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:  Distance and Area in the English system

1. Series will focus on English and Metric systems.
   a. Basic units of distance: inch, foot, yard, mile
   b. Basic units of area: square inch, square foot, acre, square mile


2. Other units of distance and area do exist
   a. Barleycorn for shoe sizes (1/3")
   b. Hand for describing horses (4")
   c. Rod for surveying (16-1/2 feet)
   d. Chain, also for surveying (4 linear rods, 66 feet, 22 yards)
   e. Furlong from horse racing and agriculture (220 yards, 10 chains)
   f. League (about an hour's walk) usually assumed to be 3 miles
      - Nautical: technical unit that's exactly 3 knots
   g. Square yard may be used in quotes for carpet installations

Table of Units: https://www.csgnetwork.com/converttable.html


Brilliant Insight #1: Units of distance were originally arbitrary.  We 
  did not standardize on inches, feet, miles, and so on because these 
  are magical units with special merits.  They were convenient at the 
  time and place where they were invented. 
  
  Standards let us talk to each other about distance without having to 
  be in the same place at the same time.  We'd have trouble if builders
  builders had to ask for boards "as long as my arm", or a plank 
  that's "Yea long". 


3. Bizarre properties of some English units explained:

   a. Rod/Chain: Used in measuring farmland and building plots
      - Rod is 5-1/2 yards, or 16-1/2 feet.
      - Chain is 4 linear rods, or the length of a surveyor's chain
      - Could have been longer or shorter.  Standard emerged from usage.
      
   b. Furlong: Longest row you can plow without resting the animals
      - Defined as 10 chains (220 yards)
      
   c. Acre: If you are on a quiz show, it's 43,560 square feet.  Huh?
      - Defined as the area of a plot that's 1 chain wide by a furlong 
      - Putting definitions together, we peek ahead to make sense of it.
      
        1 acre = 1 chain x 1 furlong x 10 chains   <--- multiply by 1
                                       ----------       (1 furlong is 
		                       1 furlong         10 chains)
      
        Cancelling out furlongs upstairs and downstairs, we get
	
            1 acre = 1 chain x 10 chains = 10 "square chains"
      
      - So the square feet in an acre is not (completely) arbitrary
      - It's just mostly arbitrary, but consistent with shorter units.

   d. Mile: Why is it 5,280 feet?  Similar story [Simplified version!]
      - Roman occupation brought in a 5,000 foot mile ("mille passus")
        Warning!  The Roman mile was defined in Roman feet, so it was
           a bit shorter than I've painted it.  
      
      - Originated as 1,000 double-steps or "paces"
      - Since 1,000 was "mille", unit naturally became "mile" in English
      
      - Elizabeth I (1603, or was it 1593?): 
        * Statute mile set to 8 furlongs (1,760 yds; 5,280 ft)
        * Why 8 furlongs?  Why not 10 furlongs?

        * Goal: Set new mile close to existing mile, but as N furlongs.
	    * New "statute mile" only about 5% longer than Roman mile
	      Note: Similar analysis could be used with other "miles".
	      
	    * Setting a mile to a even multiple of a furlong had practical 
	        benefits, and keeping it close to the old unit reduced 
	        conversion costs for "legacy users".
	        
      - That's why we've inherited a mile that measures 5,280 feet.

   To see why the story is tremendously more complicated than my account
      https://en.wikipedia.com/wiki/Mile

   Fun article on the mile.
      https://www.sizes.com/units/mile.htm
   
   High school student theme on the furlong.
      https://www.writework.com/essay/history-furlong by silverAlex2000

   Brief dictionary article on the mile, referenced by Dr. Math 
      https://www.unc.edu/~rowlett/units/dictM.html#mile 
      Referred by https://mathforum.org/library/drmath/view/61126.html

   Resource: StackExchange Physics and Maths sections ("mile" question)
      https://physics.stackexchange.com/questions/57785/difference-between-nautical-and-terrestrial-miles


4. 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

5. 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:
  https://www.regentsprep.org/regents/math/algebra/am2/leseng.htm

  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
  
   
Next time:  "Hey!  Ready to try metric?"