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  • 2640 lumens. 1 foot. 2.3 kilograms. 9 volts. Aaah!

  • I just closed the circuit with my tongue and I felt all 9 of the volts.

  • So what do all these things have in common?

  • They're units. Yes, but they're also absolutely, completely arbitrary.

  • [Theme Music]

  • You know who decides how much a kilogram weighs?

  • A hunk of platinum and iridium known as the International Prototype Kilogram or IPK.

  • The IPK isn't just how much a kilogram weighs. In a very real sense the IPK is the kilogram.

  • Every other kilogram is exactly the same as the IPK,

  • and the IPK is the lump of metal that decides what that mass is.

  • A kilogram is defined as being the same mass as the IPK.

  • We made kilograms up just like we made up seconds and weeks and volts and newtons.

  • There's nothing about these things that makes them them.

  • Someone just decided one day that that was a kilogram.

  • Now the fact that I find units fascinating probably says more about me then it does about units,

  • but I can talk about them all day.

  • For example, did you know that the International System of Units only includes seven base units

  • and every other unit is derived from those units?

  • Speed is length divided by time.

  • Acceleration is speed divided by time again, so meters per second per second.

  • Force is that acceleration multiplied by mass, cause F=ma remember?

  • Work done in joules is force multiplied by distance.

  • And power is work divided by time, so how much work can be done per unit of time. Makes sense.

  • It goes pretty deep, and it's absolutely correct to say that there are an infinite number of possible derived units,

  • just most of them aren't useful enough to name.

  • But here's a bit of trivia for you. When I say watts or hertz, those things are just regular words.

  • No special capitalization necessary.

  • But Hertz and Watt, they were real people with like last names that were capitalized.

  • So what's up with that? Well, getting a unit named after you is kind of the holy grail of science.

  • To quote Richard Hamming:

  • "True greatness is when your name - like hertz and watt - is spelled with a lowercase letter."

  • Of course when these geniuses were first piecing together how the world works

  • they had no idea that there were fundamental basic units beneath it all.

  • They were basing all of their units on arbitrary values because, well,

  • how could there possibly be a fundamental amount of mass or distance.

  • Interestingly, one of the standard base units is derived from an actual value though not a universal one.

  • The second is 1/60th of 1/60th of 1/24th of the time it takes for the Earth to rotate a single time.

  • That's something, at least but it also illustrates an interesting point.

  • As fundamental as that seems, when you get down to the dirty details things start to get kind of cloudy.

  • The Earth's rotation for example is slowing down.

  • Does that mean that seconds should also slow down?

  • No. That would mess up every calculation ever.

  • So seconds are slowly becoming less and less based on reality.

  • Now don't worry. It's gonna take forever for the Earth to slow down noticeably.

  • And when it does we'll just keep adding leap seconds to keep things balanced.

  • But units are extremely important in chemistry and in sciences in general,

  • as we learned when the Mars Climate Orbiter crashed into Mars

  • because instructions were inputted in the wrong units.

  • Next time you get a B instead of an A because you didn't keep track of your units,

  • just remember at least you didn't destroy a 300 million dollar mission to Mars.

  • But what do I mean when I say keep track of your units?

  • Well. I mean watch them. Do not let them do anything you didn't tell them to do because they're sneaky.

  • And a lot of chemistry is just converting between units.

  • So say you are in a car, and the car is going 60 miles per hour.

  • Now right now everyone who doesn't live in America is like:

  • "Boo, miles are terrible. Convert to kilometers Hank!"

  • Well I'll do you one better. From a scientific perspective, kilometers are terrible too.

  • They're just as arbitrary. We should use something more universal.

  • Like lightyears. The amount of distance light can travel in a year. And hours, hours is no fun.

  • So let's convert to lightyears per second. 60 miles per hour.

  • When you say it it sounds like a whole number with a single unit.

  • But it's not. It's actually a fraction. 60 miles over 1 hour.

  • Let's start with the easy part. Getting to the seconds.

  • So first we've got to get to minutes. So there's 60 minutes per hour. And also 1 hour per 60 minutes.

  • That fraction once we have it can flip either way.

  • We want it with the hours on the top, on the numerator. Why?

  • Because we want the units to cancel. We want to destroy the hours.

  • We don't want them in our units when we're done.

  • And then the same thing happens again with 1 minute per 60 seconds. Now we go to lightyears.

  • I asked Google, and there's 1 light-year in every 5.9 * 10^12 miles.

  • Looking at this we see that the hours cancel and the minutes cancel and the miles cancel.

  • Leaving us with lightyears per second. That's really what matters.

  • We've come out with the correct units.

  • The rest is just hammering at the calculator to discover that a car going 60 mph is also going

  • 9.3 * 10^-12 lightyears per second.

  • Now we perform an important test. The "does this make sense?" test.

  • And yes indeed it does because 9.3 * 10^-12 is a very, very, very, very small number.

  • Which makes sense because when you're traveling in a car you're going

  • a very, very, very, very, very, very, very tiny fraction of a light-year every second.

  • Now there are probably gonna be fifty to a hundred thousand people that watch this video.

  • And I'm gonna guess that maybe a solid seven of you did the math along with me with your calculator out.

  • Now I'm not giving you a hard time. That's just my guess.

  • If you want to follow along with your calculator in the future that might be helpful.

  • It would at very least be very nerdy.

  • But if you have been following along with your calculator, you might maybe have noticed something interesting.

  • I said 9.3 * 10^-12. When your calculator...

  • Your calculator probably said something like 9.3487658140029 * 10^-12.

  • So why, when I had so many more numbers to give, did I only give two? Was I trying to save time?

  • Well obviously not, because now I appear to be wasting time talking about it.

  • Do you think that it would be too hard for me to remember all those numbers?

  • Well obviously not, because I just did it. So I will tell you why.

  • When you're doing experimental calculations, there's two kinds of numbers. There's exact and measured.

  • Exact numbers are like the number of seconds in a minute or the number of eggs in a dozen.

  • They're defined that way and thus we know them in effect all the way out to an infinite number of decimal places.

  • If I say that there are a dozen eggs you know that that's 12. It's not 12.0000000001

  • or 11.9999999. It's 12.

  • But that's not true for the number of miles per hour my car was going.

  • That car wasn't going 60.0000-out into infinity mph.

  • I only know the speed of my car to two decimal places because that's all I get from the speedometer.

  • So the car could have been going 59.87390039 mph or 60.49321289 mph; the speedometer would still say 60.

  • And no matter how well I measure the car's speed,

  • I will never know it at the same level of precision that I know the number of eggs in a dozen.

  • So that's the second type of number, measured numbers.

  • Now the cool thing about measured numbers,

  • because you never ever know them exactly, is that they tell you two things at once.

  • First, they tell you the number that was measured.

  • And second, they tell you the precision at which that number was measured.

  • People often get their heads all tangled up about this,

  • but with a measured number you just have to remember that the actual number goes out to infinite decimal places,

  • you just never know all of them. You can't. It's impossible,.

  • So when my scale says 175 lbs, that doesn't mean 175.000000 lbs. It means 175.something lbs.

  • And all those numbers after the five? We don't know them.

  • And here's the thing, a measured number can be pretty unhelpful if you don't have knowledge

  • of the precision of the measurement.

  • So you have to conserve the precision through your calculations

  • or else you might end up killing someone with an imprecise dose of insulin or something.

  • So we have a set of rules for what are called significant figures:

  • these are the digits in your number that you actually know.

  • With my speedometer there are two: 6 and 0.

  • But 0 is weird, because sometimes it's just used as a placeholder.

  • Like if I said that the fastest plane can go 13,000 mph, which it can by the way.

  • An unmanned military test glider did it in 2011.

  • That's not an exact number, those zeroes are just placeholders.

  • So when a number ends in a zero, or two or three zeroes, it's hard to tell if those zeroes are significant.

  • But this all gets so much simpler when you use scientific notation, which since it's science we should.

  • So 60 mph would instead be 6.0 * 10^1. We get that zero is significant because we wrote it.

  • Otherwise it would just be 6 * 10^1. We keep that zero around because we actually know it.

  • Scientific notation is awesome by the way, once you get the hang of it.

  • If you're having trouble you can always just type it into Google or your calculator to

  • see exactly what number we're talking about,

  • but the number of the exponent just tells you how many places to move the decimal point.

  • So to the 1st power you move it one to the right and you get 60.

  • To the negative 1st power you move the decimal point one place to the left and you get 0.60.

  • To the fifth power, one, two, three, four, five, and you get six with five zeroes or 600,000.

  • Of course your significant figures get preserved, so 2.4590 * 10^-4 is 0.00024590 and you still

  • get the same five sig figs.

  • Now to the magic of figuring out how many sig figs your answer should have.

  • There are two simple rules for this.

  • If it's addition or subtraction it's only the number of figures after the decimal point that matters.

  • The number with the fewest figures after the decimal point

  • decides how many figures you can have after the decimal in your answer.

  • So 1,495.2+1.9903 you do the math.

  • First you get 1,497.1903 and then you round to the first decimal,

  • because that first number only had one figure after the decimal. So you get 1,497.2.

  • And for multiplication just make sure the answer has the same sig figs as your least precise measurement.

  • So 60 x 5.0839 = 305.034, but we only know two sig figs,

  • so everything after those first two numbers is zeroes: 300.

  • Of course then we'd have to point out to everyone that the second zero but not the third is significant,

  • so we'd write it out with scientific notation: 3.0 * 10^2. Because science!

  • Now I know it feels counterintuitive not to show all of the numbers that you have at your fingertips,

  • but you've got to realize: all of those numbers beyond the number of sig figs you have? They're lies.

  • They're big lying numbers. You don't know those numbers.

  • And if you write them down people will assume that you do know those numbers.

  • And you will have lied to them. And do you know what we do with liars in chemistry? We kill them!

  • Thank you for watching this episode of Crash Course Chemistry.

  • Today you learned some keys to understanding the mathematics of chemistry,

  • and you want to remember this episode in case you get caught up later down the road:

  • How to convert between units is a skill that you'll use even when you're not doing chemistry.

  • Scientific notation will always make you look like you know what you're talking about.

  • Being able to chastise people for using the wrong number of significant digits is basically

  • math's equivalent of being a grammar Nazi.

  • So enjoy these new powers I have bestowed upon you, and we'll see you next time.