 Hi there and welcome to this course on numbers and algebra for health data science and for biostatistics. Many of our courses in health care and public health require some level of skill with mathematics. In fact, you may also find that as you take different courses, you start becoming interested in those that require even higher skills in mathematics. It is a fact that some students don't like math that much or simply haven't done any math in some time. That is what this course is all about, a quick introduction and a refresher on the basics of numbers and algebra. Just to build up your confidence and make you familiar with enough mathematics that you do decide to take courses that require a good grounding in mathematics. After finishing this course, you can take those courses with much confidence. Now this is not middle school and we certainly don't have to do things the old fashioned way. Instead, we have very powerful computers and they are terrific at doing math. So since we do not have to do repeated example problems as in school and we have powerful computers, we are going to use a computer to help us understand the mathematics and also to do the calculations for us. Now you do not need any skill to use a computer to help you understand and do the math. I will show you how to do it step by step. You don't even have to install anything. I will show you how to do the calculations by hand with a pencil and paper and then also by using your Google Drive of all things. There is also a complete set of notes that you can read online or even download and print out. I want you to succeed at any courses that you may want to take in the future by building an intuitive understanding of mathematics. A link to all the resources are in the video description. Please read it and also subscribe to this channel where you will find more lectures on mathematics. In this first lecture, we explore numbers and simple arithmetic. So grab a pencil and paper and come along on this journey. So let's start this exciting journey by just looking at the different types of numbers that we can deal with. Because many times in data science we want to do specific tests and sometimes it will really depend on the types of numbers that we use, specifically if we think about the real number line with continuous variables or we think about discrete variables, specifically the integers. So this is, I suppose, a bit of review but it is very, very important. So what I want to start with is just the counting numbers or let's call them the natural numbers. And what we're going to do is, first of all, just give them a symbol and the symbol that we usually use is this double struck. And by that we mean it's got two strokes there, a double struck N, and the way that we write the elements or the members of the set is by using a set of braces or curly brackets. And we're going to say these are the numbers 1, 2, 3, etc. And then it just goes on. And we use these three little ellipses, hopefully you can see them there. Once we are clear, or when you write this, when you think your reader is clear about the pattern that you're trying to establish, and this should be clear that we're meaning 1, 2, 3, 4, 5, etc. Now some instances we really want to include zero in all of this, our counting numbers, because it's sitting as possible to have zero or something. And in some textbooks you'll see a little subscript there with a zero. Some textbooks will not have that, and some authors will just naturally include zero in the natural numbers, and there's debate about that, but it's not particularly important as long as you tell people what you think, you know, what you mean by these numbers. So here we have a bit of mathematics, and as much as we do have a set, and a set is contained within these curly braces, and we have members or elements of the set and they are separated by commas. One thing about the numbers that we're going to deal with is that there's some natural ordering to this. Three is larger than two, which is larger than one, and we have, you know, a minimum. It's not what we say bounded above because it goes off to infinity, of course. In general though, a set does not, its elements does not have to be ordered, but we are dealing with numbers, so we are going to stick with this. And what I'm going to try and do is be clear about, you know, what these things are when we go through them. So for me, this is clearly a definition, and I'm going to put a definition, let's put it on this side. Definitions will have a little green mark to it. That is a definition, and we're going to put a little green arrow to that. The next thing that we're going to talk about, of course, is then the integers. So let's say integers, and the integers, they have a symbol as well, and that's a double-struck z. Let me try my best at making a double-struck z. And of course, that is a z for zhalin, which is German for numbers. And again, we use set notation, and this time we start at negative infinity. And then let's perhaps put negative three, negative two, negative one, zero, one, two, three. And now it should probably be clear to everyone what the pattern is. So once again, we can make use of our little ellipses to denote the fact that this goes on from negative infinity towards positive infinity. And let's just put another green marker there just to denote that this is a definition for us. So we're always going to have these definitions in green. And I like to do this because we need to keep track of, in our notes, of what things are. And when you go through your notes, it's very easy to quickly find the thing you're looking for. Maybe I'm looking for definitions. I'm always going to find them in green there. Next up, of course, we are going to the rational numbers. Rational numbers, or the rational numbers. They also have a symbol. It's usually a q, or there's a q, double-struck again, the line there. And of course, now we have to come up with a new plan to showcase all the members or elements of that set. And what we're going to do is set, build a notation. And set, build a notation. That's fancy words for just saying we're going to have some recipe. And so we better write down our recipe. And we're going to say it's p divided by q. And then in some textbooks, you'll see a colon. I like to have just the line straight down. And we read that as such that. And it's very important in mathematics that we have these symbols for everything. We kind of want to move away from the English language, or the written or spoken language, any other written or spoken language, because language contains ambiguity. And we don't want to use, we don't want to be unclear about what we mean. So we don't want double meanings for things. And of course, in languages, many words have many different meaning. So we use this terseness by introducing these symbols. And so we say it's p divided by q such that both p and q. So we just can say p comma q, they are members of. And that's the funny looking e that we write. They are elements of a set that we've already defined. And that's the set of all these integers. Another thing we want, so we put a comma, is that q is not equal to, and that's the not equal symbol zero, because we cannot divide by zero. And so there you see a beautiful example of set builder notation. Set builder notation says give me a recipe and give me the rules for that recipe. So that's kind of like, give me the ingredients and then how to combine these ingredients. And there you have, we build all the elements instead of writing each one of them, or at least representations of them, or examples of them. We have this pattern, this recipe, it's two numbers, one divided by the others, as long as those two numbers are integers. And then we see that we cannot divide, we just have to include the fact that q cannot be zero because we cannot divide by zero. And we have many of these, of course, and we have this sort of funny notion, I don't know why I'm calling it funny, it is what it is, that we can write the same rational number in many, many ways. So a half I can also write as two fourths. I can write it as 50, 100s, many ways for me to write exactly the same number. That is the same rational number written in many ways. And of course, I can take any of my integers and just divide it by one. And that means that's also a rational number, but in fact, if I divide anything by one, nothing changes, one divided by one's one, two divided by one's two, so we just stuck with the integers again. So there's all these interesting aspects, I can say, of the rationals. Then, very importantly, we have the irrational numbers. Now we don't usually have a symbol for that. Later on, I might put one there just to show you what we do write, but there's no symbol that we use for this set. But the irrational numbers are any numbers that we cannot write as p divided by q. We cannot write it as p divided by q. And there's some famous ones. Let's do the square root, say of two. That is an irrational number that cannot be written as this fraction p divided by q. If we think of pi, pi cannot be written. We think of Euler's number E, and these are numbers that we see in data science and in statistics all the time. So usually, we have these as numerical approximations. So let's just look at this. Let's just put a, we haven't really defined it. I'll define it shortly when we get to the reals, but let's just put our little green there that is going to be a definition of sorts at the moment. And let's just think about writing here on the side. Let's write the number one-half, and when we know that's a rational number, that's an integer, that's an integer. And I can write that as a decimal value, 0.5. But let's take something like a third, and if I do that and I do a decimal value for that, I'm gonna have 0.3333, it's never going to stop. I must write three until the end of days, and that's impossible. But what we can see is that there is a pattern to these. There's repeating elements of this decimal. And the way that we would write that is to show this repeat pattern by taking the digits that are repeated and putting a little line on top of them, a bar, we call that a vinculum, and that's to say that this is the element that gets repeated. So if we write 0.333 and we truncate it there, so imagine I just stop there, we call this part a numerical approximation, and we call this part the exact representation. That's an exact number, that's a numerical approximation. Once I write it out like this, it really is a third, and it's as exact perhaps as what this is. And let's think of another one, let's do one divided by seven, do that on your calculator right now, and you'll see that's 0.14, it is 142857, 2857, and here comes my trusty Nvidia, and if you know anything about that modern artificial intelligence, you know what Nvidia is. There we go, let's put a little vinculum on top of that because this is the pattern that gets repeated, oh, 0.142857, 142857, 142857, that's the pattern that repeats, and because we have this repeated pattern, and doesn't matter how long it is, as long as it repeats, we know that it is a representation of p divided by q, and this is how we see if a decimal representation of a value is indeed an integer. If you write the square root of two as a decimal value, if you write pi as a decimal value, if you write e as a decimal value, the pattern of digits never ever repeats, and hence these are not irrational numbers. So let's look at the really important one, and that is the real numbers. We do have a double struck r for that, for the real numbers, and that is the real number line that you can think of in mathematics. On the horizontal axis, that goes off to negative infinity on this side, that goes off to positive infinity on this side, and there is an uncountable infinite number of values here, but every time, you know, in between any tiny little interval here, and doesn't matter how small you make it, there's an infinite number of values, because you can just forever and ever and ever expand, you know, add decimal values. And this is the numbers that we'll deal with most commonly, the real numbers, and let's put a little green there, we haven't really defined it, I'm just gonna use the natural intuition that you might have for the real numbers, and so just important that we do have that, and now we can finally say this is a set of numbers, and those are two sets of numbers, and actually if you take what we call the union of these two, in other words, you combine all of them into bigger set, then we get the real numbers, such that we can finally say that the irrational numbers are the set of real numbers, and then we have a set difference, which is denoted by this minus sign, if we take all the real numbers and we take away all the rational numbers, take the reals, take away the rational numbers, then you are left with the irrational number, so perhaps that would be a good symbol for the real numbers, as you say, for the irrational numbers. Just wanna show you there's perhaps one more that I want you to know about, we're not gonna do algebra with this, but there's the set of complex numbers, and we write that with a C that is double struck, and a complex number depends on the addition of something called the imaginary unit, let's write that imaginary unit, so this is just a bit of extra information, always nice to know a few extra things, the imaginary unit or the imaginary number, and that number is I, such that I squared equals negative one, and that's a bit odd because you might know that if we square any number, square means you multiply it by itself, three times three is nine, but negative three times negative three is also positive nine, so squaring something should give you a non-negative number, it's only zero squared, which would be zero, but if this number I, if you square that, you get a negative one, and then we write all these imaginary numbers, and we usually say we call them Z equals A plus BI, and so we can just ignore this little bit at the moment, but A and BI, so both A and B, they are members of the set of real numbers, and then this is that we multiply B by I, and then we have this whole new set of numbers, and you can see that real numbers are nothing other than complex numbers whenever B is zero, zero times anything is zero, so this part falls away, and you just left with the real A, and there's all your real numbers, so actually the real numbers are contained within the complex numbers, so it's actually just, you know, fun, but also important to know about the complex numbers, and those are the numbers that we're going to deal with in this course, we're definitely going to do the integers, and we're definitely going to deal with real numbers, and in many instances, many of, now what we deal with, the equations that we deal with do contain pi, and they do contain E, it's so bizarre and such a wonderful thing that we have these irrational numbers appear in so much of what we do. Good, so in this next section, we're going to talk about a little bit of arithmetic, so let's talk a little bit about arithmetic, something you can do easily on your calculator with a computer language, with spreadsheet software, and simply just doing it in your head, so the first one that I actually want to start off with is the normal idea of the addition, addition and the symbol that we use for that is plus, and we can take any of these types of numbers and we can just add them to each other, and so we'll usually just say A plus B, and A can be any of these numbers, and B can be any of these types of numbers, and we simply add them, but let's stick to the real numbers, and you can just add any two real numbers, we can all do that. And the second one in mathematics that we're usually interested in is this idea of multiplication, and of course we can use multiplication symbol, or just put a dot in between, so that we have A times B, or we write A dot B, or we simply just write AB, and that's multiplication, and the one very interesting one is if we let A equal negative one, so if I now have a value B, and I multiply that by negative one, and we'll usually do something like this, just to show it's negative one times B, well that's going to equal negative B, and so usually those are, you know the only two that we really do require, we can think about subtraction, we can think about subtraction, so if I just take A, and I add to that something, and what I'm going to add to that is negative one times B, and that's going to be equal to A minus B, and so subtraction is nothing other than addition and multiplication, so it's not something that really stands on its own, it's this idea of multiplying the second number by negative one, and have addition before that, and that's how we get subtraction, so we also have of course division, and we sort of have spoken about division, there we have division right up there, it's two numbers that I divide by each other, such that I can have a new number, and we will talk a lot more about addition a little bit later, but also in later video lectures, so watch out for addition. What you usually don't see under arithmetic, but I do want to add it here, because I have to put it somewhere, let's talk about rounding, if I want to round a number, and how rounding works is, let's just use examples, so I'm going to have zero points, say one, four, let's make it seven, and then five, and let's imagine this is where I want my rounding to be, I want to round to one, two, three decimal places, I do have to look at what the next number is, and the cutoff that we have is going to be if it's larger than or equal to five, so five, six, seven, eight, nine, if this number is five or more, we're going to increase this last digit by one, so that's going to be 0.148. If my number was, say, 0.147, and perhaps there was a three there, and I want three decimal places, this three is not five or more, so this is going to be 0.147. I think you've all seen rounding. To important types, I should call them types of rounding, but they are something in their own right, is the floor, usually the floor function, and what we're going to write is brackets that just have their bottom bits there, so imagine I have the number 3.99, and I want to take the floor of 3.99, and what you really have to think about here is think about the real number line, so here I have say one, say two, here I have three, here I have four, if I'm anywhere here, not including four, but as close to four as I want to be, the floor is what is the first integer that I find that is lower than that value, so in this instance that is going to be three, and then we also have this idea of a ceiling, a ceiling, and we're going to write this, so just with the one little line there, so what we're going to have is the ceiling there, and even if I have something like let's do 3.0001, but I want the ceiling of that, again, as long as I'm above three, the ceiling would be the higher integer, so that is going to equal four, and that's how we have the floor and the ceiling. Now none of these are that important to me that I'm going to put little definitions there, but I think these three are something that you should just take special note of. The next bit of important arithmetic I want to talk to you about is the powers, and we've spoken a little bit about powers before, but there we have powers, and that is simply where we have, let's start with an example, let's have three to the power of four, and that's going to be three times three, times three, times three, so this is our base, and this is the number that we want to multiply the base, how many times do we want to multiply it by itself, this base value, and we want it four times, so we'll have one, two, three, four of those, and so powers I think most of you are very familiar with, and you know how to probably do it on most calculators, or you can just simply do it by hand, so let's very quickly move on from that, let's not waste too much time on that, let's talk about square root, and we've seen that the square root of two is an irrational number, so the square root is taking a value and seeing if you can factor it out into another number such that when you square that number, you get back to the original, so a usual one that we can write is the square root of four, of course that is going to be positive and negative two, because if I take two times two, I get four, and if I take negative two and I multiply by another negative two, I also get four, so we always have to remember that there's a negative as well, usually we're only going to deal with a positive of these two values, another way that we can write this is to write four to the power one half, because there's actually a little two there and just because we deal with square root, so often we just leave out that four, so these would be equivalent ways to write it, which means we also have higher powers, we have higher roots I should say, so let's put higher roots, and so maybe I can have something like 27 and I take the cube root of that, that would be the same as writing something like 27 to the power one over three, and then you can see, we can get quite ridiculous with these things, let's write 10,000, and we're going to take the fourth root of that, and of course that's actually equal to 10, because why if I take 10 times 10 times 10 times 10, so 10 times 10 is 100, 1,000, 10,000, and then I'm back with my 10,000 again, so that will be the fourth root, and the root that we say is the value that we put in the denominator right there. Next one we're going to talk about is absolute value, the absolute value of something, and the absolute value we usually write with two little bars, so if I say, if I have the absolute value of three, that means the positive integer value of three, and that's just going to be three, but if I take the absolute value of negative three, that's also going to be three, I'm just getting rid of this negative number, so we actually have a nice way to write this, we say that if we take any number, and we take its absolute value, and look at this nice notation that we have, we have this big set of open braces, open curly bracket there, I say that's a, f, a is larger than or equal to zero, and that's true, look, three is larger than or equal to zero, then you just leave it alone, it's three, it's three, it's right there, but it's negative a, f, a is less than zero, so does that make sense? Yes, of course it does, because look at this, there I have a negative number, my a is now negative three, so my a there is negative three, so what I'm actually saying is minus, minus three, which is actually just minus one times minus three, and that's positive three, and there's my positive three, so this is actually how we write it, and I've written it out so nicely, I think it really, it really needs a, let's put it, let's put it here, it really needs a little green mark there, that's a definition for the absolute, the absolute number. Last, in this section, I just want to mention the logarithm of something, I think this is very important for you to know, and what we'll usually write something is the log, let's make it of base 10 of 100 equals two, so we write log, we have a base, we have the number that we're taking the log base 10 of, and then we have the solution, and I think you can see what this means, it says, well if I take 10 and I take it to the power two, I get 100, isn't it not? So what we're saying here is if we write something like this, let's write the log, let's make it the base b of a equals c, so if I have a log base b, here we had log, the base was 10, b was 10 of a, which is 100, I get c. In other words, what we're saying is, we're taking this 10, which is the b, it says this b to the power what gives me, and then we see the c is to the power two, but of what is that 100, that 100 is right there, so we put the little a there. So if I write log base b of a, what am I asking? I'm asking, what must I raise b to two, to what power must I raise the base to get a? To what power must I raise the base so that I get a? Well, that's the c, the c is what we're looking for, what must I raise 10 to so that I get a 100? Well, I have to raise 10 to the power two to get a 100, and so as long as you remember that, I think you'll be okay with logarithms, and we can use many bases, very nice to use base 10, but we actually have a special one, so I'm just gonna put this, let's put it here, because we do need to show that the logarithm is nicely defined for us, at least on this side, and nice base to use is base e, and now we've seen e, Euler's number is an irrational number, and we actually have a special way that we write that, is we say ln for natural log, and that actually just means the log with a base of e, e to the power, what gives me this value, say I want e to the power 10, log base e of 10, that would be the natural log of 10, I'm asking, what must I raise e to so that I get 10? And the natural log is just such a natural thing, and again occurs very commonly in what we do. Seeing that we are on a roll, let's just do one more thing, and that's the order of arithmetical operations, order of arithmetical operations, and you might have learned this from school, and so if I have something like three plus four times 10, we know that multiplication comes before addition, and this should actually be 40 plus three is 43, but if you just did it from the left, you would have three plus four seven, seven times 10 is 70, those are two different solutions, and we have this order. I don't like to write things like this, there's ambiguity there, it depends on someone knowing what this order is, and so I am personally, and I want you to be also just interested in the P, and so even we all know that multiplication comes before addition, or most of us know, just put there what you want people to do, it does not take much effort just to put those parentheses there, even though that's within the rules of PEMDAS, the order of PEMDAS, put those parentheses, don't try and trick people or don't write something down, and it's not clear to everyone what you mean by that, put parentheses there. Of course, if you wanted the addition first, you would have to put the parentheses, if you want three plus four times 10 to be 70, you'll have to put the parentheses there, but irrespective of the order, P is the most important, and just use parentheses everywhere, and then no one will be guessing at the order. So we're gonna continue our look into just our introduction here to algebra, and one thing I really want to discuss is the properties of real numbers. There we go, the properties of real numbers. Now, when we do some of our calculations, it is very important for us to concentrate on these properties of real numbers, and the first one I want to talk to you about is the commutative property. Now, some of these will be very, very well-known to you, I suppose, commutative property, but some of them we just have to just remind ourselves of. So we get the commutative property of addition, and we get the commutative property of multiplication of the real numbers. So of addition, what we mean by that, if I have any two real numbers, A and B, and I say A plus B, that's exactly the same as saying B plus A. And if I have two real numbers, A and B, and I multiply them, A times B, that's exactly the same as saying B times A. So simple, yet so important for us when we start doing, even the simplest of calculations, and we just have to remind ourselves of the fact that this property exists. The next one is the associative property. The associative property. Once again, we have the associative property of addition, and we have the associative property of multiplication. So let's have a look at these two. As far as addition is concerned, we have, if we have any three real numbers, we can say A plus, and first do B plus C, and that will be the same as doing A plus B first, and then adding C to the end. And when it comes to multiplication, exactly the same thing. I can multiply B and C first. Remember, our order of mathematical operations, parentheses are done first. So if I get this product B and C, and then do A times this product, that's gonna be exactly the same solution as I would get A and B, and then I multiply that by C. So again, very important for us to remember this. Now comes the real, you know, one that I suppose we don't often remember, but it's very important to us in algebra, distributive property, the distributive property, and we have distribution of, what we say, multiplication of addition. So let's again have these three real numbers, A and B, and if I have A times, and this is how we would write it, A times B plus C, now we're very lazy, we didn't put the multiplication there, but the distribution of multiplication over addition means that I have to do this multiplication with A with B and A with C, and then do the addition of that. So that's going to be equal to A times B plus A times C. So it's A times B plus A times C. And that's very, very important for us to remember. And one place where we might often see that, if I have something like this, A minus B plus C. Now you say, well, there's no multiplication sign there, but what we do have to remember is this negative there actually means negative one. So I do have this multiplication of negative one by these two, B and C. In other words, I'm going to get A, there's my A, now let's forget the A, what I just want to look at this, I have multiplication right there, distributed over addition. So negative one times B would be negative B and negative one times C would be negative C. So whenever we have something like A minus and then in parentheses B plus C, it turns out that we then have negative B and negative C and that is the result that we deal with. Next property of the real numbers I want to talk about is the additive identity. Very important, additive identity and this is this value that we use when we talk about the absence of anything and that's the number zero. And remember that is still an element of or member of the set of real numbers such that when I have A plus zero, it's going to be the same as commutative property, zero plus A and that equals A. I don't change this particular element of the real numbers. It stays exactly that element. And then I have the multiplicative identity, multiplicative identity and that of course is the number one is also an element of this set of real numbers such that if I have one times A, that's the same as A times one and that equals just that same real number. I don't change anything. So those are the properties of real numbers that we always have to be mindful of and as we do calculations, these things will be very important to us. The next thing that we're going to talk about is this idea of fractions, very important when we talk about fractions. Now though, we've actually seen fractions before and that's how we defined of course the rational numbers. And this is just a form of division and I promised you we'll talk a lot more about division in this first section, but also in the rest of this course is this thinking about division. So let's take a little example. I'm going to do three divided by four. Now we have one way that we do write it. Remember we can also say three divided by four and there's some other ways as well specifically if you use computer code, of course there's no way to write that to the computer as far as writing with your keyboard is concerned, although there are languages that can express that symbolism. And we also don't have that little stroke on our keyboard, but let's stick to these. We know what we're talking about three divided by four or three quarters. Now, one thing we have to denote that this is what we call the numerator. This is what we call the denominator. So important just for us to remember these names, the numerator and the denominator. Now I can have something like five over four and clearly here the numerator is larger than the denominator. It's five quarters and yeah, I only have three quarters. So it can be bigger, but I sort of want to us to imagine that when we do talk about fractions, we are talking about fractions of a whole, in other words something where the numerator is smaller than the denominator. So when I say the word fraction, that's usually what I'm going to talk about and we're going to look at various ways that we can compare to fractions like this and how we can add and subtract and multiply these. That's going to become very important for us. But for us to get there, let's stick to knowing that this is a numerator and denominator. And even though we are going to see fractions like this, usually when I talk about fractions, I'm going to mean three divided by four where the numerator is smaller than the denominator. The other thing I just want to mention right now and we're going to see more about ratios later. And this is something that we might write as the ratio of two different types of elements. Maybe you go into your kitchen and you have some bananas or bananas and you have some apples. And what is the ratio of one to the other? Maybe that ratio is three to four and we usually use that notation. We put a colon. So note though that in total you have seven elements. You have three of the one and four of the other. So if you want to express this as a fraction, we would say, well, there is three of the one kind and here we have three plus four, there's seven in total and we have four of the other kind also of three plus four. So I'm dealing here with three over seven and I'm dealing with four over seven, three seventh of all my elements that I have are of the one type and four seventh of the other type that gives me a ratio of three to four. So this is usually what we mean by a ratio. So just that you are aware of that. Now let's do some comparison and arithmetic. Let's write that comparison, comparison. Or let's just do some arithmetic with these fractions, with fractions. So let's look at starting in a case where we have the exact same valued numerator. So maybe I have three eighths and on this side maybe I have three fifths. And maybe I want to compare them seeing which one is larger than the other and let's just start with that. That's easy enough to do. The thing about the denominator is what I like to do is always think of one of my favorite things and let's chocolate cake and imagine then we just have a nice circular cake. The denominator is really the number of equally sized slices that I cut my cake into. And now I just want to know, you know, I really want to know how many of these pieces I have. As I say, that's my denominator and I'm just counting how many of those pieces I have and that's the numerator. And if you think about cutting your cake into eight equally sized slices, this is cutting your cake into five equally sized slices. Of course, this is the one I want. My slices are going to be much bigger. And if I have three of smaller size slices and I have three of bigger sized slices, of course, if I have three of these bigger then it's going to be larger than three of the smaller size slices. So always for me, you know, an easy way to think about the denominator is how many slices do I cut my cake into? And now I'm going to leave the additional arithmetic of that just for a couple of seconds. Let's compare the situation where we have the same denominator. So maybe I, in this instance, I have four sevens and I have three sevens. Now this one's much easier in as much as I'm cutting my cake into the same number of equally sized slices. Say, you know, two cakes exactly the same size cutting the slices equally big and all the slices are equal. That's what we mean by the denominator. Clearly if I have four of these and three of these, the four is going to be bigger. Now I just have to look at the numerator now because the denominator is exactly the same. If the numerators are the same, I just have to think if I cut that cake by the number of slices, eight slices of a cake and five slices of the same cake, if I make those equally sized, five equally sized slices or eight equally sized slices, of course those would be smaller slices than these and three of smaller slices is going to be less than three of bigger slices. And then we have the situation where everything is different. Let's have for instance, three over eight and I have something like five over nine, five over nine. Now that becomes very difficult. Not only are my cake slices different in size and I also have a different number of them. So how do we go about this? Well, let's start off with by thinking of this specific middle example where I'm going to add these things. Four sevenths plus three sevenths. And think of that. I have equally sized slices. Now imagine I took my two exactly the same size cakes. I cut them each into seven equal slices and I bring them all, which I've got 14 of now by the way and I just put them all together on a plate so that you don't know that you have two cakes that you just see all these slices. What I mean by that is they are equally sized and I can simply just count how many of them I have. And so I'm going to keep my denominator where it is and I'm just going to add my two numerators and four and three is seven. And there I have seven sevenths, which means I have one. I just have one. So what we can clearly see here, the trick of adding any two fractions is that I must have that my denominator is the same. I cannot add two fractions if I have an unequal value in my denominator. So when I want to do three eighths plus three fifths, I have to do something special. If I want three eighths plus five ninths, I have to do something special. It is only when the denominator is equal to each other that I can do simple addition like this. Four sevenths and three sevenths is seven sevenths. Think about that. If you have a half and you have another half, of course you're going to have two halves, which is this equal to one. That makes sense because I do have the same denominator. I keep that denominator and I can do that addition. So let's think about how to add these two numbers to each other. Well, one neat trick that I can remember is this idea of my multiplicative identity, my multiplicative identity one. If I multiply anything by one, that thing just stays what it is. So let's multiply this one by one, one times this one. Nothing's going to change there. And I do one times this one. Nothing is going to change there. I put my parentheses so it's still clear that I'm doing this multiplication first. One times three eighths is three eighths plus one time five ninths is five ninths. So I still have three eighths plus five ninths. Nothing there. But now I do remember if I write seven over seven or two over two or whatever number over itself, that's just one. So why don't we rewrite this one as eight eighths? Eight divided by eight is still one. That's exactly one multiplied by three eighths. And I'm going to be clever and I'm going to rewrite this one as nine over nine. Nine over nine times five over nine. And now I just have to know how do I multiply two fractions with each other? Well, that's no problem. Let's multiply two fractions. Maybe I have a divided by b and I will multiply c divided by d. What I want to show here is that these are all, these are four different numbers. All that I need to know is that b and d of course is not equal to zero. So I can't divide by zero. And this multiplication is very, very simple. I'm going to have numerator times numerator. That's a times c. And I'm going to have denominator times denominator. b times d. So what I'm going to have is ac divided by bd. Very simple. And you cannot do that with addition. But great, now that we know how to do multiplication, now this becomes very simple. Eight times three. I just have to think what eight times three is. And that's 24. And eight times eight. Am I stupid in my head? Stupid in my bloody head. I'm concentrating so hard on other things. So now I want to talk to you a little bit more about the arithmetic of fractions. Now let's look at arithmetic. And the one that I want to start off with is just the multiplication. So let's put multiplication as far as the arithmetic of fractions. Let's say arithmetic of fractions. Very important for us to be able to deal with. So let's have two fractions. I'm going to have a over b. And I want to multiply that by c over d. So what I want to say here is that a, b, c and d are all different real numbers. But I cannot have that b and d are zero. So b and d cannot equal zero. So I can't divide by zero. Now this is very simple to do. We simply keep the numerators, that's a times c, and we keep the denominators that becomes b times d. So we can write that very simply as a, c divided by b, d. Very simple to do. So let's do an example. If I have three over four and I want to multiply that by four over five, that's very simple. I have three, let's do three times four, divided by four times five, and I'm going to have 12 over 20, which I can simplify. And later on we'll learn, of course, we can just eliminate these two fours, but that's not what we're going to discuss now. What I want you to remember now is just how to do multiplication. And that's so important to me. I think that deserves our very first, for this video at least, our very first little green arrow there. Let's try and get that straighter. There we go. There we go. Okay. That is going to be how we do multiplication. So let's think about how we do addition. Now we're going to get different ways or different examples. And one is where our denominator is the same. So let's have something like A over B, and I want to add to that C over B. I just have to know that again, B does not equal zero because I cannot divide by zero. Now this is very simple to do. The denominator is the same. So this is going to equal, my B stays the same, and I'm just going to have A plus C. So let's think of an example. I have three eighths maybe, and I want to add to that two eighths. Let's B, let's B, it's not zero. It's totally legitimate this. And because the denominator is the same, I just keep the denominator, and I have three plus two, and that is just five eighths. Very simple, very simple to do. I have the fact that my denominator stays exactly the same. But what if I have the situation where the denominator is not the same? So I want to do A B plus C D, and that is not the same as multiplication. I cannot simply add B and D as I multiplied B and D. I cannot have that. If I do want to do addition, I have to have the fact that my denominator is the same. So how would I go about doing something like that? Well, let's have a little example. Maybe I have three eighths on this side, and let's add to that, say for instance, five ninths. How can I go about adding that? I cannot do that addition. This is not multiplication. I cannot simply say eight plus nine. Now, how can I turn these into something that I can add? Well, I do remember my multiplicative identity. Now, so what if I put little parentheses here, and I put little parentheses on the side, and I do multiplication here, and I do multiplication there, and I multiply by one, and I multiply by one. So what have we done here? Well, we've remembered our order of arithmetic operations, and the only one that I want you really to remember is parentheses. So I've put little parentheses there to say I want to do this first, and then I want to do that part, and I'm just going to add these two values to each other. And we do also remember that one is a multiplicative identity, so if I take one times three over eight, it's still three over eight. If I take five times nine and I multiply by one, it's still five over nine. Now, I just want to be clever about this. I want to rewrite one, and one way to rewrite one is to write it as nine over nine. Nine divided by nine is one. So I have not changed anything here, and I want to multiply by three over eight. And let's do this one. I've got five over nine multiplied by, why don't I rewrite? So why don't I rewrite? So why don't I rewrite one as eight over eight? It's still one, no problem. Now I can use this multiplication of fractions. Multiplication, I can just do numerator times numerator, denominator times denominator. Nine times three is 27. Nine times eight is 72, plus five times eight is 40, and nine times eight is 72. Guess what? I have the exact same denominator, and I can now do my addition. I've got 72, and I'm going to do 27 plus 40, and that of course equals 67 over 72. And I've done the addition of two fractions where I have different denominators. I have different denominators. And the trick was looking at these two denominators. This one was eight, that one was nine. So this three eighths I'm going to multiply by nine over nine as a representation of this nine that I have there. And this one I'm going to multiply by eight over eight because I have the representation of eight on this side. So that I have exactly the same thing in the denominator. I have nine times eight and nine times eight. Is that not a very neat trick? I think that is a very neat trick. Now one thing you'll see also, it allows me to compare two fractions to each other. I can very easily compare two fractions if I have the same denominator. Can thinking of slicing a cake into 72 equal, let's be very small. I don't want to have a slice of that cake. That's too small. I love chocolate cake. But imagine I cut it into 72 pieces. Now I can compare 27 to 40 of those pieces and clearly 40 is larger than 27. So I would say five ninths is larger than three eighths. I can do both addition and compare two fractions to each other by getting this very common denominator. So this trick might work very well in certain circumstances but what if we had this problem? Let's do another example. And in this example we're gonna say let's take the following fractions. Five over 12 plus let's make it seven over 15. Now I'm gonna employ the same trick. Let's do five over 12 and I'm going to multiply that by one. And to that I add seven over 15 and I'm going to multiply that by one as well. So I've changed nothing. I'm just writing it in slightly different order than before just to keep it a bit neater. So I've put my parentheses there so that I know I do this operation first then this operation. If I multiply anything by one I don't change that thing. I don't change the seven over 15 and in the end I add them. So we really are back to where we started. But now we're going to rewrite one and we're gonna say five over 12. We're gonna multiply that and we look at the opposite fraction. We look at its denominator 15 over 15. So I'm gonna say 15 over 15 which is just one. So no problem there. I'm gonna take seven over 15 and I'm also gonna multiply that by one but I'm going to write one slightly differently by looking at the opposite denominator. So in this term 12 over 12, 12 divided by 12 is still one. I have made, there's nothing different there. But now this becomes a very difficult multiplication because in my denominators I have to take 12 times 15 or 15 times 12. And that becomes difficult. But let's do this. I'm gonna take five and I'm going to multiply it by 15. You can use a calculator. You'll see that that's 75. And if I have 12 times 15 that gives me 180 plus seven times 12. Well, that's 84. 15 times 12, that's 180. Now I have a common denominator and I can eventually do my addition. For addition and subtraction I have to have the same denominator. Unlike multiplication where I can just do straight on multiplication. But now I have to say 75 plus 84. Well, that's equal to 159. So my fraction is quite large. And what you'll see is we can simplify this fraction. So let's think about a better way of going about this. And to be able to do this we need to discuss something called prime factorization. And it sounds difficult, but it's not. We have to remember what a prime number is. I'm not gonna write it down here, but remember the prime number is any number that's only divisible by one and itself. And because I've put and there one, for instance is not a prime. It's not a prime because it's divisible by one and by itself, which is also one. So it's divisible by one and one. That's the same thing. I need to be divisible by one and another number. And what do we mean by being divisible by? Let's take some examples. I'm taking eight and I'm dividing that by four. Another way to write this is eight divided by four. And of course that's gonna be two. That equals two. There's no remainder. I can divide eight into four. I can take eight bits divided into four. So I'll have two groups of four. But if I take something like nine divided by four, that's still going to be two, but I'm left with a remainder. And my remainder here, my remainder is one because if I take two and I multiply it by four, I have eight and that's not nine yet. I still have one left. And so when we say it divides, we mean that we have a remainder here of zero. We have a remainder of zero. Eight divided by four is two because two times four is eight. And I'm already at eight. I have nothing left. And that's what we mean by that. So let's look at some prime numbers. Prime numbers. And they are an infinite number of prime numbers. The first one is two. Two is only divisible by one and by two. If I take two and I divide it by one, well that's two, there's no remainder. If I take two and I divide it by two, well that's one and there's no remainder. Three is a prime number. Four is not a prime number because it's divisible by one, by two and by four. And so that's not a prime number. Five is a prime. Seven is a prime. Eight is not a prime. Nine is not a prime. 10 is not a prime. 11 is a prime. 12 is not a prime. It's divisible by so many numbers. Let's look at 12. 12 can be divided by one. 12 can be divided by two. 12 can be divided by three. 12 can be divided by four. And 12 can be divided by six. And 12 can be divided by 12. All of these will leave me with no remainder. So definitely 12 is not a prime. 13 is a prime. 14 is not a prime. 15 is not a prime. 16 is not a prime. And so you can carry on. Now let's re-investigate a number such as 12 and a number such as 15. Why am I doing this? Well, look at our example here. In my denominator, I have 12. In my denominator, I have 15. To use this trick to get to a common denominator, I multiplied by one. And I just expressed one as 15 over 15 and 12 over 12. But that left me with very big multiplications to do. So let's take the number 12 and the number 15 and do prime factorization. And you'll see what I mean by that. Let's start with these prime numbers. And we start with the smallest one. Can I divide 12 by two? Yes, I can. Multiply by, now if I take 12 and I divide it by two, what do I have left? I have a six left. Now let's start all over again. What's the smallest prime that I can divide six in? Well, I can divide six by another two. Well, if I take six divided by two, I've got three left, but three is a prime. And look at that. Prime factorization. 12 is equal to two times two times three. That's a prime, that's a prime, that's a prime. And prime factorization means can I break down this number into different numbers such that when I multiply them, I get to 12, but they all have to be primes. And so a simple trick to do is again, start with your number divided by the smallest prime that gives me a six. Again, divided by the smallest prime that gives me a three, but three is also a prime. And there we have it. I can divide by a two, I can divide by a two and I can divide by a three. Let's look at 15. What's the smallest prime that I can divide it into? I can't divide it by two because then I'll have a remainder. Seven times two is 14, but I have one left. Three. Yes, I can definitely divide 15 by three because now I'm left with five. So I already have a three there, multiplied by, well, five is a prime, and so it immediately goes there. And now look what I've done. I have been skillful here and I have written these numbers below each other. Certainly 15 does not have a two in it. Not a two, it does have a three, does have a five, three times five is 15. I'm factorized 15 into the product of two primes. And I will always write it as such that I write the same numbers underneath each other. And let's do the following thing. Let's just bring them all down. There's only a single two, that's great. There's a two, that's great. There's a three, that's great. And there's a five, that's great. So all I'm doing with this, I'm not adding or multiplying anything, I'm just looking down these columns. And I don't want two threes, I only want one of them because I've got two threes there in a row. And do remember, I can't put that two right there because it has nothing to do with 15. Do every row all on its own. But now let's check. Two times two times three times 50, times five. Two times two is four, four times three is 12. 12 times five is 60. That equals 60. And now we have discovered something beautiful. We have discovered what we call the LCM, least common multiple. We are saying that 60 is the least common multiple of 12 and 15. How do I get that? Well, if I take 60 and I divide that by 12, and if I take 60, this is how we would commonly write it if we want to do long division. 60 and I divide that by 15. Well, here I'm going to get five and here I'm going to get four. Okay, 12 times five is 60. 15 times four is 60. Now let's revisit this problem that we had originally. Five over 12. And I'm going to add to that seven over 15. And again, I want to do this trick the way I multiply by one and I multiply by one because now I haven't changed anything. And let's say five over 12 times, but look at 12 here. We know if I take 12 and I divide by 60, I get five. So why don't I multiply or rewrite one as five over five? You'll see where I'm going to go with this very shortly. Seven over 15 times, let's rewrite one as with a 15, 15 goes into 64 times. Why don't I rewrite this as four over four? Now I've used five over five and four over four. Why do I want to do this? Well, now this becomes a lot easier than 12 times 15. Now I'm only doing 12 times five and 12 times five is 60 and five times five is 25. Plus seven times four is 28. 12 times 15 times four is 60 and look how much easier this is now. Now I have 60 in my denominator and 25 and 28 gives me 53. And now I can see five 12s plus seven 15s is actually 53 sixtieths, which is much simpler than 159, 180th. And if you simplify this, you're actually going to get to this number and we're not going to deal with that. This is of course an algebra. So we're only looking at some parts of these adding and subtracting fractions. So instead of this rather large fraction, I'm getting a much simpler one. And why is that so? I did not have to do 15 times 15 and I did not have to do 12 over 12, 15 over 15, 12 over 12. I could make use of the fact that the least common multiple of 12 and 15 is actually 60. If I take 12 and I multiply it by five and I take 15 and I multiply it by four, I get 60. That is the least common multiple. Now take 12, for instance, you can multiply it by two, you can multiply it by three, you can multiply it by four, you can multiply it by five. Same with 15. I can multiply 15 by one, by two, by three, by four, by five, by six, by seven. It doesn't matter what I'm multiplied with, but I'm going to get various numbers. So let's just think about that. Let's just do 12 times one is 12, 12 times two is 24, 12 times three is 36, 12 times four is 48, 12 times five is 60, et cetera. Let's take 15, 15 times one is 15, 15 times two is 30, 15 times three is 45, 15 times four is 60. What I'm looking at these numbers, which one of those is common to both of those, but it's got to be the least common. 12 and 15, that's not common, 24 and 15, 24 and 30, 12 and 45, 36 and 30. Doesn't matter where I go, the smallest one that I'm going to find that is a common multiple, that's common to both, would be 60. And it's going to be the smallest one. There are going to be others, but 60 is the smallest one. And how did I get to 60, this idea of prime factorization? And that makes my addition of fractions a lot easier. Let's do another example just of prime factorization. So imagine I have two numbers that I want to add, one is over 16 and maybe the other one is over 18. So that would have been easy, whatever this was, I could have said whatever over 16 times 18 over 18, so that's just one plus whatever this number was over 18 and I'm going to multiply this by 16 over 16. And so 16 over 16 is one, 18 over 18 is one, I'm using 18 here because the opposite denominator was 18, I'm using 16 here because the opposite denominator was 16. And now if I take 16 and I multiply that by 18 and you can do that yourself, that gets to 288, that's quite big. Let's rather do prime factorization. So let's take 16 and let's just remember our primes, two, three, five, seven, 11, 13, et cetera. So the smallest one that I can divide 16 in would be a two times. Well, if I take 16 and I divide it by two, I'm going to be left with eight. Now let's do eight again. What's the smallest prime that can go into eight? Well, that's two again. So I'll have another two and that gives me four. Well, what's the smallest prime I can divide two into? A four into, well, it's two again. So there's another two and now I'm left with two and two itself is a prime. So there we go. There's prime factorization of 16, it's two times two times two times two, two times two is four, four times two is eight, eight times two is 16. And these are all prime numbers, I can't make them any smaller. Let's look at 18. Now, can I divide 18 by these primes? Yeah, the smallest primes I can divide that by is two and then I'm going to be left with a nine. So I definitely have a two multiplied by nine. Can I divide nine by two? No. Can I divide it by three? Yes, I can and then I'm going to be left with three. So there's a three. Now I want to write similar things below each other and so I can't write the three there, I'll have to write it all the way out here times, well, there was another three left, that's already a prime and there we go. Two times three is six, six times three is 18. The prime factorization of 18 is two times three times three. Three times three is nine, nine times two is 18. That's my prime factorization. Now let's bring all these down. Now there's a two in this column, so there'll be two. There's a two in this column, there's a two in this column, there's a two in this column, there's a three in this column, and there's another three in this column. So I've got to look at two times two times two times two times three times three. That is a lot of multiplication but you'll see in the end that comes to 144, which is a lot smaller than 288. In actual fact it's half of 288. So I have that the least common multiple of 16 and 18 that's going to equal 144. And so now I just have to look if I take 144 and I divide that into 16 and I take 144 and I divide that into 18, what do I get? So this is going to be nine and that's going to be eight because nine times 16 is 144, eight times 18 is 144. And so now if I have something divided by 16 and I want to multiply that by one, if I want to add to that something that's divided by 18, I'm going to multiply that by one. I know here I'm going to multiply by nine over nine and here I'm going to multiply by eight over eight. That's one, that's one, I'm not changing anything but I do know 16 times nine is 144 plus 18 times eight is 144. That's much easier to deal with. I'm using prime factorization to get the least common multiple. And again, the least common multiple of 16 and 18 is going to be 144. And by that we mean least common. I can take 16 and I can take 18. I can multiply it by one, by two, by three, by four, by five such that I get 16 times one is 16, 16 times two is 32, et cetera. 18 times one is 18, 18 times two is 36 and I'll carry on and on and on here until I get to if I multiply 16 by nine, I'm going to get 144. And if I times 18 by eight, let's just put little dots there, by eight I'm going to get 144. So that is common, common multiple of 16 and 18, 144 and that's the smallest one. I'll find others like 288 for instance, but they are going to be larger common multiples. I want the least common multiple. And the way that I'm going to find that is through prime factorization.