ok today we're going to talk about measuring and constructing segments we talked about the segment yesterday it's pretty much just a line with ends ok the metaphor used was metaphor the correct word to use in that situation okay good deal first time I want to talk about is a coordinate okay yesterday we talked about points a coordinate is just the point on a graph usually your graphs look like this a coordinate is just a point somewhere on there okay coordinates have an X at a while X tells us how far we go over to the right or left why tell us how far we go up or down okay now doesn't always have to be on this you can have a coordinate on a line as well I think that's a and that's baby those are both coordinates just pretty much a point that's later okay begging them drops now distance also heard the word distance before it's just the link from somewhere to somewhere else okay looks like a segment all right but not always just drawing a line fair enough so distance we would find how far it is from A to B let's say a is located at 2 and B is at 8 okay we would pretty much just find how far it is from there to there all right now what we do is go to absolute value that means then all that means is when you get done there's a negative take it off there that's what absolute value means just get rid of the name okay so absolute value of two – hey what would two minus eight be be negative six now those are the lines that means awesome at least two negative signs okay so the answer is six that is our distance you'll never have a negative distance okay and all just this is telling you how far it is from somewhere somewhere else okay good deal yeah next thing we talked about ordered some of y'all might have not a turnover before it's called congruent okay congruent this is the math fancy word for equal okay if two lines are congruent that means they're equal they're the exact same length if two angles are congruent they're equal they're the exact same measure all congruent is the fancy way you're saying goody gumdrops so we say that 2 X is congruent to 8 wanting to find X always to do is put an equal sign in there and then we'd solve it that's it so thanks you say two things are congruent you can just sell them equal to each other all right good next is midpoint all right midpoint is exactly what it sounds like first part sounds like mid which is middle second one is point mid point it's the middle point on a line all right so if I want to find the midpoint of a and B let's say a is that make the 3 B is at 12 okay we're on the line we want to find the midpoint of a and B all I got to do is find the length of a B so we do negative 3 minus 12 that would be negative I'm sorry that would equal negative 15 and absolute value things means that we drop the negative so the answer is 15 now that's the whole link we found the distance okay now we want to find the middle point there we want to find with the exact middle so how do I find the middle of 15 I just cut in half until I divided by two equals seven point five so I know that our point is going to be seven point five whatever's these ours between these two okay so I can just go from this way or this way I could either add seven point five two negative three or subtract from 12 either way I'm going to get the same answer okay it's going to be at four point five okay that's it that would be your answer that's your midpoint we usually handle you for midpoint okay like I said before math people weren't very creative okay look to me out I'm a bit like the destructor point because it sounds cool okie dokie now next word you need to know called boss sex all right if something bisects something that means it cuts in half okay because in the middle if that's the midpoint it bisects lines Amy M bisects that line because it cuts in half then all bisects means is that it cuts in half the end I really don't know what to say about it okay let's do an example of by said I help you pull all your hopes and dreams alright let's say we got a line here let's just make up go crazy you say B and C alright let's say a B is 5 X and B C is 3 X plus 4 so we know the length of this and this let's say we also know that B bisects it what does that make these online a see it makes it the mid one okay so it makes be the midpoint cuz it bisects it so cuts in half now if that's in the exact middle that tells us something about this line and this line if that's the exact middle that means that this line is the exact same as this line which means they are congruent so therefore you can just write out the problem of x equals 3x plus 4 all right that's our problem now we solve it using algorithm all right the first step we got to get rid of the 4 I apologize that's not the first step first step we got to get the X's together so we'd subtract 3x this cancels out 5x minus 3x equals 2x then I bring this down that's 4 and to get rid of that tube we divide by 2 so x equals we found our X now if they were asking for the length of let's say they're asking for the length of AC that's what they wanted to know we're not done yet it okay we gotta plug that in we plug in X their length of a B is now 10 because 5 times 2 is 10 now do the same thing for over here you don't even have to work this side if I trust my work here I know that's the exact same cuz that's the midpoint but just for checking sake we're plugging in 2 times 3 is 6 plus four is ten what's the length of AC altogether now C is 20 AC the whole thing so I put it all together and B 20 a lot of work alright here's a little example problem okay we didn't talk about this but it's also just something you to know okay they gave us the length of km which is the whole thing right and then they gets the link of KL and LM which whenever we put these two together what do we get the whole thing right whenever we put this length with this length we get the whole wing so what does that mean about if we put these two together they are equal to that other one so we'll put that on one side and then if we put these two together are we adding subtracting multiplying your body right we're putting things together we're adding so all we do is we take this all we bring to the table is a plus sign and then 3x minus 2 and now we just solve for X and you're thinking that looks like a long not fun problem but it's not that bad eggs combines with 3x so be for X 0.5 minus we already get rid of that that 0.5 minus 2 is negative 1 point 5 then we'll bring all this down we get the X's over here some subtract 3x from both sides and cancels out and the 1x which you can just drop the 1 and then we add 1.5 on both sides so it is 3 now that might not be what they're asking for all you did was find X if you need to find the length of one of those you got to plug it in

welcome to real analysis this is math 131 at Harvey Mudd College real analysis is an exciting subject it is one of the first courses that a math major takes and there's a good reason for it analysis as I hope to show you it involves more than just learning something about mathematics it's it's also something about the process of doing mathematics okay so in this course we're going to be thinking a lot about real numbers as the name implies but we will be also talking a lot about the process of thinking of how to communicate mathematics well write proofs well and a host of other issues related to communication of mathematics so let's begin what is real analysis what is real analysis let me start with a quote from chroniker some of you may recognize the name chroniker was a mathematician responsible for the chroniker delta which some of you might have encountered in physics is a notation chroniker said in 1886 God created the integers all else is the work of man what did he mean by that some thoughts what did he mean by that feel free to give me your speculation yes okay the integers are easy to grasp tell me your name Dylan Thank You Dylan Mary okay okay so Mary's response was that even animals can can do integers but maybe there's something more that we as humans can grasp Katie integers occur in the natural world calculus doesn't just occur in the natural world what do you mean by that because I might might say that it does okay you don't just see an integral lying around interesting other thoughts what what do you think chronic could have meant God created integers all else all else is the work of man yes integers are the basis for a lot of math and things like the rational numbers are constructed by people okay so mathematics has built up from the integers it through some construction process and we're in fact going to begin to talk about what that means in a second other thoughts as to what chroniker could have meant and the other thoughts all else is the work of man do you think he was being somehow oh I don't know somehow derogatory of what man can do you think yes Paul you showed up that things are a little more discreet he knew that at the time okay but nature is more discreet than that continuous mathematics okay so paula's is suggesting that perhaps chroniker was happier with discrete things than continuous things which one might argue don't really show up in nature interesting okay well let me tell you a little bit about the the genesis of this statement so chroniker had a very unusual point of view mathematics at least it was unusual at the time and it is in fact unusual today and that is oh these are the questions that I was going to ask do you agree or disagree chroniker had an usual point of view he was what you might call a finite test okay so this is a point of view that mathematics should only deal with finite objects finite numbers or things that could be constructed from the numbers and a finite number of steps okay so in some sense here when you hear what he's saying somehow the integers are special they're they're god-given in some sense right and everything else well it's just it's just the work of man and so the consequence of this belief is that for instance chroniker was opposed to the use of irrational numbers and doubted the significance of non-constructive existence proofs okay and so some things that we take for granted today which would include the length square root of two is something the existence of the the use of this square root of two would be something that chronica would have a real problem with okay and this particular quote was in fact a response to Linden's proof recent proof it was 1882 that he proved that pie was transcendental which in fact means that it's not the root of a algebraic the root of a polynomial with integer coefficients and his response crackers response was yeah that's beautiful it's a beautiful proof but it's of no importance because as we all know transcendental numbers don't exist and in fact chronic errs point of view met with a lot of resistance because people felt slighted often by his his comment which to them seemed like he was dismissing a whole areas of inquiry which he shouldn't be dismissing so the I think the message that I want to begin with begin this course with is that there's a lot of things that we take for granted that weren't always so obvious I mean you might come to a course like real analysis which some of you know we do a lot of things we derive calculus from its foundations in some sense and you might approach this course with a point of view where you say gosh you know isn't everything we do in this course kind of obvious well it's not and I hope to convince you that there are a lot of things which to us appear obvious because we learn things a certain way so if you think back to the Greeks you know they understood something about rational lengths I mean they were interested in constructible lengths constructible numbers okay they they were interested in what you could get by using a ruler a straightedge and a compass alone okay and so they knew how to construct rational lengths you know if you asked for a line of length for if this they could show you how to do that given a line of length one they also knew there are other lengths on the line that were not that we're constructible but not rational okay and so this of course involves you could you know you can show that the square root of two is not a rational number which we will do in this course but it's it is constructible because after all you can if you've done anything with the straightedge and compass how many people have worked with a straightedge and compass at some point okay yeah you can come two lines that are orthogonal that are perpendicular and and you know measure off a length here and a length here and then this one is the square root of two times that length constructible number the Greeks knew about other lengths like pi but they couldn't find a construction that would give a line of length pi okay and so there's a big question was whether you could construct the length pi using straight edge and compass so it turns out that turns out that you can't construct pi and that's because it's transcendental and transit constructible numbers are always algebraic and therefore not transcendental so we know that it's not possible now but pi can be obtained as Newton showed in Wallis it can be constructed through infinite process and Newton came up with an infinite series that you could sum that would yield pi but what is an infinite series so this begins to beg a question ok so right already here to 200 years before chroniker newton and leibniz in developing the calculus began to encounter the infinite okay and they didn't have a real rigorous way of dealing with the infinite in fact if you look at a lot of the history of calculus it was a toolbox at first okay gave good answers but there weren't there was not a precise notion of what it meant for a series of numbers to converge okay there wasn't a precise notion of limit and if you have a series of numbers and infinite sum that is in some sense a limit of a bunch of finite sums right but what does it mean for a set of numbers to converge are there even enough numbers available to capture all the limits right if you only have natural numbers and rational numbers yes they can be ordered on a line but why is it the case that if you start putting in let's say first all the rationals first all the integers and then all the rationals okay don't make me do them all how do we know that they that they what does it mean for these these things to fill out a lot and they don't of course because we know that there are other lengths in here but once you start talking about sequences of numbers converging is it possible maybe that a sequence of numbers converges in some sense but but doesn't have a limit there isn't a number there whatever that means is that possible what does it even mean for sequence of numbers to converge when you're not referencing a limit there's a question some some really tough questions and so even though the calculus is developed in the 1600s it really wasn't until the 1800s that people began to worry about the foundations of calculus okay so in particular Fourier series are series that you might learn about in a PDEs course or something they are infinite sums of sines and cosines okay and they did rather strange things that made the mathematicians of the day very very uneasy in fact some of them out rightly rejected for EA's work but nobody could deny that for EA's methods actually gave answers right where physicists wanted answers it gave answers and it seemed to give right answers why was that so really fourier series a dealing wrestling with series and sums infinite sums brought about a revolution in the 1800's and making these concepts precise and it was a lot of the work we're going to talk about this course is is work of Koshi that has been certainly simplified it's streamlined for for your digestion but was not necessarily so clear at the time Weierstrass and 18:50 1560s also were big players in this development many other mathematicians I could mention but I just hope to give you a sense that many of the the things we take for granted really weren't so obvious at the time and a lot of it came out of wrestling with the infinite so one of the first things we want to do in this course is actually show you how to construct the real numbers okay what does it mean to construct the real numbers so so that is actually going to happen in lectures two and and three but just to start off with something that you are familiar with and to give you some sense of what it means to construct an object we're going to start in this lecture by constructing the rational numbers okay so that's our plan let's let's begin and I'll use the board from from here on out okay so let's let's begin with some easy concepts and just to make sure we're all on the same page we'll establish some notation so let's first talk about sets and relations and I'm sure many of these concepts are going to be extremely familiar to you but we're going to make sure that we know how to use some of these ideas carefully so what is a set anybody what is a set a collection of things and as easy as that sounds the notion of set is something that mathematicians had to wrestle with very carefully in in the 1900s but we're going to call it a collection of objects okay and there's a reason we don't say a set is a set of objects there's a good reason for that here's one here's how we're going to write a set so a collection of objects might be labeled with a letter and I might notate what's in this set by putting some things in brackets okay so for instance a set could be a set containing you know a number like the number one right but it could also contain other objects right it could contain you know a smiley face that might be an object of this set right could be a parallel again parallel ax gone yes okay it could even be if you want another set containing a couple of objects okay a set might contain other sets okay oh really interesting so just a little quiz here how many objects does this set have in it for good for objects don't be fooled here because as this creature is one set as a set it is an object in this collection which is another set okay everybody with me okay good now one of the things that you should begin to do when you are learning to write mathematics carefully is to make sure that what you write is actually a complete sentence okay so of course when I do work on the board sometimes there will be shorthand but you should avoid that when you're writing mathematics carefully and I will try throughout this course to write sentences okay so a set is a collection of objects is that a sentence well yeah it is a sentence but I haven't completed it okay with a period okay now I want you to think about mathematics as being communicating mathematics well is communicating in sentences if you open up your textbooks any textbook you will notice this that mathematics is actually written in sentences so even equations so here I'd write okay so this is a command right I'm telling you to write s like this but I should complete this thought with what a period okay even displayed equations you'll see written this way in your textbooks okay okay very good here's another way we might describe a set I might describe a set by telling you some property that it satisfies so I might say let s be the set of all little X and I write a colon here which means such that little X such that P of little X is true where P is a some statement about X okay so what do I mean by that an example would be something like let's look at the set of all X such that X is less than 2 that would describe a property about X and if it's true I will put it in this set everybody with me and what should I do how about a little period there okay I'm completing the thoughts okay okay so in fact I'll let me just do this I'll say eg X is less than 2 period all right now there's lots of shorthand which will be helpful to us the only have to say to write down a lot of things so I can write some shorthand when it's convenient so we'll often use this notation X excuse me let's first do in X is in s so this means X is in s period and there's the other notation with a cross which means X is not in s okay and every once in a while I'll put these little quotes which means repeat what's above okay there's a special set which we often want to describe and that's the set with nothing in it otherwise known as the empty set very good and it has a special notation which is basically a circle with a slash through it is the empty set okay very good here's another notation of shorthand we want us to use so if I write I want there sometimes I want to say that one set one collection of things contains another collection of things once that contains another set so if I write a with a sideways symbol like this it means a is a subset of B and I have to now tell you what it means to be a subset what does it mean to be a subset or you could say it another way what this means is okay so it means this so I'll say which means what does it mean well it means if so how would you describe this definition a is a subset of B just in terms of a relation involving set what's in a set or not in a set Willie okay yeah so another way I'll write this is if X is in a then X is in B so it means this whole statement if X is in a then X is in B okay another way we can write this just give me lots of ways of saying the same thing you could notate this by saying one implies the other so you'll see this implication arrow this should be familiar to most everybody with the prereqs for this course but just being careful here one thing you do also have to be to watch out for is that with this notation some authors well generally this this also includes the possibility that that a and B are the same set okay but some authors emphasize that by placing an equals underneath but if you don't see you should assume it it could mean that one is a contained in the other so here's another thing if a is in B and B is not in a that is one is a subset of the other but not vice versa then we call a a proper subset of B okay so it's strictly smaller and if a is n B and B is in a then we will write a equals B so this is is what you'll check if you want to show two sets are equal one strategy is to show that one is contained to the other and the others contain in the first and if a is not if a is not equal to B then we'll write a with a not equal sign B okay okay just being careful about some notation here okay so let's establish a little more notation and then we'll begin to talk about relations so you can construct new sets from old so more sets if you give me a couple of sets I can there are some operations here I can do so for instance I can talk about the union of a and B so the union is a union B right a little cup between the a and the B and what is the union of a and B somebody described for that to me in words somebody described that to me and where it's Rebecca okay so if I were to write it instead of as the first way instead of listing the elements which I obviously can't do if I write in terms of a property what should I do how about the set of all X little X such that what did you say Rebecca good X and a or or is it ant good or X's and B okay okay and here's an Associated notion the intersection which is written with a upside-down cup in between a and B and it's the same definition except what or becomes and okay okay good here's another one a compliment there are many notations for this but we'll use a little C the complement is defiant of a is defined to be the set of all X such that help me X is not an A okay very good there's the – operation which is written with them it's not quite a minus symbol it's kind of a slanted minus symbol a backslash and that's to remind you that these are sets so take a guess as to what this notation might mean good X such that X is in one but not the other in particular you want to say X is in B and X is not an a X is in B oh yes sorry my this is my notes are backwards I had B minus a here so X is an A but X is not in B thank you very good okay and then this one is going to be important especially important what follows and this is the product of two sets involves a new idea and that is you can collect elements of sets together so the product here is going to be a pair little a little B such that little a is in big a and and not little B little B is in big B okay now this is a new idea this is an ordered pair it's not a sex per se it is an ordered pair where order matters because you want to indicate the first thing is in a and the second thing is in B okay product x1 so now we want to come to the notion of a a relation what is a relation when I say the word relation what do you think of what do you think it will loose when I say the word relation what do you think of good how do things interact with one another okay so for instance I might say that Milus is the sister of Paul which may or may not be a true statement okay that's a relation right or John is the father of Mary that is a relation okay so these are relations that involve two objects and so we'll refer to it as a binary relation although if you talk about a relation without saying what it is you often mean binary so we might give the relation a name we'll call it R and here's a new way to describe a relation is probably going to be new for some of you will describe it as a subset of another object let's say a subset of a cross B the subset of a product such that something is true well actually no sorry it's a set it is a subset I had it there'll be some some relations that determine what actually lives in the subset but we normally write this as follow it follows so if a and B are in R we often write it like this we'll write a R B this is a statement that basically says the pair a B R in this subset so let me give you an example relations that you are very familiar with so I might have the relation a is an ancestor of so this is the relation is an ancestor of it's a relation on what set well it's the relation on well I'm comparing sesterce so this is the product of the set of people okay with me L the relation to like somebody is also a relation on P cross P it's a different one right because you might not like your mother okay but if you look at the set of all people and you look at ordered pairs of people I might ask is the pair Bonnie and Jenny in the relation a does Bonnie a Jenny no but does Bonnie L Jenny we hope so there right it that's the right answer in this audience right okay okay here's another example s is a sibling of is a relation also on people cross people actually you know the like relation doesn't have to be just a relation on P cross P it could be a relation on P cross a whole set of objects right t another set okay okay here's a relation that you're very familiar with it's a symbol it looks like a an arrow a caret pointing one way it usually means what less than this is a relation on usually if you like since we haven't talked about all numbers we'll just stick to integers which are denoted by the letter Z and so you're used to seeing this symbol used to say seven is less than ten that's a statement where the relation symbol lies between the two objects in the binary relation are you with me okay that's all it means but the way I want you to think about this is it's actually just a subset I have described what a relation is completely in terms of something very basic membership in a set are you with me okay that's somehow somehow comforting alright good so we have some examples of relations one of the most important examples of a relation is in fact something that you might have encountered in another course and that's the concept of an equivalence relation so the real equivalence relation on a set s is well it's a relation on s process whatever that is such that and I'm going to start abbreviating such that by s T such that three things hold let's give this relation a name let me call it R so what does it mean to be an equivalence relation well we want to a word that somehow describes relations that of things being equivalent to other things okay so for instance equals is a natural equivalence relation but there might be other kinds of equivalence relations right another equivalence relation might be something like ok all the sophomores in at the colleges or somehow there's somehow equivalent right ok so what if we want to make that notion precise well maybe they're not equivalent you're debating that what what what what do you want to be true about an equivalence relation yes okay so you're saying if a are B then B are a okay that's good that is a one of the things on our list it's not the simplest thing on our list but it will it is one so we want it to be the case that if a are B this implies B are a that is if the relation goes one way then it goes the other we have a name for this what do you think we call it it's the symmetry relation okay there's a simpler one that you might demand to start yes remind me your name – Laura good very good so a are a better be true okay and there's a name for this anybody know what it is it's the reflexive condition okay so good let's look at some of these relationships is a reflexive know I'm not an ancestor of myself okay is a el reflexive I hope so I hope so okay good so is is el symmetric if if bonnie likes debt jenny does that mean Jenny likes Bonnie not necessarily right okay very good and there's a third relationship that you might hope to be true as well for an equivalence you want to say a bunch of objects are equal I know this isn't these two things are in the relation these two things for inhalation what's a third thing you might hope to be true Katie okay good if a is related to B and B is related to C then a better be related to C this is called transitivity okay the transitive property okay so these are the three things that you hope to be true and I'll put a period here and commas just to emphasize the sentence is complete okay and you often write something like this the equivalence relations are often denoted by something that looks like an equal sign right might be Tildy or might be double Tildy or you know sometimes tilde with and equals there's a lot of ways to write equivalences okay you generally avoid using the equal sign because it's reserved for actual identity okay okay good so that's an important example of an equivalence of a relation and I'll just mention one other important example though this is the one we'll want to deal with today as an aside here's another important example of a relation so I claim a function is also a relation so a function from let's say A to B is a relation so okay now of course when you think of a function you think of a maybe something like a letter F right and you think of what well you think of f you often write F : a arrow B right what's true about a function what makes a function a function yes Jenny very good so each input gets sent to one and only two to one out to a unique inputs have unique outputs right so one way to think about it is if I apply a function to a particular element here let's say you know I put in a person and I get out their bank account money the money in their bank accounts then you'd hope that when you every time you enter the person what you get out is still the same number okay which is not necessarily true about bank accounts over time right if you want to turn that into a function you probably want to add another input which is the time right okay but the idea is there is a it's a rule that assigns to every element here a unique element here let me just convince you that this is also a relation right so it's a relation such that I'm just going to write it down the way Jenny wrote it other set it and that is if if a FB and a FB prime then what do you hope to be true which should be true if it's going to be a function then B is the equals B prime it has a unique output so this is a rule this is how we capture very formally the idea that it's a rule that assigns to each a in little a in big a a unique B little B in V okay now of course we never write functions this way we usually write functions this way F of a equals B okay but that's just a notation that describes this relation alright very good everybody with me excellent so we're going to take a one-minute break two minute break stretch break and after the break we're going to construct the rational numbers okay and I'm okay welcome back from the break we are in the process of constructing the rational numbers as a route to constructing the reals but also want to show you what is some of the issues involved in construction of objects so for instance here we're going to begin by using the integers so this notation here Q means the rational numbers and there are some things I'm going to assume and one of the things I'm going to assume is that we know everything you want to know about the integers so Z here these are the integers these include the positive and negative integers whole numbers and the negative whole numbers okay we're going to assume not only that we have the integers but that they have we know about their arithmetic we know that we can add them and subtract them and multiply them and we know about their order okay okay so in other words I don't want to go too far back okay I want to assume that you guys know these things okay okay so when we say the word construction this often implies that there's some goal okay what's the goal of this construction what is Q what do we think of when you think of the rational numbers just throw out some thoughts here I mean these are things you've you've learned since grade school rational numbers what do we think of when we think of rational numbers let me hear from somebody haven't heard from before tell me your name David okay that's one thing you think of when you think of rationals and a terminating decimal has a terminating decimal expansion what's another thing you think of name please Steve Keith okay can be written as a over B oh interesting okay interesting so what is Q there's a question and a first answer as Keith suggested is maybe you write it like this perhaps so perhaps okay let me write this as a perhaps it's a certain set what's that can be written as a over B and in fact let me do this in the in using different letters here more traditional M over N such that what's true about m and n okay MN n energy is right because it wouldn't be a fraction if I put PI and PI over here right okay okay very good what else do you do do you normally do with these with this notation not only do man they be integers but I claim you demand something else about one of them yes ooh interesting that's not something I'm going to demand right away but what's another what's a more important consideration second row here good tell me your name and cannot equal zero David said so n and n are in V and n is not zero okay this is one possible answer you might give to this question but it's not very satisfactory for a few reasons but probably the most important reason is we have no idea what this notation means what do you mean by M over N right what does that mean the integers don't have a division defined on them right what does it mean okay so so I'm just going to say this is not quite good enough because we don't know what we mean by what what would this symbol what's this mean okay so let's try to be a little more careful and we can be guided by what we do know so let's think about the motivation so when we think about fractions you know we're usually thinking about trying to teach children something about dividing I don't know cakes into pieces or something like that right so you know you might for instance take a cake which looks remarkably like an interval and dividing it into oh I don't know three pieces and giving somebody one of those three pieces okay we have a name for this fraction we call it one-third right okay one-third which really means one part of three right that's one way to think about it but there are other ways that would describe the same quantity right I could have divided the cake into six pieces and picked two of those pieces right so the one-third we normally write 1 over 3 and this in this thing we might say two parts of six and we could write two over six and we see already another issue which is what we have two different fractions but they represent the same thing right sorry my microphone is it really is falling okay so this brings us to a concept here which is okay well we have two ways of representing the same thing two ways of representing the same thing these two things are in some sense we want them to be E II quit quibble its equivalent okay so maybe we want to set up a construction that's where we define fractions in terms of equivalence relations okay how are we going to do that what will the equivalence relation be so how about this let's take any construction like any picture like this and if I want the Associated fraction I will think of 1 over 3 as an ordered pair so maybe I'll do the following in the first picture I'm picking one part out of 3 and in the second picture I'm picking two parts out of six and I will think of these as equivalent ordered pairs okay and to make this a sentence I might say write that as equivalent ordered pairs period all right okay and then what I will do if I can is once I figure out what all the things that are in the that are equivalent are the idea is that these belong to some equivalence class that might have lots of other things in it right like 10 comma 30 right or 121 comma why did I torture myself 363 yes okay these belong to some equivalence class and we'll give that a name we'll call that equivalence class one-third are you with me okay now once you have that then of course you can just talk about fractions right and then everybody knows how to work with fractions which are really disguised ways of dealing with the equivalence relation that's embedded here what's the equivalence relation that everybody learns in grade school when are two fractions equivalent I'll have you think about that but the set of all such classes will be called Q which is basically the set of all such equivalence classes of pairs of these ordered pairs right what ordered pairs the ordered pairs in Z cross Z okay we might have to be a little careful here maybe Z cross Z minus zero the set containing zero period okay okay everybody with me what have we just done we've just defined what I mean by Q right it's an ordered pair and it's equivalence classes of ordered pairs now what we haven't said yet is what do you call this relation is so what is the equivalence relation by the way but before we even talk about what the equivalence relation is what's what's another thing you might want to be true about Q aside from the fact that you know I've just said that there's a set here but somehow how is it related to the set I began with I might wanted to delineate that right I might want to tell you how is it that Q is related to Z and usually I mean the way you think about Q is that it's somehow extends Z right you have a bunch of points these are whole numbers on the number line that you grew up thinking about and now we've filled them in with a bunch of other points in between right but are the points on the number line still there yes they are embedded in cue right so you might want to say how Z is embedded in Q okay so we want these pairs to extend Z so that okay which classes will correspond to Z to the to the elements of the natural numbers what will i if I give me the number five what class would you hope that it's somehow associated with yeah how about five over one so that for instance n over 1 in Q corresponds to n n Z right so this is the other thing we might hope for in our construction ok ok now if you've taken an algebra course then you what we're looking for is an isomorphism of Z into Q but if you don't know those words yet that's ok ok so tell me what the equivalence relation should be so after grade school or after enough examples we see that what Q is the set of all let's say M over N here's a def here's a these are equivalent classes such that m and r NZ and it's not 0 where M over N is an equivalence class of is the equivalence class of n comma n with the relation what relation okay tell me when key to things to ordered pairs are equivalent when will P comma Q be equivalent to M comma n Steve good this is otherwise known as the whether some name cross you cross multiply to check whether fractions are the same yes that's the equivalence relation so with this relation these are equivalent if if what conditions are true Steve suggested doing what let's take P times n and check whether it's the same as what Q times M and what else let's just demand at N and Q not be 0 okay so if these things are true then we'll say these two these two pairs are equivalent okay okay now that's a relation is it an equivalence relation there's some things to check here right we won't do them all but I do want you to think then that so here's the important thing to do once you have the construction the work is not done you should check that it's an equivalence relation check Tildy is an equivalence relation so for instance is it reflexive some of you said yes right away and some of you hesitated how would I check if it's reflexive what would I have to check to check the reflexive condition for this relation which is on pairs of pairs right you give me a pair and another pair how would I check what would I have to check for reflexivity what do I want to check what's the condition I want to check what's the what corresponds to ära over there yeah is PQ equivalent to PQ now does everybody agree this is what we have to check and it's not the only thing but it's one of the things yeah okay good now is this easy to check well then you go back to this definition this is why definitions are so important in mathematics because we know what we mean we have a definition what does it mean to check peak utility PQ it means checking that DQ equals QP is that true for integers yes okay good so I'm not going to write it out but you can write this out okay I'm going to put dot dot dot there which means you finish the argument okay the other thing of course to check is Q and Q are not zero are they not zero well by the the ordered pair but the the the set we define it's not so don't have to worry okay great what's the second thing you might check symmetry okay is Q is P comma okay what does that mean if PQ Tildy m and does that imply MN Tildy PQ first of all do you agree this is what we have to show good secondly do it can you see how the how you'd write this out which I won't bore you with question means PN equals Q M and this condition means this this equals this this M Q equals NP is that the same thing yes Adam says the same thing so again I'll let you finish that argument now the third one is perhaps the most interesting one the third thing to check because if you look at it it's actually not so obvious oh I realized I could have used that board but that's okay okay this one says if peak utility MN I've got to check that it this and if MN till the give me another pair of letters maybe a B then is is it true that PQ Tildy a be hmm there's a question is that true okay so this is this is actually where it's very important to just be a little careful okay so I'm going to give you a hint as to how this goes and you can you can verify this okay for a little bit of homework for next time so one thing that you'll have to use so try this but you'll have to use a property of the integers which is the cancellation law you don't have division but it's the next best thing to division okay so the cancellation law in Z says the following it says that if a B equals AC yeah and a is not zero then what's true B equals C right and the way of course to see that is is basically because Z doesn't have any zero divisors a B minus a C can be factored so it's a times B minus C equals zero if a is not zero then B minus C is okay but use this fact you will be able to take these two statements and turn it into this statement okay and I encourage you to try that in the privacy of your own home okay okay great so we've constructed we've constructed Q and we haven't but I'll say something about this next time we haven't yet talked about arithmetic on Q but we will have to check a few things there as well okay great we'll see you next time

the key points to an angle bisector is that it does almost lost dirt is that it does a couple things the first thing is it bisects the angle creating two congruent angles so if I have an angle that's in blue here and the red ray is my angle bisector then it has created two congruent angles so notice that this red is a rape so that's another key thing now it also could be a line segment if you're talking about something in a nice ah sleaze triangle perhaps perhaps so we could say or a line segment and every point along this bisector is the same distance from the two rays that make up the sides but how do we measure distance well the shortest distance from a point on this ray to a ray that forms the angle is along a perpendicular so if you're to construct the perpendicular from the angle bisector to a side and if you did that down here then you would say that these two segments are congruent so that's the key parts to an angle bisector is that bisects the angle creating two congruent angles that it is a liner or it is a ray or a line segment and that every point on this ray is the same distance measured along the perpendicular from the Rays that make up your angle but how do we actually construct them well to do that let's grab our compass and our straightedge and head over to this angle right here so we know that we're going to create a ray that creates two congruent angles so the first thing you're gonna do is you're gonna swing an arc just like if you were duplicating an angle so from the vertex I'm going to swing an arc so that I create two points of intersection now I want to create a point out here that is the same distance from these two points of intersection so if you want to you can change your company but you don't have to for the sake of argument I will and you're going to swing an arc from each of these end points so there's one arc from this intersection here's another point of intersection that I'm going to swing an arc from now this point right here is the same distance from both of these endpoints so I'm going to connect this point of intersection with my vertex thereby creating my angle bisector so I'm going to draw this connect my vertex and that point of intersection and what we've created are two congruent angles then when they sum you get the angle that you started with

this video is provided as supplementary material for courses taught at Howard Community College and in this video I'm going to show how to construct an amortization schedule so imagine you've got a situation that's like this you've taken out a loan for $15,000 it's at four-and-a-half percent compounded monthly and you've got monthly payments for three years and we have to make an amortization table for that situation okay so two steps we have to do before we can start filling in the table the first thing is we're going to use the present value formula and we'll figure out what the monthly payment amount should be so I've already done that and given this this situation the monthly payment would be 446 dollars and 20 cents the other thing we have to do is figure out what the interest rate for each payment period is so what we're going to do is take the angular annual interest rate which is four and a half percent we'll turn that into a decimal that would be 0.055 and then we want to divide that by the number of compounding periods every year and since we're making monthly payments we would divide that by twelve so if we do that calculation we get point zero zero three seven five that's the monthly interest rate so I'm going to put that in here now in reality in a normal table we wouldn't fill in these two amounts I wouldn't put the four hundred and forty six thousand twenty cents up here at the top where it says payment and I wouldn't feel any interest rate but I want to be able to refer to that so I'm just going to leave it in there for now okay now we can start filling in the table at zero months there are no payments you've just taken out the loan and so there's no interest on that and there's no balance reduction but we have to fill in the unpaid balance the unpaid balance in this first line is the original amount of the loan so we're going to start out with $15,000 as unpaid balance now at the end of the first month we have to make a payment the payment is four hundred and forty six dollars and twenty cents it's always going to be that amount I want to figure out how much of that payment was interest so what I have to do is multiply the interest rate this point zero three seventy five times the unpaid balance when I do that multiplication I get fifty six dollars and twenty five cents so in other words out of the payment fifty-six twenty five cents was interest the rest of it was paying off the principal so what we're going to do is find the unpaid balance reduction all that is is what happens when you take the payment and subtract the interest so we'll take the payment $446 and 20 cents subtract the interest which for this month was fifty six dollars and twenty five cents and we get three hundred eighty nine dollars and ninety five cents this is the unpaid balance reduction which means it's the amount we're reducing the unpaid balance by so we'll take this amount and subtract it from the unpaid balance which is fifteen thousand dollars and what I'm going to get is fourteen thousand six hundred and ten dollars and five cents and now we're going to begin the process again for the second month we'll fill in a payment the payment is always going to be four hundred forty six dollars and twenty cents and then we want to find out how much of that was interest so once again we'll take that monthly interest rate and multiply it by the new unpaid balance in other words we're multiplying the interest rate by whatever the bottom line is in the unpaid balance column which is now fourteen thousand six hundred ten dollars and five cents when I do that multiplication I get fifty four dollars and seventy nine cents that's the amount of interest I paid I want to find out how much of the principal I paid which is what happens when I take the payment and subtract the interest when I do that what I get is three hundred ninety one dollars and forty one cents that's the amount I'm reducing the unpaid balance by so I'm always going to be taking the bottom line in this unpaid balance reduction column and subtracting it from the bottom line in the unpaid balance column or in other words subtracting three hundred ninety one dollars and forty one cents from fourteen thousand six hundred ten dollars and five cents cents that means the new unpaid balance is fourteen thousand two hundred eighteen dollars and 64 cents okay I'm going to do one more line of this once again now we're at the end of the third month the payment is four hundred forty six dollars and 20 cents I want to find out how much of that was interest so when I get to the interest column what I'm always going to do is multiply the interest rate for that period in other words in this case is the monthly interest rate times the bottom line in the unpaid balance column when I do that multiplication I get fifty three dollars and 32 cents when I get to the unpaid balance reduction column what I'm always going to do is take the payment and subtract the bottom line in the interest column so when I do that I get three hundred ninety two dollars and eighty-eight cents and then when I'm ready for the unpaid balance column what I do is I take the unpaid balance reduction the bottom line in that column and subtract it from the bottom line in the unpaid balance column to give me the new unpaid balance so in this case we're going to get thirteen thousand eight hundred twenty-five dollars and seventy six cents and then I would begin again and before I finished I would get rid of these notes that I made for myself okay now I've got a nicer version of this this is the same numbers we did plus a couple more months I just want to make some comments about this what you should notice is that your interest the numbers in your interest column are getting smaller and smaller as you get further and further down in the schedule you should also notice that the numbers in the unpaid balance reduction column are getting bigger and bigger if you wanted to check things as you went along you could multi could add the number in the interest column and the number next to it in the unpaid balance reduction column and you should always get the payment amount when you look at the unpaid balance column that number should also be getting smaller and smaller now when you get to the very end of this process you would expect that there would be a zero in the unpaid balance column in other words there should be no more balance left to pay probably you're going to end up with either a few cents as a positive number or a few cents as a negative number the reason for that is this in a problem like this where we've got 36 payments and you're rounding numbers off each time you work to a payment it's very likely that you're going to end up end up off by one or two cents either way okay so if you're off by a couple of cents don't worry about it doesn't mean you made a mistake it just means is what we call a rounding error okay that's about it take care I'll see you next time

welcome to this video about the loci equidistance from two lines we're going to construct it but before I do just a quick summary we've done loci from a points we've done loci from online loci equidistance to two points and now the final one equidistance to two intersecting lines and we realize that loci means all the points fitting a particular description okay so make sure you've seen those videos before you have a look at this one and you can check my site explaining maps those boom well you're gonna find out my resources nicely put together for you okay now let's say I have the line a B okay over there and I'll make another line let's say of an angle of 60 degrees AC okay now that this could be the shore perhaps of so this is the land okay and then we have the C in the middle and this boat wants to make sure that it sells exactly equidistance of a C to a B so exactly in the middle yeah so just your gut feeling exactly equal is from those two lines you know this boat has to roughly seal like that do we agree with that equidistance yeah so I'm not going to silver here I'm not gonna seal here because that's near its baby in a to AC now I'm gonna seal exactly in the middle but this is not estimation yeah this is not science guys this has maps yeah we do things properly and accurately so how do we do that we have to construct the angle bisector well done yeah so before I do very quickly loci from a point you do the circle okay with a radius looking for particular amounts loci from a line parallel lines yeah but then at the vertices semi circles depending on the question anyway this is from two points in my previous video the perpendicular bisector or line bisector an equidistant from two lines it is the angle bisector now I told you this is an angle of 60 degrees it's going to be 30 30 degrees but we have to strucked it and in other videos I've shown you how to do that but I'll show you now again it's very easy you have a compass and make sure you buy a compress they're not expensive but when you open it that it stays in the same position sorry excuse me yeah so that it doesn't move okay make sure you buy a proper compass okay what do I do I open my compass doesn't matter how far but not too far because it's difficult to work with not too close possessed it puts work if I just open it a couple of centimeters but once I decide upon a particular wheel doesn't matter how wide once I've decided I gotta stay or stick to that way okay what am I going to do I'm gonna put the needle in my angle and I'm gonna intersects intersects I should say those legs that creates my my angle okay so that's there and there I'm not sure if you can see that so I'm in black I'm just going to just going to trace that with Marco so this art of the server but are okay I do not have to draw the entire circle the other this is going to confuse me I'm only interested in those intersecting parts with the arms of my angle okay very carefully I'm not going to change the width and if I accidentally change the with my compass I just have to start over again doesn't matter but if the width is still say I'm going to put my needle now at this point of intersection and draw this arc of the circle and I'm gonna do the same from this point of intersection and roll that Park over there now why again don't I not do I don't have to draw the entire circle I know that my angular bisector is going in that direction so I'm looking for this point of intersection and again I'll trace what I just did with my compass I'm looking for that point of intersection over here yeah so I don't have to draw the the rest of the circle because from my angle through that point of intersection so my estimation early on was quite accurate actually this is my Oh angular bisector and I'll tell you that a long time ago I was in the Navy and we used to do this all the time yeah creating the loci of points equidistant to two line set because that we knew how to navigate the ship and I'm pretty sure that's still doings okay so it's the angular bisector good like I can't share if this was useful guys then I can help your friends to check my site explaining my hostel Tacoma you can ask me questions there if you want in a forum I'll help you you'll find all the videos basically you need to pass your Maps ourselves yeah all for free by the way I'll see you it

geometry and constructing angles all right it sticks in one batch three men constructing angles all right the world ruler in this section is pretty much just about angles okay then one of the basic things in geometry example exhibit this right here that's an angle okay made up by two rays from meet at a youth vertex that point right there it's called a vertex all right now in an angle some of the basic things you need to know inside that angle call the interior outside everything outside of it all called the there's stuff thanks to you exterior is very good all right like I said before the point where they meet is called the town virgin all right good deal next thing you know about angles it's called measure mostly I've heard the word measure before it's talking about how big or small some is right here how far it's open it's called the in a world thanks for being in small measure comes into play all right professor all right now we measure vegetables in degrees okay a degree most time you think about what temperature oil guess what it's by this like it was 45 degrees that's the degree symbol okay that's how open or why these things are now all the way around would be how many degrees my circle is under 360 360 is all the way around so pretty much every degree is one out of that 360 okay well we have an angle and went all the way around be 360 there were times I equals there's a cute which means that it is less than 90 all right it's less than 90 degrees they begin in there listen cute is less tremendously right right angles you'll see these a lot and do a lot of stuff with them right angle means it is exactly 90 degrees in cynically 90 degrees all right now we usually mark it by putting that little box there because guess what I'm a corner piece of paper that's 90 degree angle boxes perfect squares are oh nine degree angles so we put a little square there for 90 degrees all right and then obtuse is take a log s greater stop it please greater than 90 degrees all right so you got a cute write that up too lovely now a straight angle is the last one you won't see it a lot called straight usually just called the line but that's what it is exactly all the way across so therefore it's 180 degrees okay so you got a cute which is less than I mean the way love you'll remember that is it's a cute blue angle okay you think of a cute little puppy alright Sam not at all but that's the way some people memory okay smaller right we just crack away it's perfect okay it's 90 degrees I'm too sis bigger and if it's a straight line it's always 180 degrees that all these angles one solution do the work good deal yeah last thing we talked about the words congruent and bisect yesterday congruent means it's the exact same which means easily so two angles are congruent you can just set them equal to each other all right so current means equal bisector means it cuts it in half so if you have an angle please stop all right if you have an angle and let's say the angle is 60 degrees all right you have an angle bisector if it box of X and it cuts it in the middle okay therefore 60 will become half so it be learning and urgent very good all right heart regime problems they never succeed in this math game find out on the next episode of Dragon Ball Z new label angels that is how you do it all right say we want to talk about this thing right here ABC okay we just right angle a b c the important thing when you label angles you put the little angle front of it put all three points that are involved the important thing is the middle point it has to be the vertex it has to be where you know it changes directions see we're changing directions at the B that means it has to be in the middle okay that pretty much labels how bangle is examples how do you do them all right pretty much if they just give it to you in words you got to draw it out they said L is in the interior of angle of jkm all right so first control jkm okay he has to be the vertex because it's in the middle and we know L is an interior that means it's somewhere inside this okay so we'll just draw help all right now they wants to find JK m 2j k m so we're trying to find the whole thing it's a measure of JK l is 42 degrees please don't say that every time you say next one is l km is 28 degrees now obviously if we're trying to find the whole thing and they gave us some number one part and another part you can put it together and find the whole thing's all you do is you had it 42 plus 28 now equal 70 degrees you know the whole thing is 70 degrees and you do the same thing say they gave us the whole thing and apart and we want to find the other part we were just subtracted like we do the whole thing with 70 and we knew that part was 42 we wanted to find this right here we would just do 72-42 and we would get 20 degrees good job

constructing a perpendicular bisector the first thing you need to do is draw a segment remember a segment is part of a line that starts at one point and ends at another point go ahead and put those endpoints on there to help you okay you need to set your compass length to be greater than half of your segment length okay now position the compass on one of your endpoints and draw an arc okay go over it if you need to to darken it for our demonstration we need it as dark as we can then move the compass to the other endpoint not changing the radius of the arc position it position the compass on the other endpoint and create another arc now notice the two places where those arcs intersect put a point at those places take your straight edge and connect those two points and you have created a perpendicular bisector

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