# Unit Conversion & Significant Figures: Crash Course Chemistry #2

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A unit is the frequently arbitrary designation we have given to something to convey a definite magnitude of a physical quantity and every quantity can be expressed in terms of the seven base units that are contained in the international system of units. Hank thinks this is a thrilling subject, and while you may not agree, it is a subject that is very important if you want to be a scientist and communicate with accuracy and precision with other scientists. So listen up and learn something or Hank might have to kill you! (NOT REALLY!)

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#### Transcript Provided by YouTube:

00:00

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

00:08

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

00:12

So what do all these things have in common?

00:15

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

00:20

[Theme Music]

00:29

You know who decides how much a kilogram weighs?

00:32

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

00:38

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

00:45

Every other kilogram is exactly the same as the IPK,

00:48

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

00:53

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

00:58

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

01:02

There’s nothing about these things that makes them them.

01:05

Someone just decided one day that that was a kilogram.

01:08

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

01:13

but I can talk about them all day.

01:14

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

01:20

and every other unit is derived from those units?

01:23

Speed is length divided by time.

01:25

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

01:29

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

01:35

Work done in joules is force multiplied by distance.

01:38

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

01:43

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

01:49

just most of them aren’t useful enough to name.

01:51

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

01:55

No special capitalization necessary.

01:57

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

02:01

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

02:06

To quote Richard Hamming:

02:07

“True greatness is when your name – like hertz and watt – is spelled with a lowercase letter.”

02:13

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

02:15

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

02:20

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

02:23

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

02:27

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

02:33

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

02:40

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

02:44

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

02:49

The Earth’s rotation for example is slowing down.

02:52

Does that mean that seconds should also slow down?

02:55

No. That would mess up every calculation ever.

02:58

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

03:01

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

03:04

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

03:08

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

03:12

as we learned when the Mars Climate Orbiter crashed into Mars

03:15

because instructions were inputted in the wrong units.

03:18

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

03:21

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

03:26

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

03:29

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

03:35

And a lot of chemistry is just converting between units.

03:38

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

03:42

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

03:45

“Boo, miles are terrible. Convert to kilometers Hank!”

03:48

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

03:53

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

03:56

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

04:01

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

04:04

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

04:08

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

04:12

Let’s start with the easy part. Getting to the seconds.

04:15

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

04:20

That fraction once we have it can flip either way.

04:23

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

04:27

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

04:31

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

04:33

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

04:38

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

04:43

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

04:47

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

04:51

We’ve come out with the correct units.

04:53

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

04:59

9.3 * 10^-12 lightyears per second.

05:02

Now we perform an important test. The “does this make sense?” test.

05:05

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

05:11

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

05:14

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

05:19

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

05:22

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

05:28

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

05:30

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

05:34

It would at very least be very nerdy.

05:35

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

05:40

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

05:44

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

05:53

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

05:59

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

06:02

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

06:05

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

06:08

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

06:13

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

06:17

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

06:23

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

06:31

or 11.9999999. It’s 12.

06:34

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

06:37

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

06:42

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

06:47

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

06:57

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

06:59

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

07:04

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

07:06

Now the cool thing about measured numbers,

07:08

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

07:12

First, they tell you the number that was measured.

07:14

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

07:17

People often get their heads all tangled up about this,

07:19

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

07:24

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

07:28

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

07:37

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

07:40

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

07:44

of the precision of the measurement.

07:46

So you have to conserve the precision through your calculations

07:49

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

07:53

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

07:55

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

07:58

With my speedometer there are two: 6 and 0.

08:01

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

08:04

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

08:09

An unmanned military test glider did it in 2011.

08:12

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

08:15

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

08:20

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

08:26

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

08:34

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

08:39

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

08:42

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

08:45

see exactly what number we’re talking about,

08:47

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

08:51

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

08:54

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

08:58

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

09:03

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

09:12

get the same five sig figs.

09:13

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

09:17

There are two simple rules for this.

09:19

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

09:23

The number with the fewest figures after the decimal point

09:25

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

09:29

So 1,495.2+1.9903 you do the math.

09:34

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

09:39

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

09:45

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

09:50

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

09:57

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

10:01

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

10:05

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

10:10

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

10:14

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

10:19

They’re big lying numbers. You don’t know those numbers.

10:22

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

10:26

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

10:31

Thank you for watching this episode of Crash Course Chemistry.

10:33

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

10:36

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

10:40

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

10:44

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

10:49

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

10:52

math’s equivalent of being a grammar Nazi.

10:54

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

10:58

Crash Course Chemistry was filmed, edited, and directed by Nick Jenkins.

11:01

This episode was written by me, Michael Aranda is our sound designer, and our graphics team is Thought Bubble.

11:07

If you have any questions, comments or ideas for us, we are always down in the comments.

11:11

Thank you for watching Crash Course Chemistry.

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This post was previously published on YouTube.

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Photo credit: Screenshot from video.