>> Welcome to Chem 5. I'm your instructor Doug Tobias. Looking forward to working

hard with you all quarter long and hopefully having lots of

fun learning useful things. So, the first thing that

I should tell you is that if you turn around you

will see that the lectures for Chem 5 are being filmed. And this is part of a

big project on campus which is called Open Courseware. And we're trying to get all of our required undergraduate

chemistry courses archived, both for UCI students

as well as Planet Earth. And that's an exciting project. And I've agreed to

participate in this.

One thing I should tell you is

that if you feel at all strange about having the backs of

your heads appear on film or, perhaps more embarrassingly,

if you happen to be doing something other

than Chem 5 related things like looking at your

Facebook, it may actually show up on the film, you might

want to sit on the other side of the classroom so that you're

not in the eye of the lens. Okay? All right, so what I

like to do before I get started with the course is to figure

out what's the composition of our class so I can figure

out who I can make jokes about.

So first of all,

is there anybody in the room who's not

a chemistry major? Raise your hand. Okay. Is there anybody in the room who's a

chemistry/biology double major? Ah, can't make jokes

about bio majors then. What about engineers? Any double majors

with engineering? Okay, good. We can pick on the engineers. All right, so we're going

to get right to work today. You might be thinking

that's a little strange because this is discussion. Well, in this course,

there's no distinction between the discussion

and the lecture. The only distinction is is that the discussion

is 50 minutes long and the lecture is

an hour and 20, okay? Now, you might be thinking

well that's kind of unfortunate because I actually

like to discuss. Well, the good news

is that you're free to discuss both during the

discussions and lectures, okay? So this is going

to be hopefully, a somewhat interactive course. All right? So, feel free to

discuss anytime you want.

All right, so now what I'm

going to do is I'm going to very quickly go

over the syllabus just to make sure a few

things are clear. I expect that you

all have read it as I instructed you

to do last week. All right, so first of

all, here's our homepage. And there's not much of

interest on the homepage except for this really cool picture. And you'll actually learn how

to make really cool pictures like that this quarter.

And it's not very difficult. The only thing to notice

maybe is the announcements. So every time I update the

Website, unless I forget like I did today, I'll say

when I updated it, okay? So today I posted the homework

assignment, but I forgot to update the announcements. Anyway, we'll talk

about that later. Okay, now, so if you see that

there's been a new announcement since last time you were here

and you might want to check at least once a week, you can

go and see what I had to say. So here's the announcement that you've already seen

welcoming you to the class and telling you to go

read the syllabus, okay? So let's have a quick

look at the syllabus. So first of all, this is going

to be really confusing for me, but hopefully not for you. I'm actually teaching

two sections of Chem 5 concurrently,

all right? And the schedules

are really weird.

So from time to time I'm

going to get really confused. But, the one thing I can tell

you is that this is Section B. And your TA is named

Vera Prytkova and she's sitting

right over there. And she's going to

be a very good friend of yours this quarter. She's going to be

helping you a lot and especially with

your homework. So you should be nice to her. And if you are, then

she'll treat you well. Okay, another thing

is, we're not going to hold regular office hours. We're going to be seeing a

lot of each other in here. There will be lots of

opportunities for us to talk to one another, answer

your questions. But that doesn't

mean that we are against seeing you

in our offices. So, if you have anything

you want to talk about outside the

classroom, please feel free to make an appointment either

with Vera or myself, okay? Don't feel intimidated.

Another thing is about emailing. There's probably not a

whole lot of reasons to need to be emailing us,

especially because we're going to see each other so

much in the class. If you do so, just

keep in mind that a lot of what we're doing

is fairly technical and sometimes it's hard to discuss technical

things over email. And so you know, be patient

and we'll try to figure out how to best address your

concerns via email.

But, like I say, most of

the time we should be able to take care of anything

you want to know here in the lecture. Now, if you do want to send

emails, there's a couple of ground rules so that we can

distinguish your very important emails from all the lousy

stuff that comes our way. And that is that you should send

your email from a UCI account and be sure to include your full

name and your student ID number at the bottom of your email. That will help us a lot. Okay. What else? So by now you probably

know when we meet. So we have the two

lectures — wait a minute. We're actually Section A.

What are you doing here? Sorry, Vera's not your TA.

Your TA is not here today. Her name is Krista. See, I'm already confused

about the sections. All right. Well, in any case,

you can hang out Vera. This is your class. And like I said, very

importantly, you're expected to come to both lecture

and discussion and one of the two labs. You don't have to go to both. And the labs are very

important because that's where you're going to get

your homework done, okay? So the whole purpose of the

labs is for you to do homework and get help from your TA.

And most of the time you'll get

your homework done in the lab, so you won't actually have to take it home with

you, all right? Now, if you don't finish in

the lab, and that will be rare, then you'll have opportunities

either to come here or to some other computer

lab that has the software that we're going to be

using, which I'll talk about in just a second,

all right? Now, strictly speaking, you

don't have to go to the lab to which you're assigned. You could go to the

other one, but we would like to encourage you as much

as possible to go to your lab because otherwise if there's

a lot more students in one lab versus the other, it will

be harder to give help to the larger section. Right now they're fairly

equally balanced, okay? So, but from time to time,

you may not be able to make it to one and you can

go to the other. Okay. Now, course materials. So, in this class

there's no book. The book is the notes, which

we'll get to in a minute.

And those were prepared

specifically by me for this class, okay? They're free. And the one thing that you might

consider is to purchase a copy of the software that

we're going to use. This course we're

almost exclusively going to be using a software

package called Mathematica. How many have heard of or

used Mathematica before? Okay, a few. All right. So that's what we're

going to do.

And we're going to

start from zero assuming that you don't know anything. And we'll work our way up to solving really

cool chemistry problems. And the goal of this class — well one of the goals of

this class is to prepare you to be able to use

Mathematica to do things like physical chemistry

homework, to do things like analysis and

representation of datasets from your lab classes and your

research and to introduce you to a very, very powerful

software package that is sufficiently

impressive that if you list it on your resume, which you

will have the right to do if you pass this class, then that may actually be

a good feather in your cap when you're looking for a job. Okay? It's definitely

well recognized as a very, very powerful software package. All right. Now, we don't have a book, but

some people like to have books. And so there are many, many

books written about Mathematica. There are not really any

good ones on chemistry, but if you want just general

introductions to the software, I listed three here

that I've looked at that seem somewhat useful,

especially at the beginning.

So you may want to

have a look at those. All right. The grades. So, this class is

a hands-on class. We're teaching you the

techniques and we want you to show that you can use

those techniques, all right? And most of the way that

you're going to show that is by doing your homework,

all right? And homework's extremely

important, so it's representing 50

percent of your grade. All right? We're going to have a mid-term around the middle

of the quarter. That'll be about 20 percent. And then we'll have a final

exam during finals week, and that'll be 30

percent, all right? Now, I want to say a few

things about the grades. So first of all, this is a

class for chemistry majors.

We love our majors. We want them to learn. And in this class, I don't

want you to be intimidated by worrying about

your grade, all right? If you do your work by

yourself, and that's the key, and don't cheat by

going to somebody who took the class some

years ago and get their work, and if you can show a

reasonable proficiency — and reasonable is a very

liberal definition — then you'll get a good

grade in this class. And by good grade I mean

an A or B. It's really hard in this class if you actually do

the work to get a lousy grade.

You'll have to try

really, really hard. The best way to get

a lousy grade is to not do the work yourself

and not turn it all in. That may cost you. But otherwise, you're

guaranteed to get a good grade and you shouldn't worry about

it, even if you're struggling, because some of you will. I know this from experience. Okay? All right. Now, when you're

doing the homework, you will certainly benefit

greatly from a lot of advice and consultation with the TA

in the lab sections, okay? And that's fine. You also may discuss with your

classmates certain aspects of the problems. But in the end, you

should do your own work. That's the main rule

of this course. You've got to do your own work. Okay? All right. Now, here are the exams. So you know when they are. And basically, the exams

are going to be the same as the homework assignments

except you won't get help from the TA or me, except

maybe to clarify the question.

So this is really to see if you

can take what you've learned by doing homework and do it

again but without help, okay? So not really super difficult. And one thing about all

of the work in this class, with the exception of

copying off classmates, and/or using work

from previous years, everything is available

to you at any time. If you want, you can

wheel in a rack of books to use during your

homework and exams. You can use anything

that you can find on the Internet except work

related to this course. It's all available. And in fact, you can use, and I

expect you to use and advise you to use, your homework

assignments and the notes from this class because I will

never ask you to do something that I did not teach

you how to do. All right? So the notes in particular will

be very, very useful to you.

All right? Okay, so you can read what I

have to say about cheating. I expect that you won't do it. And you can also read, if for

some reason you need to deal with enrollment, I can't do

anything with enrollment. You have to talk

to the nice people over in the chemistry office

who you probably know very well. Okay, now, a couple of

other little things. And then we'll get to work. Woops. Let's see. So when I post homework

assignments, they'll be on the homework page. And so for example, here's one. It's going to be a PDF file. It's going to have

some problems on it. They look long and

intimidating, but they're not. It's mostly because I

want to explain to you in gory detail what

you're supposed to do. We'll talk about the

homework tomorrow. Your labs are on

Wednesday, so I'm going to tell you basically how

to do your homework tomorrow at the end of lecture. If you want, you can look at it. And if you know how to do it,

you can already get started.

Now, everything that

you do, every assignment in this course including

the exams is electronic. And it's going to be

turned in electronically. All right, I haven't set it

up yet, but there's going to be a drop box

available to you. It's going to say

something like homework 1. And you're going to turn in

your assignments in drop boxes for every assignment

including exams, okay? And I assume that you

know how to do that. If you don't, there's

some instructions here that you can follow. In our class most of your

assignments are going to be what's called

Mathematica notebooks.

So, all you have to do is

name your file something like HW1, say, dot MB. That's the extension for

Mathematica Notebook. Your name will automatically be

put on it when you turn it in. So you don't need

to put your name. Just keep it simple. Homework 1, mid-term, final. That's all you have to do. Okay? All right. One last little thing. Here on the links page are

some important things, okay? So first of all, this is

basically your textbook and the auxiliary

files that go with it. All right? I've organized the material

into what I call lessons. They're somewhat arbitrary to

the way they're divided up. And we'll most likely make

it through all of them. Okay? So I assume you've

read the introduction, which just gives you a

brief overview to the class. And today we'll start

on Lesson 1. Okay.

Now, are there

any questions about what I just told you about

how the class is going to work? Okay. All right. So, I already kind of

alluded to it, but I just want to say a couple more

things very briefly about what we're

going to do here. So first of all, you

guys are really lucky. You're really lucky

because you are living on the very highly evolved tail

of what's called Moore's Law. Has anybody ever

heard of Moore's Law? No? Okay, well Gordon

Moore, who founded Intel, the company that made

the processors that are in most PC's nowadays. Back in the mid-60's,

when I was a baby and way before you

guys were born, he said every two years

the computing capacity of integrated circuits, so the CPUs in your

computers, will double.

And the price will

remain roughly constant. Moore's Law has been very, very closely followed

for about 35 years. And there's lots of talk

about it coming to an end. But in any case, what this means

is is that there's been a huge, huge improvement in

computer capabilities. And so now, you know,

computers run your cars. You have your little Smart

phones, you have your laptops. Computers are everywhere. And importantly for us,

they're in the chemistry lab. When I was an undergrad,

I used to have to make graphs on graph paper. Have you ever seen graph paper? Yeah? And we used to use

slide rules, not calculators. And then toward the end of my

undergrad we used calculators. And even programmable

calculators so we could calculate

something like a mean in about 20 minutes

from, you know, 50 data points or something. Okay? Now, everything's

computerized. It's very easy to use. And you are living in the age where that is available

to you, all right? Now, one of the —

well, there are many, many amazing software

packages that have been written over this period since I was

an undergrad that are very, very powerful at

doing a lot of things that are relevant to chemistry.

One of those is Mathematica. And hopefully in this course

you'll get a taste for some of the things that you

can do in them, okay? And the main objective of this

course is to make you familiar with the program so

that you can use it to do useful things later. All right? And what you're going to be able to do is you can use

it as a calculator. You can use it to read in,

manipulate, plot, fit data, do statistics, output data. You can use it to solve

systems of equations that would be very difficult

if not impossible by hand that are very relevant to

complicated chemical kinetics and chemical equilibrium

problems. And you can use it to do

calculus that's relevant to many aspects of

physical chemistry, especially quantum mechanics,

statistical mechanics, okay? So you're going to learn

how to do all those things.

Now, you might be thinking, man,

I'm just now taking calculus. Or I haven't had the math. Well, that's quite possible. And so, because I'm aware

of the fact that you're in different places in

your chemistry major degree completion, I'm going

to try to provide for you the appropriate

mathematical or chemical background for

the stuff that we're going to be doing because the

whole objective here is to teach you how to solve

chemistry-related problems or do chemistry-related data

manipulations using this software package. And at times that's going

to mean I'm going to have to teach you things

that you don't know. And it's going to be quickly. And you're probably going

to feel like you're lost. And I'll just tell you — give you a preview that

this is not a math class. It's not even really strictly

speaking a chemistry class.

This is a computer skills class. So if you don't get the math or if you don't get the physical

chemistry that we're going to talk about, don't worry about

it because I'm going to explain that stuff, and ultimately

all you have to do is translate

it into the software. And it may sound

intimidating at the moment, but I know from experience

that most of you will find that it's actually

not so bad, okay? And if it is bad, let us know. Because it's not

supposed to be bad. All right? Okay. So now let's go

ahead and get started. So every time, when you come

into the class at the beginning of class, log into your

computer and go ahead and fire up Mathematica.

So I'm going to show

you how to do that now. If you haven't already figured

it out, what we're going to do is go into this

folder called Apps. Double click on Wolfram

Mathematica 8. And I'm going to — you don't

have to do this, but I'm going to make my font 1 1/2 times

so that those of you who are in the back can see it. And now, you get

this blank screen. And this is what's called a

Mathematica Notebook, okay? And you may notice up in the

upper left there's a little horizontal cursor that's

blinking, waiting for you to tell the program

to do something. Now, there are many

different modes in which you can

enter information. And we'll talk about

some of those later, some of the other ones. The default mode is what

I would call command mode, which means you can execute

commands related to, you know, mathematical operations

or whatever, reading and data, et cetera, et cetera.

Okay? So we're going to start with very simple

arithmetic operations. Okay? So the first

thing I'm going to do is I'm going

to type 2 plus 2. And then what you do is

you find the enter key, which is the return

key — woops, sorry. And with a shift, shift enter, you enter this command,

all right? So when I type 2 plus

2, nothing happens. But as soon as I

hit shift enter, that means execute this command. All right? Now, what you see is

that you got a result, 4. That's the result of

adding 2 plus 2, okay? All right, so that's your first

— well, maybe not all of you, but for most of you, your

first Mathematica command. Now, I want to say a couple

of things about the format, which you will get used

to as times goes on. Okay, so first of all, every

time you're in input mode and typing in a command,

you have this thing that says in bracket number. And that number is keeping track

of how many commands you have within a given section, okay? And it keeps track

of them in order of how they're executed

regardless of where they are

on the screen, okay? And the output corresponding to that command also

gets the number 1, okay? So that's our first command.

Input was 2 plus

2, output was 4. One other thing to notice, over on the right hand side

there are some brackets. And these brackets sort of let you know how your

commands are organized within the notebook, okay? And they're useful — it's useful to be able to

keep track of those sometimes. For what we're going to do

now, you can ignore them. But I just want to

explain to you. So first of all, within a

continuous sequence of commands, there will be one

bracket on the outside that contains all

of those, okay? And every time there's an input

there will be a little bracket inside that has a little

triangle thing on the top. And every time there's an

output it'll be the same. It'll have a little

line below the triangle. Okay? So these are things

that as time goes on and we start doing more

complicated things, you may want to make use of the fact that

you can manipulate those.

So, and we'll see

that as time goes on. But for now we won't

worry about it too much. Okay, now, if I want,

I can edit. So suppose I said, well I didn't

really want to add 2 plus 2, I actually wanted

to add 2 plus 3. So I can go up here and put

my cursor there, backspace, and then if I hit

shift enter again, notice it updates the result. And now you may also notice that

it has replaced 1's with 2's because it's keeping track of all the commands I'm

making in this session.

All right. Now, if I want, I can

keep adding commands to this same what we cell. So this exterior bracket here

is denoting a single cell, okay? So, for example, I can do

a multiplication 2 times 3. So multiplication can be done

with the asterisk, all right? So if I do that, shift enter,

I get the result, 6, all right? And notice it's now 3. And there's another way that

you can do multiplications, all right? Another way is if

you say 2 space 3. Notice, when I put the

space in and type 3, Mathematica interprets

that as a multiplication. It puts in the times

sign, all right? And that's something to be

aware of because sometimes when you're typing in things,

if you have a spurious space that you didn't notice you

accidentally put in, you may or may not want that

to be interpreted as a multiplication, okay? So it's good to be

aware of that.

So if I enter that, of

course, I get the same result. Now, if I want, I can

break out of this cell. So notice — oops, sorry. These are all individual

commands. If I want, I can

start a new sequence. It doesn't really

affect things much. You know, I can go down a little

bit and — or I can go up. I'm sorry, I can go up here

in the middle somewhere and add additional commands. So for example, suppose

I want to raise 2 to the power of 3, all right? So 2 to the power of

3, use this carat. All right? So this carat means

2 to the 3rd. All right? So you enter that and

you get 8, as expected. Now notice, even though

I went back and put this in between a couple of commands,

it's still keeping track of the total number,

and this is 5.

This is something that's useful

to keep in mind because later on as we do more and

more complicated things, we may end up screwing

things up by moving around within the notebook. But don't worry about

that for now. We'll see how that works later. Okay. Now, what if you

want to do division? So I'll come down

here at the bottom. And now I'm going to

do 6 divided by 3. So division is with

the forward slash. Okay? So if I do 6 divided

by 3, I get 2, all right? Now let's try a different

division. 16 divided by 6. I got 8/3. This shows a very important

point about Mathematica, okay? This is an exact

result, all right? Mathematica in general will

return an exact result whenever it can. Okay? And we'll see

shortly how you can turn that into an approximate or a

numerical result if you want.

All right? One way to do that is to put

an inexact number in somewhere. So here we have two integers,

so this is the exact fraction, 16 divided by 6 reduced

as far as possible. If I want a decimal

representation I just can make one of my numbers a decimal. So if I say 16 point

divided by 6, now this is in principle

an inexact number. And so when I enter that

command, I get 2 and 2/3 or 2.6667, to one, two, three,

four, five decimal places.

Now, by default, Mathematica

will return five decimal places. Okay? And we'll see how to

change that soon enough. Okay? All right. Now, so far all we've done is

single arithmetic operations. What if we want to chain

a bunch of them together? All right? So let's try one. We'll do 2 times 3

plus 4, all right? So what do you thing

we should get? >> 10. >> Should we get first 2

times 3 and then plus 4? Or should we get

2 times 3 plus 4? In general, what happens is

that the operations are executed from left to right and

there is a precedent, okay? And the precedent is

exponentiation before multiplication and division

before addition and subtraction.

All right? So if you have a

complicated expression, and you want it organized

in a particular way, you should use parentheses

to group the operations that you want to be

done together, okay? And this is very important and

it's one of the major sources of mistakes or getting something

that you didn't' want to get. It has to do with this

order of operations and not putting parentheses in

the right places, all right? So my advice to you, and

I'll keep saying this again and again, is to just use

lots and lots of parentheses.

Okay? So let's try this one. We get 10. And that's the expected result of having first 2 times

3 and then plus 4. Now, what if I wanted to do

2 times quantity 3 plus 4? Well, then I put

in the parentheses. So I'd say 3 plus 4. And what this does is it

forces 3 plus 4 to be executed and then multiplied by 2. So we should get 14, and we do. Okay? Here's another example. 3 divided by 2 to

the power of 6, okay? Now, what we expect

to happen is 2 to the 6th is going

to be executed. And then 3 is going to be

divided by that, all right? So what should we get? 2 to the 6th is 64. So we should get

3 divided by 64. And we do. Okay? But what if instead

we actually wanted 3/2 to the power of 6? Well, in that case, we

should use parentheses to make sure we get the

3/2 and then power 6. In that case, we get a

quite different result, 729 divided by 64. And notice in both

cases, as advertised, Mathematica returns

the exact result. All right? Okay. Now, we're going

to be making a lot of use of so-called commonly

used numbers like e.

The base

of the natural log. We may also use — we will

also use from time to time, not very often, i,

the imaginary number. So I'm going to show you how

we actually can access those using Mathematica. Also pi. So let's start with pi. So if I want pi, I can type

capital P and then lower case I. And if I enter, notice that Mathematica says,

yes, I know about pi. And that's the exact

representation of pi. It's the symbol pi, okay? All right? And notice what happens if I do

use a lower case by accident. I get a word pi that

doesn't mean anything. Okay? So that's something

to keep in mind, that predefined numbers, and

later we'll see functions and variables in Mathematica

start with capital letters. Okay. Now another one, e.

If I want the number e, I type capital E, all right? If I enter that, notice that I

get the letter e.

So it looks like the letter e, but there's

something special about this one that I want you to

notice immediately, and that is that it's got

this little slash on it, okay? And that's to be distinguished

from the true letter e, which you can get by

typing a lower case. Lower case, you get just

e. This is the letter e, which doesn't mean anything

special to Mathematica. This is the number e that

has this little slash on it that is the actual number. So for example, we can see what

that is by saying report to me e to the power of 1 point.

And those of you

who are familiar with the numerical value of

e know that in fact it is about 2.72, all right? If I did that with

little e, I get nonsense. It just spits it back

the way I wrote it. Okay? All right, another one

is i. So if I type capital I, that's the imaginary number. And notice, as in the case

of e, it's got a little bit of that double vision

thing going on. And I can check to see that

it's the imaginary number i by squaring it. So i squared, and

notice I get minus 1. Okay? All right. All right, so those are some

commonly used predefined numbers in Mathematica. Now, I'm going to

introduce you to a few of the predefined

user functions. There are literally thousands

and thousands of functions, and as time goes on

we'll probably learn about a hundred or so. But I want to introduce you

to a few simple arithmetic or mathematical functions that

we will use commonly, okay? So one of them is

absolute value.

Absolute value is capital ABS. And so if I put in for example

minus 5 and enter, I get 5. Now notice, I used

square brackets when indicating the

argument to a function, okay? There's lots of different kinds

of brackets in Mathematica. This is another source of

many headaches and errors, but I'll try to help you to

avoid those as time goes on. But whenever you're

indicating the argument to the function you

use square brackets. Okay? So if I want

to do a square root, the command is SQRT

with a capital. And then I can put in an

argument with brackets. So for example square root of 2. And I get the square root of 2. Once again, you see,

it didn't give me 1.414 or whatever dot, dot, dot. It gave me literally

the square root of 2. That's the exact representation. If I want the numerical,

one way I could get that is to say square root of 2 point. Then I get 1.41421, okay? And notice, if I

put in lower case, it just spits it back at me. And that normally means

something's wrong. All right? So that's not square

root, obviously.

Okay. Now, what if

you want logarithms? Well, logarithms are LOG. and if you just say LOG, so

for example I say LOG of e, you get the natural

log, all right? So just plain old LOG is

understood as the natural log. So if I take the natural

log of e, what should I get? 1. And sure enough, I get 1. Well, we often like to

use other logs, right? So for example, we use

log 10 a lot in chemistry, pH being a common example. So if I want log base

10, there's a couple of ways I can do that. One is to say LOG 10.

That's a special command

to give me the log base 10. And so for example, if I put

in 100, what should I get? >> 2. >> I get 2. All right? Another way to do logs

of arbitrary bases, okay? So we'll do log 10, but you can

use this to do log of any base, is to actually use the

regular old log command, except now include

two arguments. The first is the base. So this is going

to give me log 10. And the second is the

actual argument, 100. So this is equivalent

to the previous command. And we get 2, all right? So we will see quite frequently

there are multiple ways to get the same answer,

all right? Now suppose we want log base 2? Well, there's no command

log 2, so what we have to do there is we

have to say LOG 2 and then give the argument.

So for example, if I put

in 8, what should I get? >> 3. >> 3, because 8 is 2 to the 3rd. Okay? All right. What else do we have? How about trig functions? Sine is capital Sin. So if I say Sin of

Pi, what should I get? >> 0. >> 0. What about cosine? Cosine of Pi. Minus 1. How about inverse

cosine, or arccosine? Well, the way you do that is you

say capital Arc, capital Cos. So if I put in minus

1, what should I get? Pi. And I do. Okay? All right. >> Question. >> Yeah. >> If you want to put like

degrees, like cosine of 60, it should be .5 but

it [inaudible]. >> Because the default

units for angles in Mathematica is going

to be radians, okay? And we will see soon

how to convert. But I don't want to

go there just yet. It's very easy to convert.

But we'll see that later, okay? But just keep in mind, by

default, I think it's this way with most calculators,

certainly on computer programs, the units of the

angles are radians. Okay. Now I want to introduce

you to another function that I alluded to earlier, and that is the numerical

representation function, okay? So there's a function

called N. Okay, so first let's just remind

ourselves, so if I just type Pi, I get the exact representation. Now, what if I want a

numerical representation of pi? I can say capital

N bracket Pi, okay? So what that does is it converts

my exact number, the symbol pi, into a numerical representation. And as in previous

examples, the default is to give you five decimal places. Okay? So N brackets just

means give me a number, a decimal number, of

whatever's in the brackets.

Okay? Now, what if I

want more accuracy? Well, what I do is I

say okay, N bracket. I say what I want to get the

decimal representation of. And then I say how many

decimal points do I want? So how many — anybody

in here — so some nerds like to show off

how many digits of pi they know. Anybody in here think they

know a lot of digits of pi? How many do you know? [ Inaudible Audience Response ] Yeah, I know about six or so,

so I think I'm pretty good.

Well, Mathematica is very smart. Let's try 100. There you go. There's pi to 100

decimal places. Piece of cake. All right? So you can get as

many as you want. So the whole point, though,

is it just says convert this to a numerical representation with 100 decimal

places, all right? So I can do, for example,

N bracket, and I can embed within that a function,

square root of 2. And I can say give me

eight decimal places. So there you have it, one,

two, three, four, five, six, seven, eight, okay? All right. Now, there's another way to use

this N. And the way I'm going to show you how to do this is

useful for other functions. So this is going to

be your introduction to a particular format.

So suppose I do some calculation

and I don't want to type N and put the thing in brackets. But I still want a

numerical representation. Sometimes it's convenient

to be able to do it sort of after the fact, all right? So here I do square

root of 2, all right? So there's my square root of 2. Now, if I want a numerical

representation, I could say — I could go up here and say,

oh, I have to type N bracket and put a bracket

on the other side. Well, another way I can

do the same thing is to use this notation:

slash slash N. So look what happens

when I do that. It gives me now the

numerical representation. So this is what's

called a post-fix. We'll see other examples later. And it's convenient

sometimes because you don't — maybe sometimes you have

a really beautiful thing and you don't want to make it

look ugly by putting N brackets around it, but you still want

the numerical representation.

You could just do this at

the very end, all right? What this literally means

is it means take the result of whatever's on the left side

of the two slashes and feed it into N. It's called post-fix. And you can do this

with other functions. So you could say take

the result of this and put something else here

if you want, all right? So we'll see more

examples of that later. All right. So we have just a

couple of minutes left. So I want to show

you one more thing.

All right? And this is something

that can be useful, but it also can be the source

of headaches, all right? So I'm going to show it to you,

and I will occasionally use it. And I encourage you if you think

it's useful to use it also. But just be aware that you

can get into some trouble. Okay. So what this is is this is

a shortcut to be able to refer to the result of a

previous cell, okay? So for example, the last command that I executed here

was this one, all right? And it's number 37. Now, if I want to

refer to this number, I can do that with percent. Okay? So let's see

how that works. I type percent times 10. Okay? So what this means

is it means take the result of the last, the very, very

last command that I entered, which is this one,

and multiply it by 10.

So what should I get? 14.1421 whatever, okay? So there it is. Now, if I want, I can do

other things with this, okay? So what do you think will happen

if I say percent times 10 again? Am I going to get

the same number? No. Because now the last

command is this one. So I'm going to get

10 times that, okay? All right, now, what

if I say, oh, I didn't want to

multiply it by 10.

I wanted to multiply it by 100. Can I get to this one? Yes. I can get to that one

by doing the double percent. So double percent means

go back two commands. So 39, 38, all right? And now I can multiply

that one times — well just make it different

by saying 1000, all right? And now I get 1000 times

this one, not this one. Okay? So this can be useful to avoid typing a

bunch of stuff in. But it also, you

have to keep in mind that it refers to

the last command. So for example, if I go

up here, and I say oh, I want to do 10 times this. And I say, okay,

percent times 10. Am I going to get 14? No. I'm going to get

10 times this number, because that was

the last command. And there you have it. Okay? So you have to be

careful if you're jumping around and you're using

this percent thing. Okay? Now, other things I can do

is I can say square root of 2.

And then I can use the percent

as an argument to a function. So for example I say,

okay, N bracket percent, so I don't actually have to go and type N square

root of 2, okay? So you can use it

as an argument also. It just means take

whatever was the result of the last command

and use it here. It's very general. Okay? So, that's all for today. So next time we're

going to learn how to plot, make simple plots. And then I'll tell you

how to do your homework. All right? So next time is tomorrow at 2. Okay? Have a nice day.

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