 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 and 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 it'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 it's discussion. Well, in this course there's no distinction between the discussion and the lecture. The only distinction 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 the 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've all have read it as I instructed you to do last week. All right, so first of all, here's our home page and there's not much of interest on the home page 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 announcement. 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 Pritkova and she's sitting right over there so 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. So yeah, I'm already confused about the sections. All right, well in any case, this, you can hang out there. 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 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, just 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 data sets 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 listed 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 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 is extremely important so it's representing 50% of your grade, all right? We're going to have a midterm around the middle of the quarter, that'll be about 20% and then we'll have a final exam during finals week and that'll be 30%, 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, buy 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 as 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 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. Oops. 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. All right? Now, every thing 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 one. 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, .nb. 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 one, midterm, 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 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 one. Okay. Now, are there any questions about what I just told you about how the class is going to work? Okay? All right. 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 tale of what's called Moore's Law. Has anybody ever heard of Moore's Law? Nope. Okay. Well, Gordon Moore who founded Intel, the company that made the processors that are in most PCs nowadays. Back in the mid-60s 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? 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 that, 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 chemical 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 the 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 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 you 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 and if it is bad let us know because it's not supposed to be bad alright? ok, so now let's go ahead and get started so every time when you come into the class at the beginning of the class log into your computer and go ahead and fire up a Mathematica so I'm going to show you how to do that now if you haven't already figured it out 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 one and a half 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 ok? 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 mathematical operations or whatever, reading and data etc etc ok? so we're going to start with very simple arithmetic operations ok? 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 oops sorry and with a shift shift enter you enter this command and type 2 plus 2 nothing happens but as soon as I hit shift enter that means execute this command alright? now what you see is that you got a result 4 that's the result of adding 2 plus 2 ok? alright 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 soon as it goes on ok? so first of all every time you're in input mode and typing in a command you have this thing says in bracket number and that number is keeping track of how many commands you have within a given section ok? and it keeps track of them in order of how they're executed regardless of where they are on the screen ok? and the output corresponding also gets the number 1 ok? 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 ok? 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 ok? 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 will be the same but it will have a little line below the triangle ok? 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 and we'll see that as time goes on but for now we won't worry about it too much ok? now if I want I can edit so suppose I say 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 alright now if I want I can keep adding commands to this same what we call cell so this exterior bracket here is denoting a single cell ok? so for example I can do a multiplication 2 times 3 so multiplication can be done with the asterisk alright? so if I do that shift enter I get the result 6 alright and notice it's now 3 and there's another way that you can do multiplications 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 time sign alright? and that's something to be aware of because sometimes when you're typing in things 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 ok? 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 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 3 so 2 to the power 3 I use this carrot alright? this carrot means 2 to the third alright? 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 ok 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 ok so if I do 6 divided by 3 I get 2 alright? now let's try a different division 16 divided by 6 I got 8 thirds this shows a very important point about Mathematica ok? this is an exact result alright? Mathematica in general will return an exact result whenever it can ok? and we'll see we can turn that into an approximate or numerical result if you want alright? one way to do that is to put an inexact number in somewhere so here we have 2 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 thirds or 2.6667 2, 1, 2, 3, 4, 5 decimal places now by default Mathematica will return 5 decimal places ok? and we'll see how to change that soon enough ok? now, so far all we've done is single arithmetic operations what if we want to chain a bunch of them together alright? so let's try one we'll do 2 times 3 plus 4 what do you think we should get? should we get first 2 times 3 and then plus 4 or should we get 2 times 3 plus 4 in general the operations are executed from left to right and there is a precedent ok? and the precedent is exponentiation before multiplication and division before addition and subtraction alright? 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 ok? 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 alright? so my advice to you and I'll keep saying this again and again is to just use lots and lots of parentheses ok? 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 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 ok? here's another example 3 divided by 2 to the power 6 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 alright? so what should we get? 2 to the 6 is 64 so we should get 3 and we do ok? but what if instead we actually wanted 3 halves to the power 6 well in that case we should use parentheses to make sure we get the 3 halves 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 that result alright? ok 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 from time to time not very often I 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 Mathematica says yes I know about Pi and that's the exact representation of Pi it's the symbol Pi ok and notice what happens if I do use a lower case by accident I get a word Pi that doesn't mean anything ok 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 ok now another one E if I want the number E I type capital E alright 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 ok and that's to be distinguished from the true letter E lower case lower case you get just E this is a 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 one point and those of you who are familiar with the numerical value of E but in fact it is about 2.72 alright if I did that with little E I get nonsense it just spits it back the way I wrote it ok alright 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 one alright 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 ok so one of them is absolute value absolute value is capital A B S and so if I put in for example minus five and enter I get five now notice I use square brackets when indicating the argument to a function ok 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 a function you use square brackets ok 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 two and I get the square root of two once again you see it didn't give me 1.414 whatever dot dot dot it gave me literally the square root of two that's the exact representation if I want the numerical one way I could get that is to say square root of two point then I get 1.41421 and notice if I put in some lower case it just spits it back at me and that normally means something's wrong alright so that's not square root obviously ok 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 alright 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 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 I get 2 another way to do logs of arbitrary basis 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 alright so we will see quite frequently there are multiple ways to get the same answer 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 because 8 is 2 to the third alright what else do we have big functions sine is capital S I N so if I say sine of pi what should I get what about cosine cosine of pi minus 1 how about inverse cosine or art cosine well the way you do that is you say capital A R C capital C OS pi and I do okay alright because the default units for angles in mathematics is going to be radians 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 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 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 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 so N brackets just means give me a number a decimal number of whatever is in the brackets okay now what if I want more accuracy well what I do is I say 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 yeah I know about six or so so I think I'm pretty good well mathematics is very smart let's try a hundred there you go there's pi to a hundred decimal places piece of cake alright so you can get as many as you want so the whole point though is it just says convert this numerical representation with a hundred decimal places so I can do for example N bracket and I can embed within that a function square root of two and I can say give me eight decimal places so there you have it one two three four five six seven eight okay alright 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 so here I do square root of two alright so there's my square root of two 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 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 alright what this literally means is it takes the result of whatever is 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 alright so we'll see more examples of that later alright so we have just a couple of minutes left so I want to show you one more thing alright and this is something that can be useful but it also can be the source of headaches 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 ok so what this is is this is a shortcut to be able to refer to the result of the previous cell ok so for example the last command here was this one alright and it's number 37 now if I want to refer to this number I can do that with percent ok so let's see how that works I type percent times 10 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 421 whatever ok so there it is now if I want I can do other things with this ok so what do you think will happen if I say percent uh 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 ok now what if I say I didn't want to multiply it by 10 I want 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 to commands so 39 38 alright and now I can multiply that one times well let's just make it different by saying a thousand alright and now I get a thousand times this one not this one ok 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 ok 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 ok so you have to be careful if you're jumping around and you're using this percent thing ok now other things I can do is I can say square root of 2 and then I can do I can use the percent as an argument to a function so for example I say ok n percent so I don't actually have to go and type n of square root of 2 ok so you can use it as argument also it just means take whatever was the result of the last command and use it here it's very general ok 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 alright so next time is tomorrow at 2 ok have a nice day