 Welcome back to our lecture series Math 3130, Modern Geometries for Students of the Entire University. As usual, I'll be your professor today, Dr. Andrew Misildine. In lecture 20, we're going to introduce the idea of the triangle inequality, which we will present in the next video of this lecture, not this one. But in order to talk about the triangle inequality, a very important result in congruent geometry, we have to understand what it means to add together segments. Now, I should mention that when we first discussed the six congruent axioms due to David Hilbert for Euclidean Geometry, and those six congruent axioms is what defines for us a congruent geometry, you know, when equipped with the incidence and between the axioms as well. In that setting, we had one of the axioms. It was the third one on the list, I believe. It was actually called segment addition, which essentially told us that when there's a correspondence between congruences of segments under the right assumptions, then this also means that the union of the corresponding segments would be congruent as well. And so it is appropriate to call it segment addition, but from an algebraic point of view, there was no operation defined when we talked about segment addition and likewise with the corresponding segments attraction. In this video, we are going to actually literally define an addition operation on segments. So it actually makes sense to add together segments. So we're really improving upon our notion of the segment addition axiom and realizing it as a binary operation, which then justified its original name of segment addition, as opposed to potentially something else like, oh, the segment between this axiom or something. It was a really foreshadowing to the topic we're gonna talk about right now. So in order to add together segments, what that means is we're gonna take two segments. So let's take the line segment AB and the line segment CD. And these are two segments that exist inside of a congruence geometry. What we want to explain is what it means to add together two segments and thus creating a third segment, thus forming what we would call an algebra, a binary operation. And then we'll discuss some of the algebraic properties of this binary operation. So imagine we have these segments illustrated here on the screen. So we're gonna have segment AB right there, segment CD perhaps right here. And so let me label these things. So it's easier for us to follow along. This is, we're gonna call segment AB. This will be the segment CD. And we're not assuming any congruent statements about these. These segments are not necessarily congruent to each other or anything like that. Now by segment translation and by the extension axiom, there's gonna exist a unique point E on the ray AB, such that AB, excuse me, such that B is between AE on that ray and the segment BE is congruent to CD. So essentially the extension comes into play here is because our line segment AB can be extended in the direction of B. So we get something like this. Okay, and then there's gonna be some point on this ray, which again, we've extended this ray, we've extended the ray AB there. I should say we've extended the segment to the ray extension guarantees that such thing is plausible. There's gonna be some point E on this ray AB, such that B is between AE as it's illustrated. And we're gonna have that the segment BE is then congruent to CD. So we are then gonna require that these segments are then congruent to each other. Translation allows this to happen. And so the segment translation axiom allows this to happen. Extension, all of this comes into play right here. And so then we define the sum of the segment AB plus the segment CD to be this new segment AE. And we'll denote this as AB plus CD is equal to AE right here. And so we call it a sum in this congruent sense, right? Because you have the segment AB, we have the segment BE, but the segment BE is just, it's congruent to CD. So in the usual sense of segment addition, we've put these two segments together and formed a bigger one. And that's what we mean, of course, by addition of segments here. Now, this is an equality here, as in to say that we're saying that these two sets are equal to each other as sets. These two sets contain the exact same amount of points, which of course, the set on the left is defined to be this set, so no big deal there. But what I have to explain to you, and this is what I wanna do for the next few minutes here in this video, is explain that when we talk about the notions of segment addition, this is really an operation on the equivalence class of congruence and not on the equivalence class of equality, which these are defined, right? This set is defined to be this set. So these are equal sets by definition, right? But what are some of the algebraic properties of this operation here? That's what I'm really trying to get to right now, and which will be very clear why we have to talk about congruence here. So we should mention some of these properties of this notion of segment addition. So first of all, what if we have some other segment? Let's say we have some segment PQ that is congruent to CD inside of our plan right here. Then in that situation, we're gonna have that AB plus CD is going to equal the segment AB plus PQ. Cause after all, the way we've defined segment addition is you take a copy of this segment translated onto this ray, and that then gives us this point E. E would be the same point when we translated CD onto the ray or we've translated PQ onto the ray. So we're gonna get equality in that situation, but because they're equal, that also means they're congruent, but this is a real McCoy equality in that situation. But I also wanna go in the other direction here. What if we take something like AB is congruent to PQ in that situation? Not necessarily, not necessarily, well, let's use different points. So there's no confusion here. We'll call this one RS. In that situation, we're gonna have that AB plus CD. That's one segment we could add together, but we also could take RS plus CD. These segments are not gonna be equal to each other because the first one will be a subset of the line AB and the second one will be a subset of the line RS. And if AB and RS are different lines, then there's no way these two sets can be equal to each other. So be aware that, sure, if you swap two congruent segments in the second operands there, you're gonna get equal sets. But if you swap the segments in the first operand, even if they're congruent to each other, you're not gonna get equality here, but you are going to get congruence. And so this is what I was alluding to earlier in this video that when we define this binary operation on sets, excuse me, on segments, we don't wanna think of this as a set operation. We wanna think of this as a congruence operation that this sum uniquely determines the segment up to congruence, that you could have different congruent copies, but when it comes to segment addition, we don't care which congruent copy, it's just the congruence class that we're identifying here. There's an analogous concept here if you go into the realm of algebraic topology when we talk about the notion of homotopy. One very important construction in algebraic topology, perhaps sort of like the birth, you could say of algebraic topology, at least as most students learn it, is the notion of a fundamental group, which we attach a group structure to a topological space. But we're one to find the notion of composition of the objects in the fundamental group. It's not well-defined if we just compose the functions together in the usual sense, but the classes, if you look at the homotopy classes of the loops inside of the fundamental class, then it'll be well-defined when you place an equivalence there. So that's the analogy we wanna put in this situation that yeah, when we consider segment addition, it's not well-defined if you look at equality of sets, but if you look at congruence of segments, it is gonna be well-defined, and that's what we care about because another important example here is if you take the set, if you take the set AB, notice it's equal to the set BA. As a set, these are the two exact same symbols here, in particular, they're congruent to each other, but AB as a segment is equal to BA. It's literally the same points on the line there, but if we define the segment sum, if we take the segment AB plus CD, and then you take BA plus CD in this situation, the two sets, the two operands here are literally the same set, but still these two are not equal to each other as sets because after all, the direction is implicit here, that is when you think of the segment AB, you extend it to the ray AB, but here, when you think of the segment BA, you're gonna extend it to the ray BA, and that's gonna tell you where you put this point E, do you put it so that B is between A and E, or is it gonna be over here somewhere so that A is between B and E? That's a different set, and so these two segments we add together are not the same segment with regard to equality, but with regard to congruence, it's going to be the exact same set, so that matters there. So as you switch the first operand or the second operand with something congruent, then the segment sum will still be congruent to each and every one of those. So that's an important concept there. I should also mention that if you take the segment sum AB plus the segment sum CD, and then you consider CD plus AB, which if you switch the order here again as a set, this is not gonna be the same set, but, and then the reason for that of course is in the first one, you take CD and copy it onto the ray AB, but in the second one, you're gonna take AB and copy it and put it on the ray CD. So these are, again, if these points are non-colinear, then these aren't the same rays, these aren't the same lines, you're not gonna get the same set in the end, but as I keep on emphasizing, while they're not equal as sets, they are gonna be congruent as segments. And so this operation of addition is well-defined with respect to segment congruence, and with respect to segment congruence, it's gonna be commutative. The order of the segments doesn't matter for the congruence class. It does matter on the set itself, but when we talk about things like the triangle inequality and things like that, it's the congruence class that matters, not the actual set. And basically, why do we care about the congruence class? Well, we haven't introduced any notion of measure inside of our geometry yet. We don't have things like distance or metrics or lengths of line segments, but the idea here is, we're trying to think of it in a primitive manner that if a segment had a length and this one had a length, well, the length is gonna be preserved. And so in Euclidean geometry, the congruence class of a segment is determined by its length. And therefore, this operation is just adding together positive real numbers in that setting. But we're trying to do this in a more general setting. Okay, so we have a binary operation, which is well-defined on congruence classes, and it's gonna be commutative. What other algebraic properties can we get from this? Well, this operation is in fact going to be associative. So what does that mean here? So if I take the segment AB and I add to it the segment CD, this is a binary operation. So it's only defined for two segments added together. Then if you add to it the segment EF. So I want you to be aware of what happens here. In this situation, you're gonna take a copy of CD, you're gonna glue it onto the ray AB. And this is gonna give you some longer segment that lives on the ray AB. Then you're going to take, and you're gonna have some point, you're gonna have some point D at the end of it, D prime maybe. Then you're gonna take EF and you're gonna glue that onto this ray as well. Okay, if we went the other way around, if we took AB and you add to it CD plus EF like so. So what if you switch up the operations? What if you add together CD first? So what's gonna happen, CD and EF? This one would mean you're gonna glue a copy of EF onto the ray CD. And then what that turns out to be is you're gonna glue that onto the ray AB. So I should mention that in this situation, it actually looks like we would have equality in that situation that because of the order of operations here, we're always putting preference on the first factor here. So in the end, in both considerations, you end with a segment on the ray AB. But like because of the issues we saw beforehand, I don't actually care if they're equal or not. I mean, they are, but I don't actually care. What I really care about is that they're congruent because in some of the situations above, it's not well-defined if we don't have congruence. If it's just equality, it's not well-defined. But under this, we are gonna have an associative operation here. How we do parentheses doesn't matter. And therefore it makes sense to take sums with three elements, four elements, five elements, 12 elements. It doesn't really matter because we have an associative operation. And so because of the associative property, it actually makes sense to take multiples of things inside of this binary operation. So it makes sense to take AB plus AB plus AB plus AB. You could just add it together over and over and over again because the parentheses don't matter whatsoever. And so we define this to be, let's suppose, we have N different ABs in that sum right there. We then define this to be N times AB. And so this would be an example of a multiple because if we have addition, then in some respect, we also have multiplication. Multiplication by of course, a positive integer at the moment because we could understand that just to be, we add together the segment in times. And therefore with this definition of, with this definition of multiples, then we have a distributive property in play right here. That if you have N times AB and you have CD right here, this is gonna be congruent to the segment N times AB plus N times CD. So we have the distributive property, oops, distributive property in this situation because we have a notion of that multiplication. Now, be cautious here. We're not multiplying together two different segments. We're just adding together over and over and over again. But what can N be in this situation? Well, for this definition right here makes sense that N is gonna be some type of positive integer. Could it make sense for zero? What would zero mean in this situation? You take A together zero times. We are gonna define that to be the segment AA like so, which of course, if you take the segment AA, this is actually a degenerate segment. It's really just the point. And so we'll think of zero times a segment as just a point. Now up to congruent, all points in the geometry are the same thing. So while the set definition will be, oh, you're just gonna grab the left endpoint of the segment, be aware that if you did something like zero times BA, this will give you B, points are congruent to each other. We never really defined the notion of congruence of points before this moment. But after all, if you have these degenerate segments, AA, BB, these are just points, we would then expect these to be congruent to each other because you can always just glue on a point to the end of something without changing it. And so it's very natural to consider points as congruent to each other. All points are congruent to each other. And so the zero multiple of a line segment, we then define to be just the point A itself. And this is an interesting observation here because if I take a segment AB and I add to it any point, which of course the point is just the segment PP, like so, what this means is we're gonna take the segment PP and glue it onto the end of the segment AB. But if you just glue on a point, it doesn't change anything. And therefore you're gonna get back something that was just the original segment AB. I mean, and I should be emphasized, this is actually equality in the situation. If you take AB plus a point, you're just gonna get back AB, those are equal. But like we said before, we don't care about equality, congruence is what we're saying here. So if we're equal, then we're congruent. But there are some places where you have to have congruence and equality is not possible. So this operation of addition does in fact have an identity element. If you add together, if you add on a point, which is just zero times a segment, that gives you an identity for this binary operation. So this operation is commutative, it's associative, it has an identity. So it's very tempting to be like, oh, maybe we have some type of like abelian group structure happening here. Well, do we have inverses? That's the thing to kind of consider right now. Do we have inverses? Well, we sort of do. What does that exactly mean here? Well, we now can talk about integers which have non-negative multiples. I should really should say that n could be a non-negative number. So it could be a natural number, zero or a positive. Does it make sense for n to be a negative number? Well, in essence, we can do that. And so what we're gonna do is the following. So we're gonna define negative n here. And of course it's positive. So negative n is now negative. We're gonna take negative n times a segment AB. We're going to define this to be the unique segment on the ray. So this is gonna be a subset of the ray negative AB. So remember what this notation here means. If we have some ray AB so that we have say A right here and we have some B over here and this is the ray, the negative ray that is negative AB is in the ray that is the other half of the line determined by A and B. So it's going this direction like so for which this would be negative AB as a ray. And this of course over here is the ray AB like so. Okay. And so then we're gonna define we're gonna define the segment negative n times AB as the segment that lives on the ray negative AB that's congruent to n AB, okay? So basically, so instead of n AB would be over here on the ray AB, negative n AB is just gonna be a congruent copy of it on this ray. So really what negatives here are representing is direction. Do we go in the direction of AB on that ray or do we go on its opposite ray negative AB? So in that essence congruence doesn't make much of a difference when it comes to negatives versus positives because the direction will be the same thing here and in this situation we do have that negative n times AB is in fact gonna be congruent to the segment n times AB like so the negative sign is really just giving us a direction. So if we do need to distinguish between sets then we can use negatives but if we only care about congruence classes for the most part that's what we care about the negatives don't make much of a difference but we are gonna allow it in case ever there is a possibility here. There's one other thing and then so this idea of a difference can then be described here. We do have a notion of a difference for which we can take something like AB minus CD in that essence what we're having here is we're gonna have AB okay and then we're gonna add to it negative one times CD like so and so this is all just about direction so when you have CD over here instead of putting it over here we're just gonna go the other way around we're basically gonna put it over here and so that's what it means to subtract these things and so from a set theoretic point of view then this would give us the segment subtraction that we've talked about previously in this lecture series what's the difference between them and it could be that since CD is bigger it's over here somewhere and it's in the negative realm do do do do do do do clearly when these things have measure or coordinates and such this makes a lot more sense but we're trying to do it in an abstract sense where we don't need measure we don't need coordinates to be able to do arithmetic with segments here. All right there's one last type of arithmetic that we wanna do and I'm gonna draw one more segment for that one here so imagine we have some segment AB like so well by the midpoint theorem that we discussed previously we know that there is in fact a midpoints to every segment so the segment AB has a midpoint AM let's call it the midpoint there so it's natural then to define one half a multiple of AB here we'll define this to be the set AM that is you take the left half because the midpoint by definition is gonna make the two segments congruent there so we define half of AB to be this one right here and so we can recurse this possibility here we could then take one fourth times AB because this would just be one half times one half of AB so when you cut AB in half to get AM then you cut AM in half to get whatever that midpoint is and so we could then take more and more and more of these powers so we could take one eighth, one sixteenth, one thirty second we can keep on going and going and going but you can also add that together like I could take one fourth of AB and then add it together three times to get this segment right here and so in general you can take anything of the form A over two to the K times AB here where in this situation we're gonna have that A is an integer and K is some natural number it's a power of two and these numbers right here these numbers A over two to K this is what we refer to as a dyadic rational number so rational numbers of course are ratios of integers and our numerator can be any integer you want but the denominator can only be a power of two we can't necessarily take one third of a line segment at least not in congruence geometry this will lead later on to the notion of continuity that we'll get to in a different lecture here but for this concept right here we because of the properties of congruence geometry that we've talked about so far we can add and subtract segments it's gonna be an associative it's gonna be a commutative operation there's an identity there we can use subtraction to cancel things out so from an algebraic point of view it's looking pretty good we also have a notion of multiplication we can multiply any segment by a dyadic rational number where the numerator can be any integer and the denominator can be any power of two and so we have now this arithmetic on segment congruence for which without any notion of continuity we can't do better than the dyadic rationals but at least for any congruence geometry we do have this dyadic rational number we need continuity to improve upon those scalars in the future and so I do wanna close this video by also very tersely introducing the notion of angle addition, okay where this is a binary operation on angles here so let me define it here so given two angles ABC and DEF and so if we were to sketch the picture here we would have something like here's the angle we'll call this one ABC like so and then we have some other angle DEF that might be over here which of course they don't have to be the same size or anything the orientation doesn't really matter too much we have DEF so what does it mean to add together two angles? So in that situation it's gonna be similar to what we did with segment addition so when we have two angles ABC and DEF let P be the unique point in the half plane of BC that's opposite of A such that CBD is congruent to the angle DEF so we're going to add some new point in the situation here so we have some new point over here which we call this point now P and so by angle translation and all that jazz we then have the angle CBD that was just constructed it'll be congruent to the original angle DEF like so and of course the angle P is gonna live in the same half plane that C does with respect to the line AB but of course P and A are gonna be on opposite sides of the line BC that was what's mentioned beforehand so then the sum of the angles ABC and DEF will then be defined to be the angle ABD ABP excuse me so we get that this angle here is then the sum of the two angles and we did have this binary operation on angles so analogously we can extend all the notions of segment sums to this angle sums like commutivity which I should mention that this angle sum just like the segment sum it's not well-defined in general so we really just care about congruence classes here when you add two angles together it'll be well-defined up to congruency and so with that notion if you define this as a operation on congruence classes then it'll be commutative it'll be associative you can have dyadic multiples that'll satisfy the distributive property now unlike segment sums it is possible that there are sums of two angles they could be undefined and this would be the case where because when you try to add them together you get something larger than a half angle the construction here does require some statements about half planes and such and so if you took two big angles like if we took 120 degree angle plus an 130 degree angle we add those together this definition doesn't allow for such a thing because we can't get an angle bigger than a half angle by our current construction now angle measure will allow us to get around such a thing so if we do have 120 degrees and 130 degrees maybe we do have angle measures but the idea is if we have two obtuse angles we can't add those together the way we've defined angle sums right now because obtuse angle, what's an obtuse angle? An obtuse angle is just an angle that's larger than a right angle and an acute angle is an angle that's less than a right angle those make sense in congruence geometry but we can't add together two obtuse angles because that gives us something bigger than a flat angle which is a limitation for now measure will be able to take care of that in the future but if you don't have any angle measures like in a generic congruence geometry we can still add them together and we get this nice algebraic I should say this nice arithmetic on angles just like we did with sums of segments