hpr-knowledge-base/hpr_transcripts/hpr1353.txt

Episode: 1353
Title: HPR1353: Practical Math - Introduction to Units
Source: https://hub.hackerpublicradio.org/ccdn.php?filename=/eps/hpr1353/hpr1353.mp3
Transcribed: 2025-10-18 00:02:55

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Hello, and welcome to Hacker Public Radio.
This is Charles in New Jersey, and I'm back with a new series on Practical Math.
I have no idea how long this series will run, but it's going to run at least a few shows
because I've been asked to take on a rather large topic, and that is the use of what we'll
call units, which you could think of as labels that are put on numbers to make calculations
easier in the world that we're living in.
We live in a world of objects, times, distances, things, volumes, weights.
All of those things are things that we have to keep track of, and one way to keep all
those things straight is to deal in units.
Okay.
Now to introduce the topic, I guess we should, I guess I'll give an overview of the series.
This first episode is going to be an introduction to the idea of units.
What's a unit?
And maybe give some illustrations of how you'd use it.
I'd define units somewhat more broadly because I'm including things that could be called
labels or something like that, but even those types of units are useful when you're doing
accounting, which some listeners may be interested in doing at some point for a business or just
to keep track of the things in their inventory and their kitchens.
So I've included things that you count as well as things that you measure as both being
units.
Okay.
Why are we doing this?
Well, I think, to me, the units are really the bridge from learning these tables of operations,
the addition and subtraction and multiplication tables and division.
We learn these abstract operations on numbers and we go through all the drills in our classes
in school, but those numbers really don't mean anything.
And then when we're put into a context where they do mean something, we get these, what
we used to call word problems and the kids I used to tutor would call them story problems.
Same idea, but they're still very contrived.
And I never really liked them.
I think you should actually put math to work in real life situations and solve problems
that you really have.
I suppose you can't do that in school, but for anything to really sink in, you should
really use the math or arithmetic like a tool to help you do something as opposed to having
it be a hurdle that stands between you and graduation from whatever level of school
you're in at the moment.
So I consider units to be kind of a bridge from this learning the actual mechanics of doing
arithmetic.
To actually using it in our world where you have to deal with objects that you have to
recognize and classify and count or distances or times or rates, your speed limit or the
flow rate of your gas pump in liters per minute, that kind of thing of volumes measuring
quantities for cooking, temperature, heat, electrical current, voltage, any kind of
cooking using recipes or that sort of thing.
Require you to use and have some ability to convert between units.
Now my goal for the series is to convince you, I hope, that embracing the idea of units
and carrying units along in your work with numbers as you work through real problems
in your life, carrying along these units as you go can actually help you work with more
confidence in using maths in your life.
I can tell you that when you start to use math to solve real problems, you're going to
run into units, so you might as well make them your friends and learn how to use them.
Now for this particular episode, I just want to look at what units are, what they do,
what types of units and other numbers there might be, and how to mix unit list numbers
with units, because not everything, as we'll see, not everything does have a unit attached.
So let's start segment one, what do we mean by units?
Well I'm going to give kind of a definition, but I've made it less formal than you might
find in a dictionary, so I'm going to define two types of units that are, that could be
useful in practical math, and the first would be counting units, I'm putting it first
because it's in one sense easier, because you're just counting things, but also because
we're going to be finished with this fairly quickly, there's not a lot you can do, you
can't really convert from apples to oranges without trading with someone.
So the first is a counting unit, which I'm going to call any individual thing that you
can treat as being single thing, or that you can think of as complete.
This can also apply to, let's say, an individual piece of something larger, like a larger or
more complex system, mufflers are both a thing in themselves, and they can become part of
your car.
That's the kind of thing where it's both a unit in itself, and it can become a part
of another object that's more complex as part of that unit of a different kind.
There are composite objects in the world as well as in some types of software development.
I like to think of objects that would have counting units applied to them as anything
that I could keep in an inventory and say the pantry or in a warehouse.
Things that I would want or need to count or to know how many I have.
The other type of unit, we're going to spend a lot more time on, would be measurement
units.
It's a little harder to understand where these come from because there are just so many
of them.
And any type of thing that you're trying to measure, there are just lots of them.
But what you usually have is some kind of quantity that has been chosen as a consensus standard
that you can use as a common benchmark for comparing other quantities of the same kind.
So it's a standard, usually agreed upon benchmark that you can use to compare with other
people's quantities as long as they're of the same kind.
What does that mean?
The same kind.
Well, simplest example is you wouldn't try to compare distances to times or volumes.
Simple is that.
Same kind of object or thing or phenomenon or whatever it is that you are trying to measure.
It is a little bit slipperier, but it's easier to understand it if you think of it as a
communication tool for talking about quantities because it lets you do it when you're not even
face to face.
If you can think in terms of standard units and if you have the ability to measure things
in standard units where you are, then you can avoid expressions such as I want to board
that's yay long or I believe the man was kind of tall, that sort of thing.
So standards let you talk about things more precisely and not have to be in the same location
at the same time.
Now measurement units were probably invented by people who were buying and selling things
to make sure that neither one was being cheated or perhaps by the spouse of someone who
does a lot of fishing.
You got that, right?
Okay, then there's a third kind of unit that is really just another measurement unit but
I'm going to separate it into a different class because you usually have to make a couple
of measurements and then do something with them to create these and I'm going to call these
composite units.
Now units themselves can be multiplied together or divided to create new types of units.
Now some people would skip the word unit here and call these derived quantities but since
this is hacker public radio that sounded too much like programming talk to me, it sounded
like you were making this kind of new unit as a kind of subclass of some base class unit
and that's not really what we're talking about.
I chose to use the word composite units because it creates the kind of picture of putting
things together or doing one operation after another, like multiplying and then dividing
or measuring distance dividing by the time it took.
That's the kind of thing that I'm talking about.
I suppose you could also have derived units such as for the change in dimensionality.
We have distance, let's say, that our unit of distance is a foot for those in US.
Now the unit of area could be the square foot which would be one foot by one foot.
The area of a square one foot on each side.
Likewise when I jump up to volume I could have a cubic foot which would be in effect distance
cubed or the volume of a cube that's one foot on each side.
The type of composite unit might be speed, distance over time, that would be average speed
or change in distance per unit time which would be the speed over a given interval and if
you make those intervals small enough you can get at the instantaneous rate of change,
but this is not a calculus class so you are off the hook.
The units of speed could be kilometers per hour, that kind of thing.
Low rates I think I mentioned volume per unit time like liters per minute.
Pressure would be another one which would be four supplied to a given area such as, say,
pounds per square foot.
Then there's density which would be mass per volume as in kilograms per liter.
In the counting units area you could actually have composite unit in the context of rationing
or allocating something.
If you had a bushel of apples and you were going to a portion of the amount in such a way
that you keep the doctor at a distance then you might have an apportionment that's in
apples per day and that would be a kind of composite unit.
Over a longer period of time or in a more complex situation you might have apples per
family member per day if you were sharing them amongst the group of people as opposed
to just portioning the amount over time.
Now I'm sure we'll run into more of these types of units in later shows.
I don't think we're going to call them composite units at that point.
We'll just say we're going to talk about rates today or about speed or pressure or something
like that.
But we'll run into some of these things as we go along.
Now that we've talked about units, we should recognize that there are other kinds of numbers
and not every quantity will have units attached, doesn't always make sense to worry about
whether something has units or not.
So numbers can be unitless and why would that be?
Well unitless numbers help you to make sense of quantities that do have units through
things like relative comparisons or extrapolating from something that you have, any kind of
projection or comparison, you would need unitless numbers to express the relationship between
a couple of quantities that have units.
For example, a percent change is a unitless usually floating point or rational number unless
you tie it to an elapsed time period in which case it becomes a rate or a composite unit.
And that would have units of say percent change per year.
Another example of unitless number would be if you had a number of different types of
objects or if you were splitting up quantity of cache or I don't know wheat flour or something
like that amongst the group of people, then you might want to keep track of the percentage
of the total that each one has.
And those percentage of total values would be unitless fractions as well.
You can also use unitless numbers to scale units.
Let's say you have a unitless integer, you can apply that to any kind of unit, including
a counting unit.
So let's just try a few expressions like two feet.
Could either be 24 inches or literally two feet.
Three apples, again we're in things, four quarts.
That's more in the volume realm, ten meters in distance and so on and forth.
In conversation we might talk about twice as many or ten times as far or we might talk
about doubling a recipe, which would be multiplying every single thing in the recipe by multiple
of two.
Okay.
Now counting units, usually you would think of multiplying them by an integer but if you
multiply them by unitless fraction and the result rounds to or comes close to rounding
off to an integer value, then you're probably okay.
This is where your arithmetic book has, Mary has two and a half times as many apples as
John.
Well that's fine if John has four apples and Mary has ten apples because that would
be two and a half times and still be an integer number of things.
I don't want to get into two and a half hands or three point six apples or something
like that.
You could define it and then you could use it as much as you want but I'm not going
to go there at least in this series.
Now measurement units can be applied by almost any arbitrary scale factor.
You might think in terms of how big something is by relating it to something that's more
standard.
Seems like the news always has some kind of reference to a land area that's 3.6 times
the size of New Jersey.
Maybe Luxembourg is the standard in the EU or something.
You might want to know how far something is relative to some distance that you know
by saying oh I'll meet you halfway or how much something is either in volume or weight
or something like that.
So a recipe might say if you're using white flour you'll need 30% more.
I guess that would be accounting for the different densities of the white flour versus
wheat or rye or something like that.
Okay.
How do numbers go from having units to not having units?
Well when values with units are divided by other values with the very same units the
result is a unit list number and that's going to be important because that's how we
define conversion factors.
We're going to assume that it's okay to take these numbers with units attached to them
and multiply them, add, subtract them, and divide them and have all the usual laws of
grouping and permutation so we can switch the order cancellation if it's the same in
numerator and denominator of a fraction that's the upstairs and downstairs part.
If it's upstairs and downstairs then you can cancel feet with feet.
You can cancel inches with inches or seconds with seconds.
You should not cancel feet with inches.
Think in terms of canceling like and like.
If they don't match you can't cancel them, at least not without doing some extra work.
Okay, so I've beaten that one to death.
Now this dividing of one thing by another is I'd say that the whole percent of total and
the percent change thing, those are prime examples of dividing one number with units
by another to get a unit list number that means something in the context of comparing
those units.
People frequently do this when they're comparing distances.
I'll use a very obvious example.
St. John'sbury is 45 miles away and Barton is only 15 miles away.
So you have to drive three times as far to get to St. J. See how he used the number three
which is 45 divided by 15, which makes the distance from where the speaker is to Barton,
the standard unit, and I divide the longer distance by the shorter distance to figure
out how much further away the more distant town is than the closer town.
And it appears that duty calls.
Okay, we're back.
Now conversion factors between units work on the assumption that we can just divide units
by one another and when we multiply those fractions together, we can depend on the cancellation
law of multiplication to have the like units cancel leaving us with just the unit that
we want to convert to.
We define everything correctly.
We can start with a measurement that we have, let's say two and a half feet and we want
to express that in inches.
We'll multiply by some kind of fraction that would be, say, 12 inches per foot.
The feet will cancel leaving just inches and I'll have two and a half times 12 or 30
inches is equal to two and a half feet.
We can leave that as magic for right now because there'll be people who have to divide
by the 12, it is multiply by the 12.
How does that work?
How do I know what to do?
Let's wait until next time.
We're going to go over how we create conversion factors and what we do with them, how they
can be used to clarify what we're doing when we're working with units trying to understand
them and convert between them.
Now you can actually call conversion factors by that name derived quantities that we talked
about earlier because you create them from something that's called an identity which
is just a statement of equality that you know is true either by definition or by believing
some authority such as that one I just used that 12 inches equal one foot.
We're going to talk about conversion factors next time.
Now I guess there will be those who will say why bother with counting units aren't they
just names really?
Well yes, counting units are labels or names that you can apply to the individual items
in a total count but they still can be useful in their place.
Using counting units helps us to make distinctions between items that aren't interchangeable so
we can keep track of the counts for each individual kind of item.
This is very handy when you're in a database and you use the group by keyword in a query
to roll up the counts of all different kinds of items.
But in normal life the names that you apply to things or the units that you use to describe
something really do matter because if you need two apples having even ten onions doesn't
help you unless there's somebody who's willing to trade.
So when you're dealing with objects it can help to know what kind of objects they are.
So thinking with units will help you to do things like keep inventories or to set up accounting
systems for your business or your home.
So you know how much of each item you have in stock, what you need to buy next time you're
out at the market and to determine whether you have enough in your budget to afford to buy
those things.
Now counting things by item type can also help you manage your kitchen or your budget at
home.
So that's how you define your categories by applying labels to things mentally or literally
if you want and keeping the count.
And perhaps a record of how much each thing costs but that's a separate issue.
That's another unit you're converting the item to currency to how many euros it cost
or how many dollars or yen or so on.
I'm going to push this just a little bit in segment two, it's called counting units.
Are you serious?
Counting units give context to the numbers that you're using in calculations that you have
to do because you're buying or selling or trading or just using up things that you've
stored in a beginning inventory.
Now here's what happens when you don't track units in counting problems.
And we'll go through how a lot of textbooks that are preparing you to take speed tests
in just pure arithmetic would have you solve some problems.
In fact, I'll be merciful.
I'll just solve one example problem this way.
Here's the problem.
John has nine apples in his basket.
If he gives two apples to Mary, how many does he have left?
Now if you're getting ready for a speed test, it seems like the textbook writers want
you to become some kind of a human problem compiler.
They teach you to parse the problem by looking for numbers and keywords that are embedded
in the text.
So first they'd have you fish out the numbers in their roles.
Hey, notice that nine is near the phrase in his basket and that there's a question that
says how many does he have left.
So nine must be the source.
Okay, and now let's look for the number two.
Now two is next to the words gives away.
It must be the change in quantity.
Next we parse out the operation.
Now gives away is code for subtraction.
So finally we put the numbers and the operation together and do the calculation supply a numerical
answer.
So nine minus two equals seven and we're done.
Now that's fine for a pretty trivial problem of nine minus two, but if the problem got
any more complicated, I think you're going to want more structure and a bookkeeping system
to solve the problem in a more robust way.
So let's rework the problem by tracking the units this time.
Even though it's apples, I know it's very simple, but just stay with me for a minute.
In this approach we do something radical, we read the problem.
Now I'll wait.
Okay, now we can parse it together.
And the first thing we look at is we see that John has a basket with nine apples in it.
So that's his beginning inventory.
The next thing we read is that John gives away two apples to Mary.
That means John's inventory of nine apples is reduced by two apples.
So now he has seven apples left in his basket.
So that's his ending inventory.
But those apples just didn't go into space, didn't say he ate the apples.
He gave the apples to Mary.
So what do we know now about Mary's stock?
Well, Mary now has two additional apples in her inventory.
Keeping track of where the apples came from and where the apples went helps us because
it keeps everything in balance.
Apples were not created out of nothing.
They didn't just appear in his hand, nor were they destroyed when he gave them away.
Mary now has them.
So the apples came from somewhere, from John, or John's basket.
And they went somewhere to Mary.
One way to check your work is if the apples out, the one's going away from John, does not
equal apples in, something's wrong.
So now you not only have a way to do the arithmetic problem, but you have a way to check your
work by checking to see that the number of apples involved is in balance.
So now having this information, this ledger, lets you answer questions with confidence.
And you can answer the question, John now has seven apples.
Note that John does not have seven, which is an idea, a number, a meaningless sound if
you're a two-year-old.
John has seven apples, that's important.
Now here what you're saying, that solution's exactly the same, you're picking nits.
And maybe that's true for an easy problem like this one.
It can look the same, even if there are some benefits in using units, they still appear
to be just labels in this kind of problem.
Now let's revise the problem in a couple of different ways and see if we still think
that the units were not adding anything.
Now suppose the problem had said instead that John gave two oranges to Mary.
Well if we were tracking units, we'd spot that discrepancy immediately.
It's not actually discrepancy if that's what he did.
Now giving away oranges does not really affect John's apple inventory, unless he had
the trade to get them, but there's no talk of that.
So the oranges must have come from another supply or account that we didn't necessarily
know about.
Now we can still talk about an increase in Mary's count of oranges and decrease in John's
oranges, even though we don't know the beginning or ending balance is really for either one
of them.
But we know that there were oranges that were exchanged, that they went from John to Mary.
So John must have either had two oranges in stock or he must have obtained them somehow
in a transaction that we don't know about.
Keeping track of the units and the sources and uses of those objects gives us a way to check
our answer and to know what's going on and to be able to determine whether a piece of
information is relevant or not relevant.
Suppose there was a different question.
Mary has three times as many apples as John.
How many apples would Mary have to give to John to leave each of them with the same
number of apples?
I think you're going to see that you would want to track your units now and write some
things down just to be sure that you're getting everything right.
Better yet, what if the problem was really, really crazy and it read something like this?
John has 19 apples and Mary has 14 oranges.
Now John likes oranges twice as much as he likes apples, but Mary likes apples three times
as much as she likes oranges.
How can John and Mary exchange apples and oranges to get the best equal gain in happiness?
See, this problem involves not just the tracking of apples and oranges, but you probably
have to create some kind of artificial, I'll put this in air quotes, happiness unit or
happiness function that gives a value that carries some kind of units.
Now it's the problem like this that makes most people hate economics.
And one way to solve this would be to use a method that's used in economics sometimes.
It's called defining utility function for each party.
We're not going to do that.
So you can rest assured, not going to waste your time creating artificial units.
But if you want to write abstract academic papers that no one understands, this is one
way that you can use units to enhance the incomprehensibility of your paper.
Now if you look at the statement of the problem, looks like John and Mary's preferences are
so different from the inventories that they were holding in the first place, that it's
probably best for them to just trade the baskets that they have and just have John hold
all the oranges and Mary hold the apples.
And that's pretty close to the optimal solution to the problem.
And I think I'll leave it there.
Now if you want to get even further out there, if the problem had involved trading some
of John's apples for some of Mary's oranges and maybe offering an offsetting cash payment
to correct an imbalance, any imbalance that might be left, then we would really need
units to help us make the best use of all of our information about the sources and uses
of resources and preferences expressed in terms of dollars and cents or euros and fractions
by tracking all these quantities and relationships for each object and currency in the exchange.
I guess the point is that problems can become pretty complicated and units can help you
with the bookkeeping that you need to do to work through to the answers.
I will give a piece of advice here if someone poses a problem like this to a group of people
at a dinner party that should give you the queue that it's time for you to suddenly
remember that you forgot to iron your curtains back at home.
Okay now the final properties of counting units and will be pretty much done with these
for the series.
I guess the first thing is a rule that compatible counting units can be added and subtracted.
Now compatible in this context really should be the same.
If the units are the same you can add them.
So for example six apples and four apples is six plus four or ten apples.
But if that was six apples plus two oranges that's a mixed expression and that's as simple
as it can get.
Now apples and oranges can't be added unless you are making a say a fruit salad, but
as units they can't be added.
Now in a math that's given in counting units can also be multiplied by an integer because
well that's like repeated additions.
So you can actually write out what that meant, let's say I had three apples and what would
it be like if I had six times three apples.
Well that's three apples plus three apples plus three apples plus three apples and so on
until I had six times three apples or eighteen apples.
Now you can also multiply them by a fractional amount but only if that comes out to a whole
number like that problem where Mary had two and a half times as many apples as John.
Now if you're wanting to define a multiplication of counting units by some kind of arbitrary
floating point number where the result's not an integer where you end up with six and
a half or three point six apples that's out of scope for the series and I consider it
not really worth the effort, but if you are bent on doing it you could probably figure
out some standard way to define a fractional apple and that would be an interesting if
unusual podcast.
Now counting units are not infallible either.
They have these limitations that they're denominated in integers usually just accounting numbers
but they also give you problems with classification.
This is especially true with organic items because they're usually not identical.
For example apples can vary they can vary in size like a recipe could call for three large
apples.
Well I have three apples are these large should they be larger.
Apples also vary by varieties.
Now apples in the U.S. can include macintosh, macoon, Rome, gala, all kinds of them, granny
Smith and so on and all of them can be quite different especially if you are an apple expert
then you would be quite apt to notice the difference and not even count a macintosh and
a Rome apple as being the same thing.
So classification problems are difficult with things like plants and fruits.
That's also true with animals that they can vary quite a bit within categories.
For instance cats, technically lions and lynxes and even little puff all qualify as cats
but they are very different.
So when you're counting cats you might not think to include the lion with little puff.
Even within a category like house cats, siamese, Persians and tabby cats are all just cats
until you have them living in your home and you might find that they have different personalities
or different behaviors or characteristics that make you consider them as different things.
There are some living things that are even hard to pin down as to whether they are animal
or plant or whatever.
I think of sponges and little microorganisms like paramicia and all those other things
you looked at through microscopes in school.
Now there are other things that can also create classification issues depending on your purpose.
You're using the objects for and how you value them and all of those things.
And you can get into arguments about these things but just remember that units are just tools.
We want them hopefully to work for you and not the other way around.
Second three, units of measurement.
Now units of measurement or measurement units are often what we call continuous.
You can divide them up as finally as you need to or just about.
So conceptually you can divide them into smaller and smaller subunits as many times as you like.
Although you may have trouble getting a sharp enough knife and you may have trouble seeing
that you've divided something into a smaller and smaller unit.
It's kind of unusual that the smaller the unit you want to break something into,
usually the bigger the apparatus has to be to do it and get it right and to measure it.
Not quite sure how that works but it seems that that's the case.
Just think of what size equipment you need to measure and add them.
And you can get the picture that sometimes when you go to the extremes measurements
can be very counterintuitive.
But in the range that we're dealing with, I'm going to assume that we're able to deal with them
and measure them and do simple error analysis and understand what we're looking at.
So yeah, physical limitations, yeah, they'll place practical limits on how finally we can
actually chop things up and still get a measurement.
And there are also real-world limitations on how much we can lump together.
You may not have enough things to keep lumping more and more of them together.
But I guess you get the idea.
All right, what kinds of things should we look for as being amenable to being measured in measurement units?
Well, I guess you can apply measurement units to distance or time or area,
volume, weight or more accurately mass if you're using a balance, energy, frequencies of light or radio waves,
voltages, lecture current, heat, temperature, and a lot of other things.
And we can measure these quantities with differing levels of precision.
Sometimes we can get a really good fix on something and sometimes we can just make an approximate measurement.
And it's all based on the, I guess, both the instruments that we have and our abilities to use them.
And of course, our budget.
Now, for all practical purposes, we measure within the tolerances that we can meet without spending our whole lives measuring.
If we have to spend years measuring something, well, if we have to spend years measuring something,
then what we should do is get a grant that pays us to do nothing but measure that thing.
But the rest of us have day jobs.
We measure things until they're close enough to fit the purpose.
And that's where I'm going with this series.
Some things you have to measure carefully like medicines.
Other things can be more approximate.
Like the distances in a road race.
The judges have to measure those accurately, but the spectators can be more approximate because they only want to know how long the road race is in terms that they can understand.
So what are some applications of measurement unit?
Well, one of the first things measurement units allow us to do is understand each other.
And you'll notice this if you go to another country or a different part of your own part of the world and listen to the news.
And suddenly, instead of, say, acreage or square miles, you're hearing about the hectares of forest destroyed or endangered by a wildfire.
Or if you're from an EU country, you'll hear suddenly about square miles of arable farmland in South Africa or temperatures given an unfamiliar scales like Fahrenheit.
And then there's the amusing parts such as the snowfall measurements.
If you're in a border state, you'll get quite different measurements because of the different units used in, say, Canada versus neighboring Montana.
Both numbers will probably be large, but the number in Canada will seem much larger until you realize that we're talking in inches in Montana and centimeters in Canada.
So talking with each other and understanding each other, it turns out to be the most important part of using units.
Now, another application might be following recipes to make bread or cookies or beer or any of the other items that promote world peace.
You need to understand units to be able to follow those recipes accurately and have everything come out right.
And that's really a communications process as well.
Another application would be for mixing chemicals for an old school darkroom, developer and fixer and stop bath and all that.
Or measuring chemicals for one of those very cool low-tech electronics home fab labs for etching circuit boards.
If you get that right, you'll have a circuit board that still works.
If you make the material that you're using for etching a little too strong, you end up with a very blank looking circuit board that's not very useful.
I guess in other things you can do or would be like buying gasoline in other countries and understanding their speed limits and other units of distance.
Time probably is the same, but kilometers per hour takes a little getting used to if you're used to looking at miles per hour and the other way around.
Now units can't really help you with the problem of driving on the wrong side of the road. You'll just have to make sure you master that part quickly.
Another benefit of units is you might be able to help your kids with their math homework and understand it for once.
That would be a major victory and it would feel really good if units help you to become sufficiently organized that you can actually help your kids for once instead of the other way around.
And a very practical benefit of using units. You could use units to check the dosages of your medications against your prescription.
It would be a serious problem if you and your child were on the same medication and you couldn't interpret from the dosage level whether this medicine is mine or is it my child's.
You just have to be able to get that one right. You have to.
And I'm hoping that this series will either help you to do that or inspire you to learn enough about units to be able to take care of these really important survival tasks.
And I'm hoping that we'll get to all of this and more in future episodes in this series.
I'm a little daunted by trying to cover this kind of material in an audio podcast where I can't bring in visual aids or pointed pictures.
You're going to have to visualize some things in your mind as I go along.
But I think that if we work together we can get through this and learn a lot about how to use math to help ourselves in in our lives in some kind of useful way.
And that's why I'm putting the the work into producing this series.
Hope you enjoy it and I hope that we all learned something here.
Thanks for joining me and we'll catch you next time on Hacker Public Radio. Good night.
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