hpr1497 :: Practical Math - Units - Distances and Area, Part 1

Summary: Charles in NJ continues his Practical Math series with an episode on units of distance

Series: Practical Math

Source: [http://hackerpublicradio.org/eps.php?id=1497](http://hackerpublicradio.org/eps.php?id=1497)

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?"