Pratica di Shadowing: Cursed Units 2: Curseder Units - Impara a parlare inglese con i video

After posting my previous video on cursed units, and after it got a massive spike in views for no discernible reason, I discovered that kilowatt hours are extremely polarizing.

A lot of commenters argued

that kilowatt hours was a perfectly sensible unit for ordinary people who measure their appliances in watts and hours, while the alternative unit to megajoules is much less intuitive.

But there were also a lot of other commenters who sympathized with me, saying they had given the same arguments to their friends and family and were treated like lunatics,

and that the oppressed dual appreciators should rise up against cursed units.

The truth is, I actually don't feel strongly one way or the other.

I just find it fun to play a round of notational standards given to us by history, and make a caricature out of it.

My goal is to teach you something about mathematics, not to claim that these are signs of societal dysfunction.

Wait a second, people are saying they've seen kilowatt hours per thousand hours.

Isn't that just watts?

Wait, I need to check this for myself.

You're up!

The best explanation of this I've found is from a user on the Electrical Engineering Stack Exchange who suggests

that for a long time people conflated wattage with lightbulb brightness.

This meant that when converting from older incandescent bulbs to more efficient LED bulbs, someone might swap out their old 40 watt lamp with a new 40 watt LED lamp and get flash banged ranked.

So manufacturers describe the brightness in watts, and the efficiency in kilowatt hours per thousand hours.

This of course would never have been a problem if anyone cared about Lumens or Candela, the actual units for brightness.

But you know what?

It simplifies nicely, so it's not the worst.

There's also kilowatt hours per annum, which is what my fridge has.

That's seconds, hours and years all in the same unit.

You know what else was polarizing in the last video?

The music!

And I intend to keep it that way.

Anyway, here's some more cursed units.

The radian is a unit of angle such that 360 degrees is equal to 2 pi radians.

This is so that if you have a circle with a radius of 1 unit of length,

let's say 1 meter, then an angle of X radians subtends an arc length of X meters along the circumference.

That's pretty neat, and what's also neat is that if the radius of the circle changes, then this arc length changes proportionally,

so we could think of a radian as being 1 meter of arc length per meter of radius.

A meter per meter.

Hmm.

So, a radian is just 1.

When units completely cancel out like this, we call them dimensionless.

Radians are quite special here.

If we used a different measure of angle like degrees, we would get a different number.

The fact that the radian is the unique angle

that gives 1 is the reason why in mathematics we commonly assume

that everything is in radians and don't write any units for angle at all.

This has some weird run on effects.

There are angular versions of momentum and force, which are called angular momentum and torque, and since these depend on the radius as well, they pick up an extra unit of length.

If we continue ignoring radians, then torque inexplicably has the same units as energy.

There's also a 3D variant of radians.

A Steradian is a solid angle subtending an area on the surface of a sphere equal to the square of the radius.

So by the same reasoning, Steradians should be measured in square meters per meter squared.

Which is a different thing, but also the same thing.

But are radians really dimensionless?

I don't think so, because there's another conflicting way that we could make angles dimensionless.

Just take the number 1 to be 1 revolution.

This is the more sensible option for describing the frequency of something rotating, where we use units like RPM, revolutions per minute.

So we have two different standards, which are off by a factor of 2 pi, and this results in 2 pi's popping up all over the place in any equations relating to rotation.

So angles aren't really dimensionless, we just like to pretend that they are.

But now let's look at a unit that is dimensionless, but we like to pretend that it isn't.

A mole is a unit of quantity equal to the number of atoms in 12 grams of carbon.

If you're thinking, isn't that just a number?

You're correct!

It's called Avogadro's number, and it's approximately 6 times 10 to the 23.

Despite the fact that the mole is very clearly dimensionless, it's still counted as an SI base unit.

The perpetrator here is chemistry, and when it comes to shoddy units in chemistry, trust me, the mole is the tip of the iceberg.

First, let me state that while you can multiply, divide and take powers of units, more complicated functions like exponentials and logarithms only apply to dimensionless quantities.

You can see this in complicated equations in other areas of science, the units have to cancel before we can apply weird functions to them.

After all, it doesn't make sense to take the logarithm of a liter, right?

Except that's exactly what pH is!

The acidity of a solution in water is determined by the concentration of hydrogen ions.

You need to measure this in moles per liter, and then take minus the base 10 logarithm of that, and that's the pH.

Uhhh...

So, what's actually going on here is

that we're secretly dividing by a constant 1 mole per liter to make the units work out.

And this is very important because if we measured the concentration in different units, we'd have to convert this one mole per liter to those units as well.

This is why the whole study of dimensional analysis exists.

When you conflate units, you get the wrong results.

Chemistry literature has a bad habit of randomly forgetting figures of moles per liter, which makes the dimensional analysis really annoying.

This all culminates in the reaction quotient, a quantity which, by some interpretations, has different units depending on the chemical reaction.

Now, apparently the pH of pure water is 7, and this figure confused me for a long time pH is determined by the concentration of hydrogen ions in a solution,

but surely pure water doesn't have any ions, right?

Well, as it turns out, water self-ionizes, meaning water molecules randomly split up into ions and then randomly recombine.

The presence of these temporary ions is what gives water a pH of 7.

But why 7?

What's special about 7?

Wouldn't that change with temperature or pressure?

Well, it turns out the pH of pure water is exactly 7, when the water is somewhere around 24.87 degrees C.

So the choice of 7 as neutral is completely artificial.

Going back to moles, the fact that we use a special unit for what's actually just a number isn't too weird.

It's quite common for two units with the same dimension to behave differently because of context.

For instance, Hertz and Becquerels are units of frequency that are both equal to inverse seconds.

The only difference between them is that Hertz is used to describe periodic processes like waves, where the time between each event is the same,

while Becquerels is used to describe random processes like radioactive decay, where the time needs to be averaged over many events.

They're technically the same unit, but they have different contextual meanings, and so you would never conflate them.

Now, becquerels originate from the study of radiation, and radiation dosage is another situation where you have multiple units with the same dimension measuring incompatible things.

Grays and Sievert both have dimensions of joules per kilogram, that is radiation energy per unit mass, but the former is absorbed dose,

which is simply total energy, while the latter is equivalent dose, which has an extra factor depending on the type of radiation to describe how much it damages living things.

Again, these units have the same dimensions, but you can't just interchange one for the other.

However, the study of radiation does also have some redundant units.

A rad, for example, is one hundredth of a gray, and they both measure absorbed dose.

We could have just called the rad a centigray, but no, it gets its own special name.

The historical reason for this is the separation between two competing systems of units, the SI system and the CGS system.

This is a whole can of worms.

A can of worms that deserves its own section.

CGS is a system of units based on the centimeter, gram, and second.

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It was an old attempt at standardization that was eventually supplanted by the SI system, which instead established meters and kilograms as the base units of length and mass.

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But the usage of centimeters and grams is not the biggest difference between CGS and SI.

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The biggest difference is that while the SI system now has about seven base units, CGS insists on describing everything in terms of centimeters,

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grams, and seconds, even when it reasonably shouldn't.

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This causes a big problem with electricity.

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First, I need to give some background on how we measure electricity.

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There's two ways that we could connect it to length, mass, and time.

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The first is to look at the electrostatic force between two charged objects.

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This force is modeled by Coulomb's law, which we can use to define the Coulomb, the SI unit of charge.

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The second is to look at the magnetic force between two current carrying wires.

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This force is modelled by Ampere's force law, not to be confused with the two other laws named after Ampere, which we can use to define the Ampere,

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the SI unit of current.

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Now, current is just moving charges, so we want to define these in such a way that an Ampere is just a coulomb per second.

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One coulombs worth of electrons travelling through a wire every second.

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In order to do that, we need to introduce some physical constants to make the units work out,

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called the electric permittivity and permeability. But CGS works differently.

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In CGS, there are no constants, because everything is centimeters, grams, and seconds.

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The CGS unit of charge is the Franklin, also called the Statcoulomb, which is the charge on these objects, such that if they are placed one centimeter apart,

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then the force between them is one Dyn, the dyn being the derived unit of force in CGS.

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Rearranging, we get that one Franklin is a centimeter times the square root of a dyne.

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This approach is called electrostatic units, or ESU.

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Meanwhile, the CGS unit of current is the BO, also called the abampere, which is the current through these wires such that if they are placed one centimeter apart,

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the force between them is is two dynes per centimeter of length

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because for some reason physicists have an obsession with throwing factors of two around.

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Rearranging, we get that one BO is just the square root of a dime.

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This approach is called electromagnetic units, or EMU.

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Okay, we now have CGS variants of the Coulomb and Ampere.

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We know that an Ampere is a Coulomb per second, so naturally a BO should be a Franklin per second, right?

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Well, no. The units don't work out.

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A Franklin per second is a new unit called a stat-ampere, a dine to the half centimeter second to the minus one,

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and a B0 second is a new unit called an ab-coulomb, a dine to the half second.

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That's okay, it just means that we need to multiply by some constant in order to get between these two schemes.

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This constant must have units of of centimetres per second, which means it's a speed.

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Can you guess what speed this is?

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It's the speed of light!

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This fact was discovered around the mid-1800s, and no one could make sense of it since the entire study of electricity was in a serious state of disarray,

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probably in no small part due to the confusion between the two variations of units I've just described.

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This was eventually disentangled in a series of papers by James Clerk Maxwell, who connected all of the results of the time into a single theory which we now call Maxwell's equations.

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Using this theory, Maxwell derived a model for a wave of propagating electric and magnetic fields, and derived that the speed of this wave should be this ratio between the static and magnetic units,

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which by this point in time had been measured to be very close to the speed of light.

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It was already well known that light had wave-like properties, so this was the smoking gun that confirmed that light was in fact an electromagnetic wave.

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Now that's a fascinating bit of history that's totally not cursed, but what is cursed is that the speed of light is 300 million meters per second.

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Oh, uh, sorry, CGS.

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That's 30 billion centimeters per second.

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Which means the two variants of CGS are different by 10 orders of magnitude!

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This massive disparity of scales is part of what makes electricity so counterintuitive.

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I think this is at the core of why electric quantities have all sorts of special names, unlike, say, velocity and acceleration, which are always expressed as derived units.

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In addition to amperes and coulombs, we have volts for voltage, ohms for resistance, farads for capacitance, Henrys for inductance, Vabors for magnetic flux, and Teslas for magnetic field strength.

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And each one of these units can be thought of in about ten different ways.

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To be honest, I respect the boldness of CGS, trying to reduce everything down to three units.

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It reminds me of Planck units, where you define the speed of light to have a value of 1, so that length and time collapse to the same dimension.

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And then you do the same with three other constants, and as long as you're consistent, you get a system that's entirely dimensionless numbers.

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These are pretty useful for simplifying calculations in certain parts of physics, but be careful, because if you pick different constants, you get different results.

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CGS has for the most part been supplanted by SI, but it's still quite common to see centimeters

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and grams being used in a couple of smaller fields of science that like to bask in their old-fashionedness.

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Which leads me of course to materials science.

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I'm going to build for you a unit that only a couple of people suggested in the comments of the previous video, because only a couple of people know about it.

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This is going to be a unit for describing a property of a substance.

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So before I go into it, just as a warm-up, let me go through the example of heat capacity.

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Specific heat capacity is measured in joules per gram degree C.

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So, given a certain number of grams of a substance, how many joules of energy do we need to put into

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it to raise its temperature by a certain number of degrees C?

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Each substance has its own number attached to this unit, for example, water has a heat capacity of 4.18 joules per gram degree C,

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C, while tin has a heat capacity of 0.21 joules per gram degree C.

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Make sense?

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Okay.

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Let's now build a unit to describe the diffusion of gas through a substance.

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How easy is it for air to get through a membrane?

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Well, the amount of air that gets through can be measured as a number of molecules, so we'll use moles, right?

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Eh, I'll use moles for now, but we'll We'll come back to this later.

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The next thing to consider is surface area.

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The bigger the surface area of the membrane, the more air can get through.

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Let's measure this in square centimeters.

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So our unit is now moles per square centimeter.

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Make sense so far?

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The next unit is time.

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If we keep the air pressure on each side consistent, air will flow through at a uniform rate.

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So our unit should be in moles per second per square centimeter.

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Okay.

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The next unit is thickness.

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The thicker the membrane, the less air will get through.

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This is an inverse relationship, so it goes on top of the fraction instead of on the bottom.

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We're going to measure the thickness in centimeters.

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Okay, I know a lot of people who watched the previous video are going to be shouting at their screen saying, CANCEL THE CENTIMETERS!

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If you are one of those people, then I think you missed the point.

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We saw that quantities like fuel efficiency and the Hubble parameter could have their units simplified, and that sometimes led to interesting results.

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But that doesn't mean we should simplify them.

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It's way more intuitive to measure fuel efficiency in gallons per mile than in square millimeters, and for the Hubble parameter,

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kilometers per second per megaparsecs is a direct description of how it's measured.

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At the end of that video, I mentioned that I used nanometers per square root nanometer in a research paper, and a lot of commenters suggested writing it as just square root nanometers, unironically.

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Yes, it's certainly interesting that you can write it as square root nanometers, but actually writing that in a research paper would be silly.

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I was trying to describe a relationship between two lengths, So why would I obscure that relationship by cancelling those lengths?

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It's fun to play a round of units, but in practice, you should always use the units that best describe the physical quantities that they represent.

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Usually, it's the most commonplace units that are the ones that are best at doing this.

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The only exception being kilowatt hours per thousand hours, which is an inexcusable abomination.

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So no, we're not going to cancel the centimeters.

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You can do that in your own time, but right now we're building a sensible unit for sensible people.

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Right, the next thing to consider is pressure.

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If we increase the difference in air pressure between the two sides of the membrane, more air will get forced through.

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This turns out to be a linear relationship, so we need to put pressure on the bottom of the fraction.

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Now, the CGS unit of pressure is the bari, not to be confused with the bar, which is a million times bigger, but we're not going to use either of these,

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or the SI Standard Pascal, because this is a sensible unit for sensible people, So we're going to use the only sensible option...

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...centimetres of mercury.

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What is a centimetre of mercury, I hear you ask?

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It's the amount of pressure that will push liquid mercury one centimetre up a tube.

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Great!

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This unit is a holdover from that time in history when we used mercury for basically everything, before realising that it's both toxic and volatile.

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The best combination of properties.

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And finally, moles.

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We don't like moles.

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Moles are silly.

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So let's replace them with centimeters cubed SDP, the number of air molecules that fill a cubic centimeter at some standard temperature and pressure.

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Oh, and, uh, multiply everything by 10 to the minus 10.

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This is the BARA, a unit of gas permeability used in the study of membranes, particularly contact lenses.

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And I had to check through some research papers to make sure that this unit is actually used, because when I first heard about it, I was certain it had to be a joke.

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It contains four different variations of centimeters, two of which aren't even related to length.

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And centimeters of mercury isn't even the most common unit of mercury!

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It's way more standard to use millimeters of mercury!

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Which would be called tors if they were the same thing, which for some reason they aren't?

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I've intentionally avoided talking about joke units like Smoots or Beards seconds in this video.

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I'm not just gonna repeat Yan Misal is a joke about measurement.

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No, the whole point of this is to understand how we use units in practice, in the real world.

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And this, this monstrosity, is a real unit used by real people.

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That being said, I do genuinely think the bara has its place.

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If it were completely absurd, people wouldn't use it.

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The main reason the bara is so complicated is that permeability is related to a lot of different factors.

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Sure, using centimeters of mercury and cubic centimeters STP is weird and esoteric, but even if we replaced everything with SI units,

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we'd still have something pretty cursed.

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Notation standards only make up half of the cursedness, the other half is the complexity of the science itself.

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There are so many other interesting ways units can get frighteningly bizarre that I haven't touched on yet, so there may be a Cursed Units Part 3 in the distant future.

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Maybe then I'll finally talk about Imperial Units!

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For now though, I'm just going to go ahead

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and crown the Hubble Parameter and the Barra as the King and Queen of Cursed Units, and then go and hit my head against a rock for a while.

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I'm not sure that's the loudest note before the C sharp.

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Oh, that was the B flat.

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