 So welcome to lecture number two. Now, this is not going to be a long lecture, but it's very important. We're going to explore the language of algebra. We're going to define some terms and that is going to be the common language that we use throughout. I'm also going to talk to you about signs, whether something is positive or negative. Variables and expressions, and so as mentioned, we're talking about the language of algebra and together with that, I also want to just discuss the sign of numbers. You know positive and negative? We're going to also discuss the sign. So there we go. The language of mathematics. The first definition that we really want to talk about is a constant. Now a constant is an actual value. It's a number, numbers that we are familiar with and we'll have the constant negative 3 and we'll have the constant 4. We'll have the constant square root of 2. We'll have the constant pi. These are actual real numbers. So just any number that you can think of and we just write them down. The second one is a variable. So we want to talk about variables. Now a variable is just this placeholder. So you can really think of it as just this box and anything can go inside of that box. And until I put a value inside of that box, that box is multi-potential. It can really be anything that we can think of. And so I'm just getting some of my stickers here. So just to remind us we have our first definition, a constant. And we're just going to say that it's just any old number, an actual number that we can put down. No matter how complicated that number is, such as pi or the square root of 2, those irrational numbers. Remember? And variables are just these placeholders. Now, typically we will use symbols. So we can use symbols. We won't use this square or rectangular box there. We're going to use symbols. And the symbols that we commonly want to use are symbols such as x and y and z, etc. And so those can be any value. Until I put a value inside or assign it to x, which then makes it a constant, while it is in this form, it is variable. It can take on any value. We also typically use things like a and b and c. Sometimes we use k and we use m and we use n. And why would we use some of these? Sometimes even p and q. And sometimes we just say x, y and z. Sometimes even v and w. And then what we distinguish, how we distinguish these two, is that here we're really talking about integers. Integers. So if we want integers, these are typically the symbols that we're going to use. And here if we talk about, if we talk about the real numbers, if we're talking about the real numbers, so any thing with decimal places really, we use x, y, z, etc. It is not a rule of the universe. It is just something that we commonly do. So a, b, and c, v commonly, if we're talking about integers, something like x's or y's or z's, those are the symbols that we would use for real numbers as far as variables are concerned. So let's have our first little example. So let's put an example here. And I'm going to write down 3 times x times y. Now what we'll typically do is we just write these right next to each other. You can put little dots there or you can use the multiplication symbol, which is sometimes difficult because x looks like a multiplication symbol. But here we are just being lazy as we usually are when it comes to our notation and we just write 3 times x times y. Maybe I have minus 4x. Maybe I have plus 2 times a. Maybe I have this a also being times the sine of something like, let's make it b times pi. Now that is quite, you know, a mouthful there. And the task that you have is just to tell me which ones of these are constants and which ones of these are variables. So let's write out all the constants. Let's write out all the constants. And also let's do all the variables. So the constants, let's have a look at these. Definitely 3 is going to be a constant. We have negative 4 there is a constant. Now I'm going to put negative 4 there, including the fact that there's a negative there would be a positive, which we typically don't write. 2 is going to be another constant. And pi right there is a constant as well. So let's look at these variables. x is going to be a variable, y is going to be a variable. There's another x there that's also a variable. Let's just put all of them there. Definitely a is a variable and b is a variable. So clearly values that are actual numbers that we can point it on the real number line and then variables these placeholders. I can pop in any of the values. I can pop in any constant into any one of these. The next definition in building up our vocabulary is going to be what we call an expression. An expression. Now what we're going to say expression we're going to call that a mathematical combination. So let's write that it's a mathematical combination. So we're going to combine things and that's going to be of constants and variables with or without addition and subtraction. Let's write that out addition or subtraction. So it's the same way we are combining in any form constants and variables with or without addition or subtraction. Now let's put our little green marker there because right here we have a brand new definition. So we just remember that we now have constants and variables. Now we have expressions. So let's have a look at this. I am combining here some constants and variables and this time I have addition and subtraction in between. So I'm combining these things here by multiplication and then I have these different bits of my expression bound together with addition and subtraction and this whole thing as long as I just have multiplication, division, subtraction and addition I can have parentheses as well that will be an expression. And we have many expressions that we commonly use. You might be asked to take a look at temperature and we express ourselves in degrees Fahrenheit but maybe we're working in science and we just want to now convert that to degrees Celsius and so an expression would be something like this five divided by nine. So this is going to be our example by the way. So let's put an example there five divided by nine and we're going to multiply that by the degrees Fahrenheit that we have minus 32. So there's my parentheses. I'm going to do this first. Take Fahrenheit subtract from that 32 and multiply by five and divide it by nine. So use your calculator. Let's have a look at what is 100 degrees Fahrenheit. What is that in Celsius? Well do that 100 minus 32 times five divided by nine. If you use a calculator you're going to get about and these two little squiggly lines as opposed to you know two straight lines which is equal. This is approximately equal to 37.8 degrees Celsius. So you can check on that. Now there's more decimal places here but because remember last time we spoke about this being a numerical approximation so we're going to say it's an approximation of this value. So 100 degrees Fahrenheit is approximately 37.8 degrees Celsius. You can work out a couple more examples of those but we're going to move on now to the next definition in this expansion of our vocabulary as far as algebra is concerned and the thing that we're talking about now is a term. Now a term really is that is a part or a whole of an expression and the whole thing about these is they are separated if I take an expression in these parts what they really are they are separated by addition or subtraction. So if we go back to our example of an expression here there's a negative and there's a so subtraction there an addition there so this is going to become a term this becomes a term and this becomes a term. Anything that has multiplication division stays one term but we can take an expression and we can divide it up in terms by looking at subtraction and addition and that is going to give us the different terms in an expression. So let's take our let's put this one right there just to remind ourselves that we have a new definition so that is going to be a term. Next one is an equation and very simply we're going to say that an equation is just two expressions that are equal to each other so two expressions that are equal to each other and so that deserves another little marker there we go we have another one there we're going to say that an equation is just those two expressions that we said equal to each other now remember an expression a term can be a whole expression so it doesn't really have to be that it's two full expressions but just two terms now typically this consists of a left hand side we usually write LHS left hand side a right hand side and the equal sign let's say that something is equal to something else and so sometimes we would have something like this here's a little example example maybe I have that 3x equals 9 now that 9 on itself that's a whole expression it only is a single term I have my equal symbol that's my right hand side my equal symbol is my left hand side and this left hand side is also a single term as long as I have a left hand side and the right hand side that are equal in value I have an equation next thing that we want to talk about is an operation and we're going to apply an operation to a constant or to a variable and that uses an operator so I'm just going to let's just say we operate on we operate on a constant we can operate on a constant we can operate on a variable and we can even operate on many more things we can operate on a whole expression we can operate on terms so let's look at a couple of operators maybe I operate on a constant maybe I use the negative operator and I multiply that by a so a is any variable but I'm taking the negative of whatever value I put in there that is an operation on this very on this variable a I can take something like the operation between two variables a plus b here plus is my operator it is operating on two separate variables so we can also see that as an operation if I take the square root of a number I'm taking the square root of x I'm operating on that value maybe I want to take the absolute value of a number I'm operating on that number I can take the sign of maybe I'm taking the sign of a single number I'm still operating on that number so it's a very general thing to think about this operation operating on a number on an on a constant a variable an expression a term I can operate on many things and so there's a little extra little green there and so now we have constants we have variables we have expressions we have terms we have equations and we have operations and we all talk the same language this is common language for algebra and the second part of this lecture I want to talk to you about the signs signs so let's talk about sign and by sign I mean positive and negative something larger than zero or something smaller than zero and so there's a few common things that we just have to kind of remember how these things work and so let's do a couple of these I'm going to say something like a and I'm going to use a and b in other words I'm going to think about variables you can pop in any number inside of there and I'm going to say something like a minus the positive version of b and I can just write that out by remembering that if I take a negative times a positive I'm going to get a negative and this will be the same as writing a minus b so that one was quite easy what if I have a minus negative b well a negative times a negative is a positive and I'll write these out a little bit later but that means this would be the same as a plus b what about having negative a minus positive b now that's going to be minus a minus b minus times a positive is just a negative what about negative a minus negative b well that's going to be negative a plus b so those are the things that I really want you to remember because we have a couple of rules so let's look at those rules we're going to say and I'm going to write that out as such I have a positive number and I multiply that by another positive number and that is going to give me a positive number what if I have a positive number and I multiply that by a negative number well that's going to give me a negative number if I have a negative number and I multiply that by a positive number that's also going to give me a negative if I have a negative number and I multiply that by another negative I'm going to get a positive number I'm going to get a positive. And the same thing happens with division. So let's take a positive number and we're going to divide it by another positive number. Well, that is going to give us a positive number. If I take a positive number and divide by a negative number, that's going to give me a negative number. Now, remember, I'm not dividing by zero. That value is larger than zero. This value is less than zero. If I take a negative number and I divide by a positive number, that's going to give me a negative number. And if I take a negative number and I divide it by a negative number, that's going to give me a positive number. So let's look at some examples. If I say three times four, that equals positive 12 because it's a positive times a positive. What if I say three times negative four, that's a positive times a negative, that's going to give me a negative, that's negative 12. What if I have negative three and I multiply that by four? That's again a negative times a positive. And we don't put the positive there. That's going to give me a negative 12. And if I have negative three and I multiply by negative four, that's going to give me a positive 12. And you can do the same exact thing here. You are going to get those values as far as your positives and negatives are concerned. And so let's just do a couple of examples. I'm going to write an expression. And my expression is going to be, so let's say that we're doing an example. So here's my example. I've got negative three. I'm going to multiply that by two. I'm going to multiply that by another two. And let's divide by, let's divide by, I'm going to have a negative three. What else can I add? Let's do a negative four. Let's do a two. And let's do a negative four. And the only thing I'm asking you is what is the sign of this solution going to be? What is the sign of all of this? And so let's have a look at this. I'm going to have a negative times a positive, that's negative, times another positive. I'm going to have a negative in my numerator. And I'm going to divide that by a negative. And the times a negative is a positive. Times a positive is a positive. Times a negative is a negative. And there we go. I have a negative divided by a negative. And I know before doing any of that, that I should get a positive value. Let's do one more example. Let's do one more example. And this example is going to be slightly more sophisticated. And I'm going to say let x and y and z all be larger than zero. So they're all positive variables. I can only put in values there that are positive. I can't put zero in there. And I cannot put a negative number in there. And so let's do another example. I'm going to have negative x. And let's do a y. And let's do a negative z. And I'm going to divide all of this by let's make it a positive x. Let's make it a negative y. Let's make it a z. Let's make it another negative x. And why not? Let's make it a negative y. So what am I going to get? A negative times a positive times a negative. I'm going to get a positive in the numerator. Let's have a look at what we get in the denominator. Positive times a negative is a negative. Times a positive is still a negative. Times a negative is a positive. And a positive times a negative is a negative. And so my result is going to be a negative value. And so do a few of these. See if you can very quickly determine if the result of multiplication and division is going to give you positive or negative results.