out because they appear in both 
        numerator and denominator (top/bottom, upstairs/downstairs) of
	    the fraction in the next to last term, leaving only "feet".

  Why people hate units and conversion problems:

  Comment: The "algebraic" approach suggested here is ugly, ad hoc in
    nature, and unnecessarily complicated.  Forget about setting up
    equations and going through formal operations to solve them.
    Choose your conversion factors so that the units work out properly
    as a straight multiplication problem with cancellation of all the
    units you don't want.  You may have to "divide" numbers, but you
    can use your calculator for working through the numbers.
  Cranky Summary: You should not have to solve equations to convert
    between units. Phooey on anyone who says otherwise. :-)

Segment 2: Conversions using compound conversion factors.

1. Suppose I want to find the number of inches in a furlong, or the 
   number of acres (or hectares) in a square mile?
   - My almanac doesn't carry these conversion factors, so I start with 
       what I do have and work my way through it.
                                       4 rods    16.5 ft    12 in
   1 furlong = 10 chains = 10 chains * ------- * ------- * -------
                                       1 chain    1 rod      1 ft
	     = 10 * 4 * 16.5 * 12 inches = ... = 7920 inches 

2. For acres in a square mile (1 mi^2), we have a bit more to do.

     Abbreviations used: miles = mi, furlong = fur, chain = ch
     Area means that we are dealing in two dimensions, so we have to 
        convert the lengths in each dimension.  An acre is already a 
        measure of area, so we're good.
                       1 acre    10 ch    8 fur   10 ch   8 fur
   1 sq mi = 1 mi^2 * -------- * ------ * ----- * ----- * -----
                       10 ch^2   1 fur    1 mi    1 fur   1 mi
	   = (1 mi * 1 mi) * 1 acre * 10 ch * 10 ch   8 fur * 8 fur
	                              ------------- * -------------
                                       10 ch * ch     1 mi * 1 mi
   Units cancel, leaving this:
   1 sq mi = 1 acre * (100/10) * (8 * 8) = 10 * 64 acres = 640 acres

Segment 3: Hey!  Ready to try metric?

1. Metric system never caught on in the US, although most of English-
     speaking world has adopted it.  Units conversion is easy in the 
     metric system, because everything is in powers of 10.
   - But you still need to carry along units in calculations!
2. Area and distance units in the metric system

   - Basics of distance: Centimeter is easy for us to see, and now the 
       factor to convert centimeters to inches is exact.
       1 inch = 2.54 centimeters (cm) exactly
                                    1 inch
       1 meter = 100 cm = 100 cm * --------- = 39.37 in (approximate)
                                    2.54 cm 

       1 kilometer = 1,000 meters

   - Basics of area:
       1 are = 100 sq meters  (area of a square that's 10m on each side)
                                          100 sq m
       1 hectare = 100 ares = 100 ares * ---------- = 10,000 sq meters
                                           1 are

3. For short distances, we should do our conversions fairly precisely.

   - There's usually a higher relative error from rounding off too soon.
   - If you measure wood for a small project, you want to be "close".

                        2.54 cm
   So 1 foot = 12 in * --------- = 30.48 cm exactly.  Cut carefully!
                        1 in
4. For larger distances, like distances covered in track and field, or
     the length of a football pitch (to a spectator), approximations can 
     give you a nice intuition for comparing units you know and a new
     set of units that you don't know as well.
   - 1 meter is around 39.37 inches.  Suppose I call it about 1.10 yards
        as a kind of approximate benchmark (39.60 in), so each meter in 
        my reckoning is about a quarter of an inch too long?  
   - If I'm planning a space mission, I could be in trouble. 
       But how bad would this be for getting an intuitive feel of the 
       distances covered by the athletes in the Olympic Games?

   - Error at  100 meters is about 0.23 in (0.6 cm) * 100 = 60 cm over
           at  200 meters, it's 1.1 m over.
           at 1 kilometer, it's 5.6 m over.
     Unless you're a long-range sharpshooter, 5.6m off in 1 km seems OK.

5. Bonus: The news talked about a wildfire that burned 100,000 hectares.
      What kind of area are we talking about?

   - Let's use our approxmation of 1 meter is about 1.1 yards.
   - Acres are defined in terms of "square chains", so let's look at 
        meters vs chains to see what we get.
	                                               1 m
     1 chain = 4 linear rods = 22 yds = 22 yds * -------- = about 20m
                                                  1.1 yd

                                                 20m     20m
     1 acre = 10 square chains = 10 ch * 1 ch * ----- * ------
                                                1 ch     1 ch
	    = 10 * 1 * 20m * 20m = 4,000 square meters, or 0.4 hectares

   - Wow!  An acre's about 0.4 hectares, or 1 hectare's about 2.5 acres.

   So what's the answer?
                                               2.5 acres   
      a) 100,000 hectares = 100,000 hectares * --------- = 250,000 acres
                                               1 hectare 

                                            1 sq. mi    250,000
      b) 100,000 hectares = 250,000 acres * --------- = ------- sq. mi
                                            640 acres     640
	                  = 391 sq. miles (about 400 sq miles)

   Note: This suggests a shortcut conversion (hectares to square miles).

                                 640 acres   1 hectare
    1 square mile = 1 sq. mile * --------- * --------- = 256 hectares
                                 1 sq. mi.   2.5 acres

6. Final check: Error analysis on this approximate conversion from 
      hectares to acres or square miles.

   - Using Google or 'units' in the shell, we have:
     1 sq mi = 259 hectares to 6 significant digits, versus 256 (1%)

     Note: If we used 250 hectares per square mile, the relative error 
       is 3.5%.  That's less than the error in the news report.

     1 hectare = 2.47105 acres, versus 2.5 (1% error)

   Units shell command: Dann Washko did a really nice job on 'units' for HPR.
    * Linux in the Shell #26:
    * HPR Episode #1213:

Final word: Unless you are buying, selling or cultivating land, use the 
  cruder approximations here to understand the relationships between 
  acres, hectares and square miles.  It will make you seem smarter.
  - If someone calls you out and says it's wrong, blame "that guy on HPR."

Next Topic?  Volumes and recipes, other than medicines (separate topic)
  - Volumes are the bottom line in cooking, unless they aren't.
  - Hint: You should weigh some items, like some kinds of flour.


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