Episode: 1497
Title: HPR1497: Practical Math - Units - Distances and Area, Part 1
Source: https://hub.hackerpublicradio.org/ccdn.php?filename=/eps/hpr1497/hpr1497.mp3
Transcribed: 2025-10-18 04:15:45

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Now let's open up the governments.
Hello and welcome to Hacker Public Radio. This is Charles in New Jersey. I'm back with another
series on Maths. This time it's not recreational math. It's what I would call practical math.
Today's show is going to focus on using and converting between units of distance and area.
I'm going to start with an example to tell you how the whole thing is going to be treated.
And if you only listen to this part, you'll have the general idea, but I'm about to discuss,
in your mind, and it might help you as you go forward in problems where units and units conversion
come up. Now suppose John has nine apples in his basket. If he gives away two apples to marry,
how many does he have left? Of course the answer is seven, right? Well in school I suppose you
could get away with saying just seven because the teacher would know what you meant. They'd know
that you said, okay nine apples and gives away two apples and okay that's really just code for
a subtraction problem of nine minus two and get seven, right? Well I'm hoping to change the way
you look at this. What I want to emphasize today is keeping the units in the math equation.
So when you're doing any kind of reasoning or calculation on physical objects or distances or
times or volumes or areas or speeds or voltages, what have you? That you'll always give the answer
with the units attached because if the units are correct, you have a much better chance
of having the right numerical answer, the right number of units, so that you won't be the one
to send a spaceship to Mars and crash it into the planet because you supplied a number
for a parameter to a subroutine that was for English measurements when the subroutine
was expecting metric for the other way around. So if John has nine apples and he gives away two
apples from his basket to marry, he doesn't have seven left, he has seven apples left. Now you're
going to get sick of me saying that seven is in some way a wrong answer and the answer is seven
apples, but you will thank me when we start talking about things like square miles being converted
to O hectares. I think you're going to appreciate the importance of carrying the units in even the
simplest of calculations involving something physical, something that you can count or measure
or experience or see. Very important that you get the units right. If you don't get the units
right, you may not get the answer right and if you give the wrong answer as just a number, someone
could use it and get hurt and we don't want that. So here's segment one. I'm going to talk about
distance and area in the English system and some of these things are pretty wild. I left out
quite a number of them because there's just so many. Unless you consider the history, they're just
very strange. So I'll give a little synopsis of some English units, how they relate to one another,
where some of them came from and how you would convert from one to the other when you need to.
Now I'm going to talk about the English system and the metric system eventually, but in some
sense measurements are really kind of arbitrary. I could have used a distance of measure that was the
width of my grandson's hand and on the day he turned two years old and I could have called it the
dexter because I used his right hand or I could use another alternative measure, which is the
the span of his left hand the day he turned three. I could call that the sinister. It's probably
just as valid as whatever originally gave us the idea of the inch and the foot and the yard
and even the mile that we've used for without thinking about it really for a long time.
So let's get into the English system. Now the basic units of distance without getting into the
micro distances are our familiar friends, the inch, the foot, the yard, and the mile. Now the
units of area are the square inch, the square foot, the acre, and square mile. There are of course
others. If you're buying carpet you probably have talked about buying a certain number of square
yards. You might even have called it yards. Now there are other units of distance and an area
of course. The barley corn is still used in some shoe sizes. That's a third of an inch.
There's the hand which is four inches for describing the size of a horse. There's the rod that's
used in surveying. That's 16 and a half feet or five and a half yards. There's a chain which is
also from surveying and building which is four linear rods which you can convert to 66 feet or
22 yards, a lot of inches. And it probably originally referred to an actual chain that was used,
that was a standard length so that you could measure a field without having to keep moving the
ends of the chain hundreds of times. It was probably convenient for measuring frontage with only a
few measurements and getting it right. The fewer times you have to play around with the end points,
the less error correction you'll have to do when you finally look at the data in your diagrams
and figure out the area of the field or piece of property. Let's see what are some others.
Well, there's the fur long that some of us know from horse racing, although it's really a more
general agricultural term from plowing. That's 220 yards or 40 rods or 10 chains. I guess the fur
long was originally the greatest length of a furrow that you could plow without resting your
animals. So it might have come from furrow long but furlong is what we have and it's about an
eighth of a mile. In fact, it is an eighth of a mile and I'll get into that in a moment. Another
unit of measure would be the league which was supposed to be about an hour's walk, I guess,
through the woods because it's assumed to be about three miles. So if you're looking at 20,000
leagues under the sea, that's really, really deep, just kidding. I'm really just kind of being silly
there. 20,000 leagues refer to distance traveled in a submarine which travels under the sea.
For ships at sea, one league would be three nautical miles which is more consistent and useful.
So I'm going to now share with you what I'm calling brilliant insight number one units of
distance are somewhat arbitrary. Now we did not standardize on inches or feet or miles or
these others because they're in some sense magical numbers handed to us by some deity or whatever
you pray to. We used them because they were convenient and we standardized on them because they
let us talk to each other. When you have a standard unit of distance, you can start talking about
how far something is or how tall someone is or the signs of a room or the length of the
piece of wood that you're buying without having to be there in the same place at the same time.
I don't think we'd get very far in building houses if builders had to ask for
boards that are as long as my arm or a plank that's oh yay long. If you're not standing in the same
room if you're talking on the telephone, yay long doesn't buy you anything. So you really want to
have standard units. So you can order things by writing down the measurement on a piece of paper
and sending it with a messenger or in these days you could order it online because you have measured
it and both people on each end of the communication will know what's meant when you say something is
six feet long. Okay that's the English standard unit six feet. Okay I know I know how to measure that
so I'll give you the accurate measure. Okay now some of the some of the units that I talked about
earlier seem a little odd right seem a little just a little bizarre. Now the rod and the chain. Now
these are these were I guess used in measuring farmland of building plots and other things that
surveyors need to measure. Now a rod is five and a half yards or 16 and a half feet. What is that?
I guess it was convenient because that's probably a the size of an actual rod you could use to measure
in certain places. It's not so short that you have to lay it down 150 times to get across
someone's frontage but it's not so long that the stick starts to bend and warp and collapse under
its own weight. Maybe it's kind of a convenient link. Now for longer distances you could use
the chain which is four linear rods or the length of some surveyors chain back in history.
Now I guess suppose you could have defined these originally to have been longer or shorter
but this is a standard that emerged from usage over time and that's what we have. Okay now a fur
as I said before that's the an agreement that it's the longest row you can plow without resting the
animals and it happens to be ten chains long. So you've got even though these are somewhat weird
sounding units of measure or distance they are related and they're consistent with one another.
So you can use them together and not have to worry about this fractional piece left over because
rod and the furlong aren't exactly in sync not an even number of rods in a furlong
no they're consistent and it works. Now if you want to get into bizarre let's go to the acre
which is the measure of area and if you're ever on a quiz show an acre is 43,560 square feet.
How about that huh? But if you're ever on a quiz you should remember that if the category is
English units of measure. Okay now it's defined as an area of a plot that's one chain wide by a furlong
in length and if you remember that a furlong is ten chains you will see that an acre is really
the area of a rectangle that's one chain in width by ten chains in length so you could call it
ten square chains and square chains is as good as square feet or square yards it's a unit of area
just as one square foot is the area of a square that's a foot on each side a square chain would be
the area of a square that is one chain on each side so it's not really that hard to see where they
might have come to this unit of measure that turns out to be this weird number but it came from
something that was consistent to the people who are measuring fields using instruments like chains
and it actually makes sense that an acre is the area of a field that has sides that are integer
numbers of these chains so when you multiply it together you get a weird looking number but it's
actually quite easy to see where it came from and if you're not convinced I'm not going to be able
to convince you I'll just move on now let's see let me back up a bit how did I get to that bit
about an acre being ten square chains because it was defined as an acre is one chain by one furlong
now I also know that a furlong is ten chains and I can set up a conversion factor by comparing
ten chains and one furlong because if I take two things that are equal let's imagine an equation
one furlong equals ten chains and I divide them both by the same quantity let's say I divide each
side of that equation by one furlong well then I'll get one furlong over one furlong equals one
and ten chains over one furlong must also be equal to one because equals divided by the same
thing if a divisor is not zero will give you equals so when I'm doing unit conversion I'm going to
get to this in just a bit I'm really in a sense I'm multiplying by one because one furlong
is ten chains so if I multiply by a quantity that is a ratio of ten chains to one furlong
I'm really multiplying by one so I get an acre is one chain times one furlong times one but
instead of one I'll use ten chains over one furlong furlong's cancel I get an acre is equal to
one chain times ten chains or ten square chains so keeping the units in the equation is a kind of
magic it really helps I think we should go forward I'll explain a couple of other things and then
I'll get into converting between units the mile has a similar story except there's a historical
development that explains why is it that a mile is 5,280 feet well the mile came into use in the
culture that gave us the English system of units during the Roman occupation because they marched
a lot the Romans had a unit of measure that the English people began to call the mile they
standardized on five thousand feet which was about the length of a thousand double steps
or paces a person from that period two steps would take you about five feet so they decided okay
five thousand feet a thousand melee paces let's call it a mile and it worked for them for
probably fifteen hundred years now the Roman mile was a little short for practical use
partly because the Roman foot was shorter than our modern English foot so we ended up getting into
a lot of different measures that were all called the mile some of them persist to this day
it was the old English mile the Irish mile the Scottish mile and there was probably a Welsh
mile and other miles depending on where you lived now going forward if you were trying to
standardize what would be a good standard well I was thinking that I would speak of this to
simplify things using a Roman inspired mile which would be 5,000 English feet
which would make things a little bit simpler than referring to all the different legacy miles that
we were talking about before so I'm going to call 5,000 English feet the Roman inspired mile
now Elizabeth the first came in and she created something through parliament called the statute
and the statute mile was set equal to eight furlongs which is our current mile is seventeen hundred
and sixty yards or five thousand two hundred and eighty feet I know that metric users are probably
looking at this and saying all right you have a unit that was five thousand feet is a mile
and you made it five thousand two hundred and eighty feet to make it equal to eight furlongs
what gives why not at least make it ten furlongs or something well I'm sure that even though
Elizabeth was the queen she still didn't want to create huge disruptions in society the goal
presumably was to set the new mile equal to some integer number of furlongs because the furlong
was really in use it related to agriculture in many different ways as we've already seen
how we measure farmland and even how we set work rules on how much to plow without resting
and people were using and they liked the mile as it was but it would be convenient to have
an even number of furlongs in a mile so as a compromise this new statute mile was pretty close
to my Roman inspired mile of five thousand feet only about five percent longer and within
striking distance of the miles already in use and yet it was equal to an even number of furlongs
so you didn't have to say it was oh eight furlongs less two hundred and eighty feet
that doesn't really doesn't really work but if you say it's eight furlongs then okay you can
even work with that because then you can have you can quote distances as a quarter mile a half a
mile and if somebody's trying to picture that they can say well that's a quarter mile is two
furlongs a half a mile is four furlongs so it does kind of make sense it has practical benefits
because you can talk about how far something is in terms of furlongs which you might know
or the new mile which people are getting used to it's it's defined in terms of something familiar
and yet it's close to the old unit that people used to use and I guess anybody who didn't like it
very much could either move or be very confused and they talked to other people so now we have
this this unit called a mile that measures five thousand two hundred and eighty feet so if we're
gonna have units that are these crazy multiples of something that we do know like a foot it would
probably be helpful if we knew reliable ways to convert between units so that we're comfortable
that we're getting the answer right if you've ever had to convert between temperature scales you're
going to like distances in about a minute because remember that the whole thing about converting
between Fahrenheit and Celsius where they had zero in different places and there were negative
temperatures and positive temperatures and they they started in different places you had to do
this offset thing and then adjust to the scales well distances don't do that zero is zero and there
are no negative distances unless you start talking about vector quantities which have direction
as well as length and if you understand vectors you don't really need to listen to this podcast
because you probably will understand all of this at least as well as I do so just think of it this
way you don't have to worry about shifting anything units of distance you can define them in terms
of the scale factor you know there's a foot and if you want to use a larger basic unit you could
use a yard three feet cool there's an inch but if you want to use a more refined unit or talk to
somebody in Britain or in Canada you could use a centimeter which thank goodness is now exactly
2.54 centimeters to the inch it used to be approximately that but somebody changed the length of
one or the other of the platinum bars that they use as the standard for these things so that the
centimeter is defined so that it is exactly 2.54 centimeters to the inch but don't be afraid of the
2.54 bit because a mile is defined as 5,280 feet a foot is 12 inches a hand is four inches a yard is
36 inches they're just arbitrary fixed units of distance they're just of different lengths
so that if you take a distance measured in one unit and you want to convert it to another unit
all you need is the scale factor that converts the first unit to the second and all of this works
because we have agreed it's easy to agree in a natural way on what is zero distances so we don't
have to adjust for shifts and origin as we will have to do when we play around with non absolute
temperature scales and so on we'll get to temperature scales and non absolute scales soon enough
but it'll be in another show so you don't have to tune out now we're not going to go there now for
absolute scales like distances all we need is a conversion factor and a calculator if you need one
I don't think I'll be doing any calculations that really need a calculator but if you need one
get it out now okay yeah for absolute scales like distances we can convert from from any one unit
to another one using a conversion factor and I'm going to show you how to set these up
because when you're first looking at conversion factors the question that I get all the time
from newbies is all right I know that an inch is 2.54 centimeters so I want to convert from
centimeters to inches or inches to centimeters how do I know whether I'm going to divide or multiply
by that 2.54 and people ask us all the time I'm serious I mean it's it's a problem that you have
to think through or did you get it right but once you have a system for doing it you don't have to
think about it every single time you approach it if you're just doing it at all every time you might
have to go through this whole thought process of oh my goodness how do I do that and do I divide
if they bigger they smaller all that's good but if you have a system for doing it that takes care
of all the accounting for which unit I'm in now you'll have a much better chance of having the
right intuition getting the right number and getting the right units so that nobody's embarrassed
or getting hurt I think we ought to fix this in your minds by working through a couple of examples
okay first I guess I'll pose a couple of problems now I know that a foot is 12 inches so how many
inches would there be in say 10 feet or look at the other way how many feet might there be in
660 inches two different problems I warn you in advance so you can't say oh just you can't just
pick off the numbers that I've quoted because they are two different problems but there I chose the
two problems because ones going from inches to feet the other ones going from feet to inches now
it's clear that going either direction that a factor of 12 should really be involved because the
foot is 12 inches and how do I know when I'm going to either multiply or divide by 12 in the conversion
well let's take a look at it if we do it with a naive setup then I'll answer the first one by
saying 10 feet well that's 12 times 10 inches or 120 inches and 660 inches is 660 divided by 12 or
let's see 600 divided by 12 is 50 60 divided by 12 is 5 55 feet well how do you know that you did
it the right way in each case I sort of wrote down these numbers and you can see it in the show
notes that without the units it looks like magic because I just I just sort of knew which I had to
do I'm going feet inches I I multiply going inches to feet I divide inches are not feet and the
only way to make sure you're doing the right thing when you go from under the other is to develop a
system and this system is fairly simple to write down the calculation in such a way that you
cannot get lost unless you make it all messy and everything but if you work through the calculations
and cancel units against identical units and numbers against numbers and multiply everything
together if you come out with the right units at the end all you have to check is your arithmetic
much easier than having to check whether you did the right calculation that's checking my math
the process by which I got from one to the other if I got from inches did a conversion factor
with that that whole thing it relates the the source unit and the target unit in the right way
and I do all my canceling and I get the right unit at the end I've done the right process
and all I have to check is whether my multiplication was right and I can do that with a calculator
and have some confidence that I got it right without having to step it off um if I just write down
numbers I'm yeah I might get the arithmetic right but if I'm not keeping track of everything that I
did I may not be able to be sure that I worked out the procedure quickly that I got the math right
so here's a system for creating factors conversion factors that tell you exactly what's going on
at each step so when you're doing the conversion you can really be sure that you know what's going
on and that you've got it all right and the basis of it I think of already said is the very obvious fact
that when I multiply any number any quantity by one the answer or the value of that quantity remains
the same how do I turn that into a system for doing successful unit conversions well let's start with
the other idea that I already told you about let's start with identities that we know are true
in this case we're talking about inches and feet so let's say let's start with 12 inches equals
one foot I've already said if I divide two equal values by the same quantity I'm not saying number
here I'm saying quantity because the quantity includes the units and that's the the leap that we're
making we're going to carry the units with us and if we do it correctly we can use the cancelation
laws that anything divided by itself will be one to work through the conversion and make sure that
the quantity that we end up with including the units is in the right units and it's the right number
so let's turn that equation 12 inches equals one foot into conversion factors that work in either
direction okay to go from inches to feet I can divide both sides of this equation 12 inches equals
one foot by the quantity 12 inches the left hand side 12 inches over 12 inches is equal to one
the right hand side is one foot divided by 12 inches well since I started out with an equality
I divided by the same quantity that's not zero that's undefined I have to have equals so that one
foot divided by 12 inches is equal to one now if I go the other way I could divide both sides by
one foot that actually gives me a conversion factor to go from feet to inches and I'll tell you why
that works in a second so 12 inches divided by one foot is the left hand side and that's equal to
one foot divided by one foot which we know is equal to one feet cancel with feet one canceled with
one and I get one so that I know that 12 inches divided by one foot is equal to one now if I multiply
12 inches divided by one foot by any quantity that's in feet I'm multiplying let's say it's
three feet that I'm multiplying by that well the three feet times 12 inches over one foot is equal
to well just rearrange terms and I get 12 inches times three feet over one foot and feet cancel
up and down so I get a unitless value of three so then I can say 12 inches times three is 12 times
inches and I get 36 inches which I know is three feet because I've used the yardstick I know that
three feet is a yard and it's also 36 inches so that accords with what we already know it accords
with intuition and it uses very simple techniques like multiplying by one in a way that the unwanted
thing cancels out leaving you with the quantity the units that you want so let's use this in the
problems that we've already talked about so let's say that I want to convert from feet two inches
and I want to go back to my problem how many inches in 10 feet well 10 feet is equal to 10 feet
times one which is equal to 10 feet times 12 inches over one foot which we know to be one because
we've just done that and that is 10 feet divided by one foot times 12 inches now feet cancel I get a
unitless number 10 so I can say that 12 inches times 10 is the same thing as 12 times 10
quantity inches and that's 120 inches which is what we had calculated before except that the
magic is gone it's all very systematic I start with the units I have I multiply by a factor
conversion factor which has the units I want upstairs the units I want to get rid of downstairs I get
cancellation and I'm left with a multiplication problem that's all in the unit that I want very cool
it it's hard to imagine getting that wrong now suppose I'd use the conversion factor that's
equal to the one I used except upside down so that I multiply 10 feet my starting point and that's
equal to 10 feet times one equal to 10 feet times one foot over 12 inches and what that is is 10
square feet divided by 12 inches so it's 10 12 so the square foot thing divided by an inch what is
that well the equation is actually correct but it's stupid because it's not in a unit that I want
and that I can interpret now if I put in enough conversion factors to cancel out the square feet
and put inches back and everything I would get the right answer but I know that I'm I know that I'm
doing something wrong when I put in this conversion factor because the units are crazy they're
not what I want so I know the number can't be right unless some really big coincidence where
everything I don't want cancels but so when I use the form of that conversion factor where the
units don't cancel each other I can't know what I'm looking at I can't tell whether I'm right
because my units are wrong so I can't just look at the number and check it I can't just multiply
numbers blindly I need to look at the units so it actually helps you to carry the the units along
so that you know whether you're multiplying or dividing by that 12 because what you're really
multiplying by in this case is not 12 you are multiplying by one foot divided by 12 inches
that's why you couldn't say that's 10-12 of an inch is 10 feet no it's 10-12 of the ratio of
square feet to inches so that if you don't get what you want in the units on the right hand side
when you have done all your cancelling go back and check your conversion factors and make sure
that you apply them correctly when you do that you can unwind all this stuff and get to the right
answer answer that you want in the units that you want so here's brilliant insight number two when
you use unit conversion factor you can help your cause by carrying along both sets of units in the
form of a fraction as you are writing down your your problem if the right hand side of the equation
doesn't have the units that you were looking for your numerical answer is almost certainly wrong
now what can we derive from that well the implication is that to convert units of distance
you need to multiply by a conversion factor that's in the form x target units divided by y original
units because when you do that and you write the conversion factor out in its full fractional form
with the units and you carry out all the multiplications and cancellations you can see whether you've
got the right answer whether you've done it right it makes it hard to do the wrong calculation
because you have this crazy set of units on the other side that tell you that you've done something
that didn't make sense this is going to revolutionize your life if you've always depended on
calling someone who's good at math to do unit conversions for you because if you do it this way
and the units match you're good if you've got the wrong number it's because you did the wrong
arithmetic and that's easy to check if your units are not right you're solving the wrong problem
now the equation may still be correct because you may have done all the multiplications but it's not
expressed in the units you want so it's not very useful so let's use this the same system to solve
the second example i wanted to convert six hundred and sixty inches to feet so i start with six
hundred and sixty inches which is equal to six hundred and sixty inches times one i multiply that by
the fraction one foot divided by twelve inches because i'm cleverly setting myself up to be able to
cancel inches with inches and be left with feet that's why i put feet on top that's the one i want
at the end inches on the bottom that's the one i want to cancel and i think i'm gonna get the right
answer when i just multiply through the numbers and cancel the units and that's exactly what happens
when we rearrange terms this whole thing on the the second step putting in the conversion factor
can be rearranged so they've got the original number six hundred and sixty inches i was multiplying by
one foot over twelve inches so i can put that twelve inches directly under the six hundred and sixty
inches and then all i'm left with some multiplying by one foot oh that looks good because it's clear
when i've got inches over inches that those cancel and i get an unit list number that's six hundred
and sixty divided by twelve and that ratio that answer six sixty by divided by twelve which is
fifty five is now multiplied by one foot so it's clear that that fifty five times one foot is
fifty five feet and feel pretty confident that that's probably the answer if i did the division
of those two numbers correctly at least i know i'm in feet so this gives you a real sense of
confidence which you're going to need because sometimes you don't have direct conversion factors
and you have to actually combine sets of factors may have to take one conversion step using one
identity a second one using a second conversion factor maybe even the third conversion factor
but if you do this step by step by step aiming for conversion factors that cancel out units you don't
want and put in units you do to go to the next step you will zero in on the right answer in the
proper units and if you carry the units along with it you're much more likely to get the right answer
or at least you have done the right process so that you can go back over your arithmetic and make
sure that it wasn't some silly mistake that you made along the way you're like calling nine times
six fifty six instead of fifty four that that kind of mistake you can fix because you can see
how you did okay great so i just have to check some arithmetic that's much easier than determining
at each step do i divide or do i multiply because that gives you two choices to make on every
conversion factor you use if you had to use four of them then you've got sixteen combinations to
try you don't want to go down that road make your life easier carry the units and you'll see that
you can convert between units of distance and we're going to get into area in just a minute and
you can do this with confidence says great okay that's it for today's show you back with more
practical math and more units here on hacker public radio thanks for listening bye
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