In this course we're going to present data in the form of a flat data set. The most common example of this is a spreadsheet and you can see an example on the screen now. I'm using Microsoft Excel but you can also use other spreadsheet software. LibreOffice comes to mind it is a free and open source package that can do nearly the exact same things as can be done with Microsoft Excel. Here we do have Microsoft Excel and I want to show you what the idea is behind what we call a flat file and how we store data in there and what the philosophies are around doing this. What you'll notice of course is that there are some column letters up here A, B, C, D, E etc. and then down on the left hand side we see the what we call row numbers. So row 1, row 2, row 3, row 4, row 5. So these are the columns so if I were to click on D up here that highlights the whole column that is column D. If I were to click on 5 here that would be the whole row 5. That gives each and every cell an address. If I clicked on this cell it would be in row 5 and column D. That would be its address D5 or when we start working in software we're more concerned with the row value first and then the column value doesn't matter it has that address column D row 5 and here's the philosophy. We place the variable names inside of the head of each column in row 1. So we see in the first column we have a header called column headers we refer to it sometimes as ID and we've already populated a bunch here it says study underscore XR3 underscore 1. So this simulates data from some study called XR3 and what we have on each of these rows would be the entry for all the variables for a single patient. So this was patient number one, patient number two, patient number three. Now throughout the courses that I do make for healthcare statistics and research we're going to assume that the patient's identity and all of the issues, the legal issues surrounding that has been taken care of. It's different for every country in every region and everyone should be familiar with what the legalities are of what can be done inside of these files what data can be kept, what the permissions are, what the ethical rules are for keeping this data. I'm assuming that that is all sorted out. So we have in this case our patients by number that is their number that data is kept separately so we can have access to perhaps their patient folders if we need extra information. Next on we see the age column. So age would be a variable and we can type in an age for this patient. Let's make this patient 35 years of age. Now age is a numerical variable. It's a continuous numerical variable and that means we can do certain statistical tests on that. The referral for us would mean where the patient was referred from and let's suggest that this patient was referred from what we call the city region. Now that city region would be a nominal categorical variable. We have three regions in this study. It's city and then inland and coastal. Those are the three regions from which a patient can be referred and those are nominal categorical variables but there's no order to them and they are most definitely not numbers. Now let's move on to the next variable and that is temperature. Let's assume this to be the admission temperature and let's suggest that this patient had an admission temperature of 37.0 degrees. Once again we would see this as a continuous numerical variable. The admission heart rate let's make that 80. Once again continuous numerical variable as would go for the white cell count. Let's say suggest that this patient's admission white cell count was 10.1 and then we want to know whether they have diabetes yes or no. Let's suggest that this first patient of us did not have diabetes. The only two data point values that we can add there's yes and no. There's no actual natural order to yes and no so that would be nominal categorical variable. What grade of disease that they have? Let's assume that the disease that we are testing here comes in four grades and let's suggest that this patient had grade two disease. Now what is that grade one two three four? Is that numerical? I would suggest no. I would suggest that this is ordinal categorical. The reason why it's not numerical is that there isn't a fixed difference between the data point values one two three and four because I cannot say that someone who has grade four disease is twice as worse off as someone with grade two disease because two times two is four or the difference between grade one and two is exactly the same as the difference between three and four. That has no meaning. Those statements have no meaning. So the numbers are actually just placeholders. I could have put A, B, C and D there. They are not real numbers. They are not numerical values. They do have some natural order to them. We would accept that there is some natural order between someone who's got grade one grade two grade three or grade four disease and so we would call that nominal, I should say, ordinal categorical variables. The patients received three groups of treatment and there was a placebo group. Let's suggest that the first patient was in the placebo group which we code as P in our example here and then the outcome. The outcome is whether they survived or did not survive. So at the moment I'm going to just say code this as this patient survived. This patient survived. And lastly is months. That would be for us a variable, again numerical variable.  Let's suggest that this patient was last seen 20 months after they were randomized into the treatment group and they survived. That's the last time we saw them which means they might have died at month 21 but that was the end of the study we do not know. So as a recap, these columns are all variables. The variables contain data point values and those data point values are all of a certain type. Here we have numerical, we have ordinal categorical, continuous numerical, continuous numerical, continuous numerical and this is nominal categorical again, ordinal categorical, nominal categorical, nominal categorical and again, continuous numerical. And that is what we would do. We would fill in a row for each patient. So that would be patient number one. That would be for patient number two. And this is what we refer to as a flat file. We have columns which represent our variables and rows which represent a subject in our study. And what we want is for all these values down a variable to be of the same type. I should not type in 20 year for the next patient. That would just be a word. We want to keep this at 20 for the next patient and only choose from the data point values for that specific variable. Also for referral, it'll be city, coastal or inland. So all of these should be either city, coastal, inland, city, coastal, inland, etc. So we keep the data point values down these columns exactly the same, but each row would be one subject or one patient in our data set. And this is a data set that we would then import into our software to do our statistical analysis. In the next video, I'm going to show you a very easy way to create some simulated data right inside of Excel here or any spreadsheet software for that matter. How to create some simulated data so that you have something that you can work with, that you can use as data to do your exercises with.


Hi, good afternoon everyone. My name is Denise and I will be taking you through the basics of Digimap for Schools today. And so today webinar will give you an overview of the mapping tools available to you on Digimap for Schools. But we also offer the Digibytes which are smaller, more detailed webinars that focus on different areas of Digimap for Schools. For example, if you want to learn more about drawing tools, you will join the webinar, the Digibytes webinar for drawing tools. If you want to know about layers, there is one for layers, one for adding your own data. So make sure to check those webinars and the past webinars are available on YouTube and the webinars who are going to come soon. They are on our website. I will show you at the end, but make sure to sign in and take advantage of this free service. So when you first join Digimap for Schools, your whole school receives a pin code and a password. So the password is what you use to login and the pin code is what you use to change settings and arrange your preferences on Digimap for Schools. We will talk a little more about the pin code later on in the passwords. So when you first open your Digimap for Schools, that's what you see. This is the main mapping area of Digimap for Schools. So you can zoom in and out by using the scrolling zoom on your mouse, or you can use the bar on the right of your screen and either drag the bar up and down, or you can use the minus and plus. And if you find that you zoomed in too much, you got lost, don't know where you are. This box underneath with the arrows pointing in four directions. If you click on that, you'll go back to the original map and the original screen where you started from. So before we go ahead, let me zoom in and concentrate a little on the UK. And here you can see I'm using the scrolling button. If you have a touch screen, you can also use the pinch your screen up and down, and that will take you to zoom in or out on your screen. Okay, since I'm already zooming in a lot here, let's go to a place here called the Holy Island, which is on the north of England. So, and I will show a very interesting icon, or sorry, a very interesting function set of tools that you can use with younger students, for example. So on the left panel, you have different tools that you can use with digital map for schools, and you will have also some of the tools on the top bar, and we are going through each one of them. So let me start with the key tool. And the key tool as you zoom in is going to show me different symbols in a map. So if I buy a map on the gas station, for example, it will show me some keys. And those keys are exactly the same keys we have with the map for schools. And that is a good way of working with your students, comparing maps and showing them what is available and what each of the symbols in a map, each of the keys will mean. For example, if I go here and look at the water features, I will have a set of features that I can find in my map. And if I go to the land features, I will see that, for example, I have a wind turbine, and it is right here. I have a mast for electricity, which is here, and I have also a lighthouse, which is down here. And if I change then and go to, let's say, tourist information, I will have another set of symbols, and I will have a castle, and I will have a garden and a museum, and you can scroll down and nature reserve to show this little bird here. So there are lots of things you can do and lots of activities you can work with your students. For example, if I'm concentrating on the wind turbine, I can study the distribution of wind turbines across the region in the UK, for example. Or I can also use the windmills and study the concentration of windmills and just really play around and use the keys to what best suits your studies with your students. So I will zoom out of here a little, and while I can find places, if I know where they are, zooming in and out, I can also use search. And I don't know if you notice as I was zooming out, my key changed as well. And it changed because instead of showing specific points in more details and symbols, it is going to start showing me what I can see in the map. The map will tell me, like I can see red, pink lines, blue lines, green lines, and of course those are roads, those are motorways, and the junctions, and you can analyze those with your students as well. So there is a lot to play and a lot to investigate with the keys. But let's say I don't know where the place I'm looking for is. I can go to the search and I can either click on the magnifying glass here, or I can tap and type if I have a touchscreen, or I can just click with my mouse and then look for a place. So let's say I'm searching for Aberdeen. So Aberdeen, and then I press enter, and I will have two different results, two different sets of results. I have Aberdeen in Scotland, and I have four places, or places in the world, and which will bring me also everything that is called Aberdeen in the world that the map could identify. So the first Aberdeen I have here is the one in Scotland, the same one that I can press from here. But let's press from the word map. And it will bring me straight to Aberdeen. So I can zoom in. A pointer will appear in the place that I'm looking for. And if you don't want that pointer there, as soon as you click on the X button here, the pointer will disappear. I like to keep the pointer while I'm zooming in because that guides me to the correct place. As you can see, the maps in the UK, they are very detailed, very rich in all that you can see, you know where the roads are, you know what the houses are, and you can even identify, if you go to the key, you can identify from the color of those marks here which type of building it is. If it's a building with a glass roof, the key will let you know as well. Let's see, if I go for example to Aberdeen in Mississippi, USA, you can see that the map here of Aberdeen in USA is already not so rich in details. So depending on where you are looking for the places in the world, some countries will give more details to their maps and others will not have so many details. So I can see the streets but I cannot necessarily see the houses. And if I go to Canada, I will have even less details. So I'm just in a map lost in the middle of a big empty area. Of course the area should probably not empty but the information that we are giving in the map is not much. And if I decide to go to Aberdeen in Hong Kong, China, and then you find the city of Aberdeen, you'll find that the Chinese map is also has more details, although there are clusters of houses rather than individual houses in some places. And you'll find that the names are also written in Chinese. There is some translations but not always you'll find the translation so you have to bear in mind that depending on where you want to find a place in the world, it is going to have translation or it's not going to have a translation. And you can use the search button to search for cities, towns, touristic points. So if I go to the Taj Mahal, it will take me to the Taj Mahal in India. And yes, it can guide me to historic points, places and touristic places. So big landmarks as well. Okay, the next thing we are going to talk about and we are going as we are going through the icons and the tools on the left is we are going to look at the measuring tools. So the measuring tools, they are quite good. Okay, I am somewhere in India and I don't want to be in India. So let it's a good time for me to show you here on the top these two arrows that go around.  And then I will have to zoom out or in again as I want. So let's search, for example, for Newton Grange Primary School. Okay, and that is a good one because I only have one. This one is wrong. So I only have one Newton Grange Primary School in the world and that one is in Midlothian. So I will leave the pointer there while I'm zooming in so I don't get lost because as you can see it moves a little and now I can get rid of it. So I have a Newton Grange Primary School. And I can measure. So I will use my measuring tool and I can measure the distance. So let's say I'm going from the hall to the main entrance of the building in the school. If I click and leave and click and leave, I click again at the end and I have, so 200 meters. 205 meters and I have also the measurements in yard. If I want to measure in area, for example, I can just click, drag, click, click, click, click and click at the end. And I know the measurement in the square meters as well. The only issue with measurements is that it is something that you can do with your students on the spot. They can use it for calculating different measurements. They can have as many measurements on the screen as they want. Let's say the distance, compare the distance between Johnny House and the school and the distance between Peter's house and the school. And you can let them play around and do some calculations with the measurements. However, at the moment I leave this map, all the measurements will disappear. I can't print them. I can't do anything with them. And what I'm going to do then is I'm going to delete them all. I just click on deleted measurements, delete all. And then I can go to the drawing tools and then you have a measurement area in the drawing tools, which will behave exactly in the same way. But this time it will be saved. And this time I will be able to print or download my map with all the measurements and all the drawings. So let's take a look at the drawing tools. So the first tool that I have here is the marker. And the markers are quite popular with young children because there are all the icons they can choose. They can also change the colors of their icons. So let's say, for example, I want a tree and I want my tree to have a bright green line and a dark green fill. I don't want the fill to be solid. I want it to be transparent. So that's how I make it transparent or not. So I will get it completely transparent or just slightly visible. And at the moment I touch and click on my map, the tree that I designed is going to appear in the colors that I chose. And as many times as I click, you have a tree. So if you are asking the students to design a playground, for example, they can add trees, they can add children playing, they love the stickman, and they can also go around the school and map the area and say, like, how do you like this play area? So it is a happy play area or is it a play area that needs some improvement? So you can do also lots of interesting studies with your students on with the markers. The second one is the shape. And for the shape you can draw polygons, you can draw freehand. I won't even attempt that because I'm terrible at freehand drawing. You can make a triangle, square, rectangle, pentagon, circle, oval, and I think the most popular one is the polygon because with that one you can make, you can shape any building or can shape around any building. And the process is the same as in the measurements. Click, move on, click, move on. And as you click on the place, you will be designing your polygon. And I'm going just to go around this school building, the main building here. I will ignore everything else. And I'll go here and click at the end. Now I have my polygon. And if I don't like the color and I forgot to change that beforehand as I did with the marker, I can always change it afterwards. So all I need to do is to select. So I will click on select the building. And then I can click on the color and change to the color I want. So I want it blue. It is the line so it is okay if it's not transparent. But for the fields, I want also blue. And this time, yeah, halfway, not completely transparent, but transparent enough that I can see the building underneath it. Because if I make it solid, I can see the building underneath. And then when I click outside, this is the polygon as I wanted it to be. I can do the line in the same way. So I want this time a red line and my line is going to be a dotted line. And it is going to be slightly thicker at four points line. And I want it again from the church hall and all the way to the entrance of my building. Click again. And now what do I do with these drawings? I can measure them. I can actually ask my students to measure those drawings and to calculate. So let's say if each of the students will at the distance between the house and the school, they can calculate who's got the shortest who's going to go to the longer distance. And for example, I would just click on measure. And then I click on the building and I have my measurement in the square meters. And I could actually have chosen imperial unit. So if I go again here and click on measure. And I click on my line measure. I can choose this time. I'm going to choose imperial measurement. And as I click on the line, it will appear there. So in yards. So those are some of the tools you can use with the drawing tools. You can also add a text. And for example, if I want to add a label Newton range primary. And that is where it's going to be the little hand that appears when I hover over the, the, the name or the label. I, if I move my little hand, I will move label to whatever I want. So let's say this is the type of my page. And that's when I would use the text. If I just want a label, I can when I modify the drawing, I can add a label as well. And they can also add a text box. So I student can add a text box to make an explanation of what it is. So I can add a text here. So I'm designed a play part. Okay. So the teacher can leave this as a note for the students or the students can use that to explain what they've been doing and what they like and to enhance their discussions as well. So it's entirely up to you. But it is a good way of making notes on on your work as well. And you can add image to your drawing as well. So if I add an image and I want to add an image here, I will just click on image, click on where I want the image to go and choose from a file that I have on my computer already. So I have a picture of Newton Green's primary school. I would just click on that open and upload and my major up here here when the little hand is over it. I can change and drag around and put it where I want it to be. I can also use the grid reference and get the grid reference of my drawings and they can use the buffer tool. The buffer tool, I think it's a very powerful tool for elder students because it is going to help them to analyze, for example, these structures and identify, for example, let's say I'm doing a buffer tool and I want a point buffer tool. So at the moment I click on the, so I want a point I will have to add here. So I want that half a kilometer, half a kilometer. I want bother changing the colors now I am fine with as they are. And we are going to discuss on how to where we want. Let's say a football field to be. So when I click here I have a buffer tool that is half a kilometer in every way from the center from where I clicked. I can also delete the buffer point that I've just made. And let's try again. And this time I'm going to get a line buffer tool, and I wanted 200 meters from the train line. So let's say I want to know if the noise pollution from when a train passes by would affect the school in any way. So if I'm to understand where to build a school and I understand that let's say in that region, the noise from the train would travel 300 meters. When at the moment I create my buffer tool, I will know that in this line from that is closer to the school. My buffer tool, my noise pollution will be fine. It won't affect the school in any way. And you can use that also for rivers like if you want to if you know the flooding, like I'm not supposed to build a house 100 meters from that river because that river has a flooding extension of 100 meters. So it is a good buffer tool for you to work with your students and to identify. So again, what do I do with all this? So once you play around and once you work on your drawings and you go around and do as many activities and drawings. that you want, you can also save your drawings and for saving them, you can press on save map. And you can save them here. So I have one saved so everything that is saved will be collected here. So I have one new tongue range primary school saved and it when I press save it because I have it saved and I'm opening the one that was saved already. It is going to ask me either to replace or merge drawings. I can merge drawings so I can have drawings from different activities. So let's say each of these students will do one part of the activity. I can merge all the drawings and print it as one big activity to save my maps. I will just click on save map here. And I will be asked to add a title. You can add the class name that's optional. You can add the pupil name. The only thing you need to bear in mind is if you add your students name or if you add sensitive information like the child's address and the distance to school and or the way they go to school and so on. You have to be aware that everyone who has the password will be able to to to see those sensitive information in the same way that I was able to see all that my colleagues have saved here. Your students will be able to see each other's saved work and everyone else with the password. So giving a password to to parents, for example, we will allow them to help the students, but we will also allow them to see information about other students if they are sensitive information. So there are two ways we can go around that. As if you go to the arrow here. And you can go to the settings and from the settings you will be prompted to add your pin number. And with the pin number you can actually disable the saving. So your students wouldn't be saving able to save. So that information that is sensitive information wouldn't be seen by others. That's one way. And the other way is you can work around and give students fictitious names like working groups and add their group names and give characters like fairy tale characters. So you can imagine a way around that would not necessarily expose sensitive information. But yeah, it is one way that you can work around. And one thing that we recommend is why it's okay to give the password to parents. Ask the parents not to give it to someone else because it's only digital map for schools is only meant to use for educational purpose. The pin number is a different story. The pin number should not be given to school to students and parents. It is something that should be only between the teachers and whoever is responsible for the settings of the digital map for schools in your school because if you have the pin number you are able to change the settings. And imagine if I still didn't change the settings from able to save to disable saving and no one would be able to save anything and also your preferences and everything will be tied to your pin number. So very mind that the pin number is restricted because it can change the settings of your digital map for schools. And so that's going back to the drawing tools. I can delete the tools if I don't want them anymore. So I can either delete on click or delete selected to delete selected. I click on select select here my buffer to and then delete selected the buffer to we will be gone. And if I want to delete in a different way I can delete on click.  If I can also just decide that I don't want any of that and they can delete all my drawings.  Everything goes back to where they were before. So that is the drawing tools. And now I showed you the saving as well. Let's see the map selectors. So I have different maps and it depends on how zoomed in or out my map is. I can select so this is now gray. That means I can select the map. But as I zoom in, it is going to change into green. And if it wants it becomes green, yeah, I can change different maps. So I have here area of view. And I can change I have two different historical maps as well. So let's say let's do something else. Let's go to Darrington. And I quite like to work with Darrington for the map selector. And I want Darrington in wheelchair. Darrington is nowadays quite a big town. And let's see how it was in 1890s. So I have it ticked here. So I can either tick here where my dot on the top is and it will change the map. Or I can leave it as it was. Let's say Ordnance Survey in the 1890s. And I can just drag this dot here and it will move and transition between maps. So that's how Darrington was in 1890s. And I zoom out a little so that you see there was absolutely nothing here. It was a little parish village, which is here. And right above Darrington, there is a place called Milston. And Milston is this place here with those three houses. In 1950s, Milston is still the same place with the three houses. And Darrington is still quite small. And that is, I'm talking about 1950s already. But you can see that something started growing around Darrington. And when it gets to, and they're moving here again. And you can see that it changed a lot. And as you can see now, Darrington is quite big and quite quick. So you can actually use the map selector to study the growth of towns and cities. You might know a city nearby or in the UK that grew too fast in a short period of time. I think Milton Keynes is a good example of that. And you can study the reasons for that and the infrastructure is the city coping. Actually, Darrington is not coping and they have infrastructure issues because of the speed of growth that they had. And it is all due to these barracks here. They are military barracks that were built during and after the Second World War. And that is what caused sudden growth in this town of Darrington. But Milston is still here.   So that's why it never became a big town because it's been a farm since the 1890s. Why your Darrington grew a lot in that time. And here are the barracks. So that is the map selector. And you can use for doing historical analysis and seeing how places changed from time to time. And now let's talk about the geography. You can let me see if I am in Darrington. I can look for an image. So I will just go here to image search. And if I click on let's say just delete the stars, it will bring me everything. Every single picture, every single image that is in that map. But I don't want that. I just want a church. And it will bring me every church and everything that is a reference to a church. So basically I have some church here at the bottom of my page. But Darrington seems to have only one church, which is this one here. The number three means there are three pictures. I can either click on the tree or I can click on the pictures here and it will show me. And the only thing is the image in the photograph, they are added by people. And although they are secure and discretized, so they are safe for your students to use. Sometimes you need to be creative about how you go, what title you give, how you search. Like sometimes you look for church and you won't find. Then try again, try with the places of worship. Or try if you are looking for a synagogue, you will try to look specifically for a synagogue. And because, as I said, people who add the pictures, they will give the titles and the information as they feel that it's fit. But not necessarily the correct one or the conventional one or the one that you would use instead. So be aware of that. And if you are looking for something on photograph, make sure to look again in different ways. So there are lots of things you can do. You can look for images and you can study those images and discuss with your students. And, but mainly it's quite attractive for the younger students as well. So they can discover features in their own town, for example. Now, adding your own data, I'm going to start over again and we are going to add our own data to the map for school. So to add our own data, which is this icon here. I will just click and here there are four different types of files that I can use. And the most common one is actually, I don't know why it's not here is the one on top is the CSV file. So as a CSV file, I will show you in a minute, it is this file here. You will have a label. And that is the names of the places you want to add to your map and they are postcode. Don't forget the titles you need to give is label and postcode. Because if you put name here, it might not work. So always use label instead of name. It is very similar to an Excel file. But when you save it, you save as an SV file or CSV file, sorry. So, and I will show you how it works. This one is one I prepared earlier on. So I will choose file and in my folder here, London Touristic Attractions. So let's say you are doing a field work with your students and you want them to go and import. Go and find places in London, for example, and add them to the map. And or you can add that data to the map and then ask students to use the drawing tools to measure the distance. So it is quite confusing here because it's pink that you can also change the markers. So let's see the area of view. Okay, that makes it easier. So you have here London and you have all the, I think I have here 20 different tourist attractions. So you can actually go to the drawing tools and make a line between Kensington policy and the Tower of London, for example, and see the distance. So you can plan and map a nice day out with your students. They can talk about the transportation. You can also have a map of the London subways. You can add here so you can actually add different data to your map all together at the same time. And they will come with the same pointers, but you can always go here and so click on your pointer. I don't want to change anything. And you can change the color of your pointer or even the icon if you prefer. And you can, as I said, draw lines, make measurement, and you can actually plan your day out. You can have different topics and work with them. And you can study with the older students. You can start, I have a set of activities for waste recycling centers around London, so you can see the distance between them. You can use for hospitals. You just need the labels and the postcode, and you can add any data to the map for schools. And you can also add, and I'm going to refresh it again, you can also add a bigger set of data, for example. I have here the Jill Jason, and usually they'll come in a zip file like this one. And I want to know where the national reserves, nature reserves are in the UK. So at the moment I import. So this data I get extracted from our website already in the zip file. So I don't need to do anything with that. I just transfer that directly to my map. It takes a while to load because they are bigger chunk of information than just the label and the postcode. But when they load, you will see you'll be able to identify every place in UK where there are nature reserves. And I have one also for the national parks, which you can identify, and it all depends on the topic you are working on. So you can also print your map. And to print your map, you are going to click on this printer on the top banner, add a title to your map. Let's say test and you can actually have the grid lines or not this time it's not offering, but you can get your drawings in your printing as well. And generate a print file. When you click on generate a print file, it is going to appear on your screen, and you can just accept and print. It will always print what is appearing on your screen. So if you want a smaller image or a bigger image, you have to adjust the image to the size you want to print. So that is my file as it's going to be printed. And I don't want that now. I'm not printing it. So going and reloading again, I'm conscious of time and now I need to go and show you the map information. So every time you move your mouse on your map as you can see the numbers are going crazy. You can actually see the coordinates of your map. If you want to capture the coordinates, you will click on this button here. And then you can at the moment when you click, you have the exactly coordinated. Now I can hover around the numbers won't change because that is the only number that matters to me. And I can again have print my map. So that is all for the coordinate. Now I want to show you the overlays. And the overlays are our last activity here. So let me go back again and refresh. As you can see, I'm quite fond of the refresh button. So for the overlays, you have two types of overlays mainly the GB overlays. And there you can it's only for the Great Britain and it's not appearing. It's great. It gets black as I zoom in. So and other stuff appear. So you can have roads, place names. You can have post codes boundaries and the let's see. As we zoom in more and more, then you are more able to do to use more of the layers so I can use the road names. I can use the national grid. I can use the post codes. And I can use boundaries as well. But my favorite here, principally for the smallest children is the coloring in. So for the coloring in you have to zoom in a lot. So you will stop zooming when it becomes black now. And I don't have a town here. Let me see a place with a busy place. Okay, here. So let's. Yeah, that's the town. And as I zoom in color, I can color in. So if I click on coloring in my map goes black. And I can just print this blank map. This is taking a while. Okay, I can print this blank map and ask my students to color in. So that is a good activity for when you are specifying different buildings and the significance of them. And again, what I can do is press on print. And I can generate a print file. And my print file is going to look like this in here. So I can give to my students and ask them to print. This only works for the Great Britain maps, because it's in the GB overlay. And as I said, there are fundamentally two types of overlay, the GB overlays and the word overlays. So the word overlays, they are fantastic for the more mature students, because with the word overlays, you can discuss all sorts of issues. And like for example, word climate, you can do a comparison in the temperature when between 1970s and 2000 and between 2010 and 2018. And you can even put an overlay on top of the other and make one more transparent than the other. And also check the projected temperature until 2040 from 2020 to 2040. And that is quite interesting so they can actually work with predictions, you can even use these formats, for example, for average predictions and so on. And the same for precipitations. You can have the word human geography, which will show word places names and populations and word places names is still great out. Don't understand why population density. So if you click on population density, you will see, for example, you can study with your students why some parts of the world are heavily populated, why others are not and sometimes in the same country in the same continent. So they can understand, for example, the concentration of people along the Nile River and why that concentration over there in the, yeah, there are loads of things you can discuss with your students on the population density. And you can also go to the time zones and study different time zones. One thing I only learned once I started working here is that there is such a thing as three quarters of a time time zone. And so you can discuss that with your students as well. So if you go to World Physical Geography, then you can have the biomes, which is a fantastic resource for discussing where the forests are, what type of forest, what type of vegetation. And you can have mountain range. You can get overlay on top of overlay. I'm not doing that now because I want to show you them individually. So we have a mountain range. So you can start that as well. Volcanoes can check with your students where volcanoes are and even compare them with tectonic plates. And if the tectonic plates are too strong, we can have just the boundaries as well. So yeah, I suggest you just go away and play with the overlays.  And you can have the latitude, longitude grid and the major lines if you don't want too many lines. So depending on the age and the stage of your student and the top you are going about. Okay. And the last thing I want to show you is this question mark on the top banner. If you click on this question mark, it will take you to Digimap for Schools, how to guides. You can go around those guides and look at everything that you need to know. Everything will be here and will be explained to you. And some of the guides are with videos as well so that you can understand better. You can also click on webinars. And here you have a list of the future webinars with Digimap for Schools and you can sign up just by clicking. And you can also look at the learning resources. There are over 150 learning resources for primary, secondary and early years. And you can just go and play with those resources and check and see what suits you, what you would like to explore. So they are divided by topics. And some of them will explore different drawing tools, the others overlays and so on. So they will work with different areas, different tools in Digimap for Schools. So back to category. The secondary will also have a similar range. So you can start climate change and earthquakes, for example, emergency rescues, exploring the world biomes. There is an array of activities you can do with your students with Digimap for Schools. And that is us in a nutshell. I will stop sharing now. And if you have any questions, please let me know on the chat or you can even ask in person and I will answer to the best of my knowledge. Thank you very much. And if you want to stay for the questions, please feel free. If you don't, thank you very much for attending and hope to see you soon.


We've had Atari versus Coleco and Mattel. We've had Sinclair versus Commodore and Amstrad. We've had Commodore versus Atari. The history of the early 8-bits is strewn with conflict. And in each case, a clear victor has emerged. But who was the best bond? Who was the best 007 agent? The clear winner for most would be Sean Connery. But for me, I'm talking Timothy Dalton. He was the most believable of all the 007 agents. He was grittier. He was darker. Essentially, he was a madman. Sadly, we only got two films from him, but those are two of the best in the franchise. Ah, don't get me wrong, we all love a bit of Sean Connery and Pussy Galore from Goldfinger. But Timothy Dalton's movies were edgier. But then we come to the ZX Spectrum, and if Gold and I is the best James Bond game ever made, what is the best James Bond game on the humble ZX Spectrum? Right, pay attention Bond. This is the Sinclair ZX Spectrum Plus 2. It's a fully operational computer with 128k memory. But it comes with three James Bond games and a light gun that fires armour piercing shells. Now that's your assignment. No, no, don't sit in that chair! Sorry Bond, haven't perfected that yet. The Sinclair ZX Spectrum Plus 2. Well, this is my pick. Licensed to kill. They should have probably gone with the name from Dumbark with love. Wait, I'm not ready yet, so let's add Bond Girl, to Lisa Soto, and the equally mesmerizing Carrie Lowell for your eyes only. Whoa, steady on, old boy. If you've ever had the unfortunate experience of playing a view to a kill, you'll have some understanding as to why 8-bit gamers have found previous Bond games on their computers. Disappointing. There were other Bond games as well, but many had all the ingredients of a poop-eye. But all credit to Dumbark then. There's little to criticise in this entertaining, exciting, over-the-top action title. In Licensed to Kill, the world-conquering megalomaniac, the then largely unknown George Davie, who played Sanchez. He's a really good, bad guy, and in this movie he plays a cocaine drug smuggler. So Bond is out for revenge when his CIA buddy, Felix, is done in by Sanchez. After the drug baron escapes from custody in Florida, Bond is on his own, disowned by M and the service, but decides to chase Sanchez across Latin America and through the heart of the drug business. All the essential Bond ingredients are there, including a massive lack in originality when it comes to the story.  So let's have a quick look at the spy who loved me. It certainly left me shaken, not stirred. For me, it was worth £1.99, but not full price. There's not enough here to keep you entertained long-term. So it's definitely one of the better early Bond games.  For me, it's all a bit too samey. It's not very original, and once I'd played and completed it, I wouldn't play it again. But definitely a step in the right direction. Ah, the living daylights. Now, this was Dumark's second Bond game after the travesty that was viewed to a kill, and it was released as a light gun shooter on the ZX Spectrum with the 007 action pack. Crash awarded this one 63%, and then they said, by no means a disaster, but the lack of real variation means lasting appeal is on the short side. But a second opinion turned around and said it's just the same thing over and over again, the same idea with different backgrounds. Now, I had high hopes for Live and Let Die, because it was one of my favourite Roger Moore movies. Now, this didn't start out as a Bond game. It was a rebranding after Dumark saw elite systems working on a game named Aquablast. Dumark thought that this title resembled the speedboat chase of Live and Let Die, and the rest is history. But can you imagine what could have been if Ocean Software would have got the license instead of Dumark? We could be talking about some of the best movie to home conversions. Up there with Hudson Hawk and Robocop, worst case, Miami Vice. So I found license to kill to be fast and satisfying, and I'd go as far to say that it's the best ever ZX Spectrum Bond. On the second screen, weaponry, i.e. ammunition can sometimes be hard to come by, but you're very rarely short. I don't know, I just really like this game, especially the first two levels. The helicopter chase scene, it's over really quickly, which is a shame. But then this style of level, yes, it's a commando style run and gun, but I'd never seen it implemented like this before. The aim system was similar to Rebel Star or later Laser Squad, but I'd love to know where the programmer found the inspiration for the control system. And it must be a good game as well, because I often come back to it. I had it on the Amstrad CPC as well, and that's just as good as this. Perhaps a tad slower, but in 16 colour. Regardless, the game on all platforms follows the movie quite well, albeit simplified. And let's face it, it's a massive improvement over the other three titles from Domark. And I personally believe it captures the spirit of the movie. You've got six levels, and there's diversity in the gameplay. One minute you're on a jet ski, next you're on foot, and on the final stage, you're driving an articulated lorry. But better still, after completing the game, the difficulty ramps up and you can go again. So as a 007 Bond game, this one seriously raised the bar. There's also some great little things that people probably aren't aware of. If you destroy the Jeep on the first stage, the second stage is then a lot easier. And if you're playing the game for a second time round, on the water ski section, people now fire at you with machine guns from the boat. Now I almost forgot there was a Tex adventure on the ZX Spectrum in 1982, but I don't think it was official. You basically had to travel the globe and put an end to Doctor Death. Well, I hope you liked this video. And that's it, Bob's your uncle, fan of your art. I'm not sure what else I can say about the game. Other than watch it until the end, let me know in the comments what you think. Did you own this game? Did you ever play it? Or is it one that passed you by? Please share this video, please like, please subscribe. And I think it's best viewed on a big TV, like a 50 or 60 inch. So, until next time, Tararabit!


Welcome folks, we have the now industrial straightened down at 49 Nasdaq's of 78 S&Ps down 14.5 gold gold contract of $8.80 straightened at 17.64oz. We have silver up 29 cents $21.42oz. Light sweet crude up $1.16. $78.40 a barrel notes and bonds. So 10-year note. Down six ticks $1.12.23 the 30-year off 21 at $1.26.24 a king dollar. King dollars up $137 ticks trade in $106.819. The year is at $103. The yen is out here at $138.00 and the British pound is at $119.00 to $1.00 at U.S. dollar. We get over and take a look at the S&P folks bottom line is that, you know, get a lower market. Volume looks like it's going to be light again. We have 40 million shares out here. You know, yesterday we did 68. I think we're doing the same thing. Now you got, you get a swing point here at $284.00. So we'll see how that shakes out. That's in the S&P. NDX as we said, you know, yesterday also is that this is a different situation. The NDX is already inside its lower range. Once it got back inside the 284, it's game now to fill that gap. The gap is at $268.00. Volume today is $29 million. That's going to do more than yesterday. We're going to do about $40 million. The swing point's $55.00. So it's still going to be less than that, but the bottom line was lower price. And it's all about $30 million. So I suspect what we're going to have here tomorrow, no matter what pile Powell says, King Dollar wants to go higher. What we had is that first off, you know, if you take a look at this, you're going to see where there's a straight line move down. Well, the straight line move goes like this. The last leg down. Last leg started at 113, 148 straight down to 105, 34. You have small bounce. You tested that area. You rejected it. We rejected that on Friday. I mean, yesterday bottom line. You're inside the range. Once they were inside this range, 108 is game. It's going to be actually, it's higher than 108. It's 108.00, I think. Yeah, it's 108. Actually 108. Is that 108? Yeah, 108. Approximately $292. That's only the .382. If it goes to 50, a bottom line is that you're dealing with 109, 251. So that's the way the market looks to me right now, how this is shaping up. If we go into the, well, here, watch this. If you go into the gold contract, this is going to be interesting because, you know, if that's what we get, I suspect this little gold contract here, going to not only pull back, okay? You're going up today. So going up today with 121,000 contracts. We take a look at this. Now, this is the roll. So that's good contract volume. But watch this. When you go to the GLD, okay, the GLD looks like a sweet ABC structure down. So that's a big heads up, man. The GLD got to 163.63 today. Gave it up. Light of volume. That looks to me like a small ABC down. Stay right there, folks. Come right back.


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All right, so today I'm going to start out by finishing up some examples of doing integrals that are relevant to chemistry. And I'm just going to go ahead and write on the board what we're going to do and then we'll crank them out. Okay, so I've mentioned to you previously that if you have a probability distribution of say a variable, continuous variable say X and you want to calculate the average of some function of X, the way you do that is you integrate the product of the probability distribution times the function over all of the range of X. Okay? And so what we're going to do today is we're going to take the Maxwell-Boltzmann distribution of speeds of molecules or atoms in an ideal gas so that's f of u where u is the speed, it's equal to 4 pi and then we have m, the molar mass over 2 pi rt, squeaky, 3 halves and then e to the minus m u squared over rt. And then what we're going to do with that is we're going to calculate the mean speed so that's going to be the integral of u times f of u, I'm sorry, there should be a u squared in here too, okay, du. Now what's the range of the speed for a molecule in a gas? We have to know that so we can do the integral over the correct range. What's the slowest a molecule can go? Zero and what's the fastest? Infinity in principle so we'll do that and we should get a known example either from GCAM you might have seen this or you will see it in physical chemistry which is that this should come out, it should have a closed form and it will be 8m rt divided by, well sorry, 8 rt divided by pi times m to the 1 half, okay? So this will be the average speed and then in addition I want to calculate the so-called root mean squared speed which you've probably also seen before and the definition of that is that it's the square root of the mean squared speed so calculate the mean squared speed to take the square root and so that's going to be the integral from 0 to infinity of u squared times f of u times du and this one you may also recognize that's going to be 3 rt divided by m to the 1 half, okay? All right so those are the first couple of things that we're going to do and then while I have the lights and I'll tell you what we'll do after that, okay? And so the next thing I want to do is I want to calculate the probability that the speed is between two values so let's say u1 and u2. Anybody know how I do that? Well what I want to do is I want to add up all the infinitesimal probabilities ranging from u1 to u2. Okay so what I do there is I'll call this p that u1 is less than or equal to u which is less than or equal to u2 so to get that probability what I should do is take the integral from u1 to u2 of f of u du, okay? So that's just like I have my Maxwell-Boltzmann distribution say u1 here and u2 here. What we're calculating is just the area under the curve here, all right? So that's another thing we can do. And then what we'll do is we'll consider a different system but a similar sort of problem so what we're going to actually do is for the hydrogen atom we're going to take an orbital in the hydrogen atom and we're going to look at the 1s orbital and we're going to calculate the probability that the electron is closer to the nucleus than the Bohr radius so it's within a zero of the nucleus. And what that's going to be is first of all we need to know as in this example we need to know what's the relevant probability distribution and that turns out to be because we're in spherical coordinates it's going to be r squared and then times the wave function, radial part of the wave function squared which I'll call r10 of r squared, okay? And then we need to integrate that between r equals zero and r equals the Bohr radius, okay? So these are some examples where we'll use Mathematica to calculate probabilities and then average properties from probability distributions, okay? And you will definitely see this in physical chemistry and those of you who are in physical chemistry right now are probably very close to seeing calculations of this sort. Started the hydrogen atom yet? Nope, not yet, okay, maybe next week. Anyway, we'll see how that works here but mostly for us this is just practice using Mathematica to do integrals. Okay, so for the first three things that I listed there we need to have the Maxwell-Boltzmann distribution typed in, all right?  Then close that parentheses and that one. All right, so let's go ahead and we can enter that and now we'll do the mean speed, okay? So we could say u average equals, all right? And what we want to do is integrate u times f of u and we want to integrate from u goes from 0 to infinity. All right, so let's let that rip and we get a nice conditional expression. So what is that telling us? We should put in some assumptions, all right? So what can we assume about this guy? I'm sorry, I can't hear you. Yeah, so let's put in assuming and then put some brackets around here and then we'll put in some curly brackets here for our assumptions and a comma and now we can list, I don't know what this, I put the comma in the wrong place. So comma i, there we go. Okay, so we know that m is greater than 0, r is greater than 0 and t is greater than 0. So we can try that and see if that'll give us a nice, tidy expression and you see that we get a nice, tidy expression that we could use later in subsequent calculations if we want and it doesn't look exactly like what I wrote there but if you bring this 2 inside it becomes a 4 so that gives us a 8, rt over pi m all to the square root. Now that's one you could probably crank out by hand if you wanted to but I suspect most of you would prefer to do it like this. Okay, so let's go ahead now and calculate urms so we'll just mouse that in since we're going to use the same assumptions and now what do we need to do? Well, we need to change two things. First is we want to put u squared in the integral and then the second is we want to take the square root of the whole thing and we'll call it something else, urms. So let's try that and sure enough we get the equation that you find in your textbooks which is the square root of 3rt over m, okay? And now if you wanted you could put some numbers in there so for example we could say do in SI units we could say r equals 8.314 and let's say t equals 300k and then m let's put in 0.028 kilograms per mole which is n2 and we've seen this number before. Now we can ask for urms and evaluate that, okay? So we get about 517 meters per second for the rms speed of a nitrogen molecule at room temperature. Any questions? Okay, now let's go ahead and do the probability that a nitrogen, well let's have a look here just for fun. Let's go ahead and plot this guy here with the values that we have so I think we ought to be able to plot f of u and then we'll go from u goes from 0 to 1,000 meters per second. Okay, so there's our Maxwell-Boltzmann distribution. Now what if we want to calculate the fraction of molecules, n2 molecules that have speeds between say 300 and 500 meters per second? What should we do? Well that's going to be the area under the curve between lines drawn vertically at 300 and 500 and we can calculate it by doing an integral so I'll just go ahead and grab this guy here and put it in and now we don't want u times we just want the probability and now we'll put in specific values for the lower and upper limits. Now what do you think the answer should be? Look at the curve here. Is it 10 percent, 50 percent, 90 percent? All right, what we want is the area going basically from here over to here so this area in here. Probably about a third or so wouldn't you say? It's good to know the answer sometimes in advance and so when we calculate we see that in fact it's about 38 percent. All right, so you can imagine that in this room about 38 percent of the nitrogen molecules are going between 300 and 500 meters per second. All right, any questions on this calculation? Okay, well let's now go to the hydrogen atom and we'll calculate the probability that the electron in a 1s orbital in the hydrogen atom is within the Bohr radius of the nucleus. All right, so the first thing we need to do is type in the wave function for the 1s orbital, we've seen this before and I'm going to call it f of r underscore colon equals and so what this is is 2 times 1 over a 0, the Bohr radius and that's to the power 3 halves and then it's times an exponential decay which is e to the power parentheses minus r over a 0. Okay, so we can go ahead and enter that, all right and let's just, what do we want to do here? Well let's go ahead and calculate the probability and then we'll see whether or not it makes sense. Okay, so then what I want to do is, I'll say prob equals integrate and this one has a volume element that involves r squared. You'll see this when you learn about spherical polar coordinates in physical chemistry class. So what we actually have to do to turn the wave function into a probability is square it and then multiply by r squared. So we'll say r squared times f of r squared and then I said we want the probability that the electron is between 0, the nucleus, and the Bohr radius, a 0, all right? So we can go ahead and try that and what do we see? What do we see is that I should clear r because I assigned r to a value and that screws everything up. So I'm going to go ahead and say clear r here, re-enter and then try again and notice we got an exact answer for this. If we want a numerical value we could say n percent and it says there's about a 1 and 3 chance of finding the electron in the 1s orbital between 0 and the Bohr radius. Does that make sense? Well, let's go ahead and put in a value for the Bohr radius. It happens to be 0.0529 nanometers and then we'll plot r squared times f of r quantity squared and we'll do that from r goes from 0 to 3 times the Bohr radius, all right? So let that rip. So the probability we were asking for was r goes from 0 to about 0.0529 so out here somewhere which is this area over here. Does that look like it's about a third of the area under the curve? Maybe it's easier if you go out a little bit further like to 5. Does this look like about a third of the area? Yeah, okay. So that's sensible. All right, so the electron spending about a third of its time in that region there and notice as we previously had noticed that the peak, the most probable place of finding the electron is actually at the Bohr radius. That's the significance of the Bohr radius, all right? So that does it for integration. Are there any questions on using integration in Mathematica? All right, well you'll get to test and see how to use it by doing the homework which is what I want to talk about right now. So let's go ahead and have a look at our homework for the week. Okay, so there's three problems in the homework this week and the first one you're going to use different plotting techniques to have a look at the 4FZ squared orbital, hydrogen like orbital, okay? Which in Cartesian coordinates has this form where R in the exponent is given in the Cartesian coordinates like that. So this will be very similar to examples we did in class. You define this function as function of x, y, and z and that one and then you're going to use contour plot 3D to plot the orbital and you're going to plot two contours, one corresponding to positive amplitude or amplitude equals 0.2. That'll give you the positive lobe and then another minus 0.2.  And what I'd also like you to do in part A here is you can choose your two favorite colors to plot each of the two contours in different colors, all right? So we can tell which one is which. All right, and then in parts B and C, what I want you to do is look at slices through that orbital, all right? So the particular slices that I want you to look at are the slice through the YZ plane, okay? So that corresponds to x equals 0. And you can go to the notes to see how you do that. I want you to take it from a four-dimensional function to a three-dimensional function by eliminating the appropriate variable. And then in part B, make a contour plot of that projection. I give you the ranges to use. Go ahead and use 20 contours so you can see the orbital in detail and use plot range arrow all to make sure you get the whole thing. And then in part C, what I want you to do is actually plot a projection, the same projection of R squared times the function squared and plot that as a density plot. So this will give you a function that's proportional to the probability of finding the electron in that orbital in the YZ plane, okay? All right, so that should be straightforward if you follow the examples that we did last week in class.  Now, what you'll learn in Chem 131C, you'll be learning about a branch of physical chemistry called statistical mechanics. And what you will learn there is that there's a very important function which is called the partition function from which, if you know it, you can derive all of the macroscopic properties of a system, okay? And here in this problem, I'm calling that function Q which is a function of temperature, all right? So here's a couple of examples of things that you can calculate if you know the partition function. So one of them is the thermodynamic quantity called the internal energy which I use the symbol U for here. And that's Boltzmann's constant times T squared divided by the partition function and then times the derivative of the partition function with respect to temperature, all right? So you'll derive this formula when you take Chem 131C. And then another thing you'll learn in thermodynamics is once you know how the internal energy depends on the temperature, you can calculate a quantity called the heat capacity which is how much heat something absorbs, a substance absorbs when you increase the temperature. And the relation is that it's just simply the temperature derivative of the internal energy, okay? All right, so forgetting about all the statistical mechanics part, what I want you to do is actually evaluate these two things for a function that I'm going to show you on the next page. So basically it's just a differentiation exercise. And then I'll have you plot the results as a function of temperature, all right? So the particular system that you're going to study here is the so-called two-level system which is a model system that's used a lot when you talk about NMR. So for example, NMR of spin one-half systems, maybe you'll learn this in organic chemistry. There's two spin states and they're separated by an energy which I'm calling epsilon here, all right? And so what I want you to do is define this function and you can go ahead and set epsilon equal one and Boltzmann constant equal to one because the important part here is the temperature dependence. And then use the formulas on the previous page to define the internal energy as a function of temperature and then I want you to plot it over the range shown there and then plot the heat capacity versus temperature for the same range, okay? So this problem is to find a function, take its derivative and multiply by some numbers, then take the derivative again and make separate plots of those two things, okay? And maybe you can ponder the results and see if they make sense. But in any case, it's a derivative and plotting problem. And be aware that you may have to be careful with how you treat the derivative in the plots and there are examples in the notes showing you how to deal with some of the problems that arise when you do that. Okay, last problem is, so we've been talking about kinetic theory and so for example on the board there, we have some results from kinetic theory, namely we have the Maxwell Boltzmann distribution which tells us what's the probability of finding a gas molecule in an ideal gas with a speed u. And then we have the average and RMS speed that we can derive once we know the probability distribution. Now there's another important probability distribution that comes from kinetic theory and that's called the Maxwell distribution which is different from Maxwell Boltzmann. And what the Maxwell distribution is is the probability of finding a velocity component within a certain range and that velocity component might be the velocity in the x direction, the y direction, or the z direction. And velocity components range from minus infinity to infinity. It's not the speed, it's not the square root of the velocity squared. Okay, now it turns out that for an ideal gas that distribution is a Gaussian, it's shown here. This is the Maxwell distribution for the x component of the velocity of a gas molecule. And I think you can see over here it's got e to the minus something times vx squared which is Gaussian. So this is one of the places where our friend the Gaussian distribution shows up in an important context. All right, so what I want you to do here is basically derive the ideal gas law from kinetic theory. Okay, so kinetic theory tells us that the pressure is equal to the number of moles of gas times the molar mass times the mean square of one of the components of the velocity and then divided by the volume. Okay, so this is what you call a microscopic formula. Telling you how you, well, it relates a macroscopic quantity of the pressure to microscopic behavior. In other words, the average of the velocity components. All right, so this is defined in the same way as we defined averages over there on the board. So the average of vx squared, you take the corresponding probability distribution, f of vx, so that's that, and you put in vx squared in here and you integrate over the whole range which is minus infinity to infinity. So what I want you to do for this problem is define this and then plug this into here and evaluate that. And if you did it right, you should get the ideal gas law and hopefully you'll find that somehow satisfying. What it is is it's a proof that the kinetic theory is consistent with the macroscopic ideal gas law. Okay, so that's part A. And then for part B, I just want you to show using an integration that this guy is normalized, meaning if I integrate it over its full range, I should get one. Okay, so this is a problem where you get to practice using integration on some chemistry related examples. Any questions on the homework? And if you're like freaked out by all the physical chemistry stuff in here, just worry about the formulas and put them into Mathematica. And then later on in physical chemistry, you'll be like, oh, I remember that. Okay, all right, so we're onto a new subject now and our next subject is equation solving. Okay, so let me motivate that with a couple of examples. What we're going to see is that the technique of equation solving, so we can solve one equation or multiple equations simultaneously, we're going to see that this is very, very useful for doing calculations for complex equilibria, which is stuff you normally don't touch in general chemistry and you get a little taste of in Chem 151. And then also some gnarly kinetics examples where you have much more complicated things than what you saw in general chemistry where you can treat several rate equations at the same time and get the concentrations as a function of time for many species. Okay, so this is a very powerful thing that Mathematica is very, very useful for. So I'm going to start with a very, very simple example, excuse me, to show you how it works. So suppose I have an equation x plus 1 equals 0. What's the solution to that equation? What value of x makes that true? X equals negative 1, right? So you just do very, very simple algebra that you learn in, I don't know, fifth grade or something and you have the solution for x. Okay, so how can we do that problem using Mathematica? Well, the first thing we do is we turn it into an expression that basically asks the question, is that true? And the way you do that in Mathematica is you make a double equals, okay? And then the next thing you do is you say solve that. So solve, bracket. And then in general, there may be more than one thing that looks like a variable, so you want to actually say which variable do you want to solve for? Okay, so in this case that would be x. And then you enter it and you get the answer in a somewhat strange looking form. Okay? So this looks kind of like a replacement rule sort of object, all right? So what it is is it's a list of lists. In this case, the list inside contains one element, but in general we may have more than one solution in which case they'll all be listed. And we'll see soon enough how we could actually pull out the value if we wanted it, all right? We'll worry about that in a second, all right? But this is the basic form of using the solve command and seeing what the format of the results is, all right? So let's try another familiar example or a familiar example.  So I can say solve. And I'm going to say I have an expression A times x squared. Plus B times x plus C, all right? So that's a general quadratic equation and solve that or set that equal to zero and then solve for x. Now, what should I get out?  So that hopefully should look familiar. Now, let's use this as a way to actually get out, to get access to those formulas, okay? Now, let me go ahead and mouse this in because I want to show you how this works in comparison to this format here. So what I'm going to do is I'm going to define an output here, roots equals, all right? Now, if I just enter that, I get the same thing so that doesn't help me. But if I want to actually extract out the formulas from the list, notice we have a list of lists, right? So there's two elements in the list, two things inside. So if I want to get the interior brackets off of there, what I can do is say x slash dot, okay? So what that's going to do is it's going to unpack the guys in the solutions. And now notice I have those two formulas, okay? Now, what if I wanted just one of them? Well, now I can use the usual way of addressing a list. If I want the first one, I would say that and notice I have a formula that I can use now. So if I wanted, I could set it equal to something like root one, okay? And now I could use root one in a calculation and similarly, I could pull out the second root. Call it root two, say, all right? So there's the, yes, notice plus sign, minus sign. Okay, so that's pretty cool. What if I want to put in some actual numbers? Well, I could say now that I've got roots here, I could say give me roots for particular values of A. So let's say A arrow one, B arrow three, and C arrow one, all right? If you enter that, you get some numbers. And in this case, they're exact. So if you wanted numerical values, you could say N percent. And then you get some numerical values. And notice, if I use those replacement rules, I still have my formulas intact. So if I want to reuse them for another set of coefficients, I can do it, all right?  But you can quickly get pretty sophisticated.  How was that? Yeah, it was hard, wasn't it? Well, I'll show you an easier way to do it. So let's solve a general cubic equation, A times X cubed and then plus B times X squared plus C times X plus D equals, equals zero, and solve for X. Enter. Now notice, I can't even fit it on the screen. In fact, I'm going to shrink it down just for a second here. So you can see how impressive that is. Do those formulas look familiar? Now, you probably did a particular case. All right, so the thing I wanted to point out to you, though, on this one is that we have three roots. It's a cubic equation, so we should have three roots. One here, one here, and one here. And notice that the first one looks like it's real, whereas the second two could be complex. They have I's in them. See the I and the I. All right? So when you're solving equations, you oftentimes have to make a choice at the end to choose the root that makes most sense for your particular problem. And we'll see some examples of this later on. Okay, so there you have it. But that's kind of nice, right? You can crank out exact formulas for the cubic equation very simply using the solve command. All right, so I'm going to go back to the big screen here. And I'll just finish up, since we're almost done for today. We'll evaluate for a particular set of numbers. So one way I could do that is just say percent. That means the last result. And then use replacement rules to put in some values. So A arrow 1, B arrow 2, C arrow 3. Whoops, my B doesn't have an arrow here. And D arrow 4. All right? So I get a big mess here. But I could convert if I want to numbers and percent. Okay, so there you have it. Now you could see you have the first root is real and the second two are complex. Okay, so let me just give you a preview of what we're going to do next time. So what we did here was just solve single polynomial equations. So next time we'll see it works perfectly well a lot of the time for non-polynomial equations. We'll see what you can do when the solve command won't work because sometimes it doesn't work. There's no closed form solution. And then we'll see how you can solve multiple equations simultaneously and then we'll start doing examples relevant to chemical equilibria. All right, so some fun to look forward to. See you tomorrow.


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