Defining Every Number Ever

show me three things four things minus five things pie things I things have you ever noticed how weird negative numbers are well we tend to think of positive numbers as representing tangible quantities of things we can't get our hands on a negative amount of objects and yet they have an infinitude of uses the concept of debt is made easier also Concepts like temperature and altitude whether they're clearly defined zero point and even Concepts like electric charge where quantities can be equal in magnitude but opposite in sine the negativeness is just defining the direction of the number but is that how mathematicians Define negatives and what about the other numbers we would never question usefulness of fractions or numbers like Pi which we can never actually fully write down or numbers like I which seems so abstract as to Bear the unfortunate moniker imaginary do mathematicians have a concrete tangible way of defining numbers like these the answer is yes and that's what I'll share with you today but strap in it's about to get complex literally this will be an Epic Journey of many mathematical steps through the weird and wonderful world of abstractions and axioms of Foundations and functions of sets and structures of real and rational and irrational and integer through infinity and the imaginary but stick with me I'll guide you through this mathematical maze one step at a time and together we'll gain a greater understanding of every number there is we have a real mountain to climb here but have patience I'm so excited to share these concepts with you and I'll do my best to communicate them regardless of your background just remember the most rewarding Journeys are the most challenging to undertake and the view from the top of this mountain is ah unparalleled but every great journey starts with planning and preparation so first the plan it's easy to think of this process as filling in the gaps so far we've defined the natural numbers 0 1 2 3 Etc sometimes the set of natural numbers doesn't include zero but mine does um then we have the negative numbers to include so we'll put these down here and then we need to fill in the gaps between these numbers with the rational numbers now the rational numbers are like this kind of foamy line that appears between them and I say foamy because it stretches between the numbers filling in the gaps but it's patchy yeah so there's still some holes in this number system which are the irrational numbers and then we fill those in next and then there's the complex numbers which we'll get to this is the overall goal but the process will look a little bit different instead we will layer new definitions on top of the old one so for example we will use the concept of a natural number to define the integers but our definition of integer will include the naturals thereby replacing them and then we will use the integers to define the rational numbers but our concept of rational number will include the integers each time we're coming up with a definition that includes more numbers and also replaces the previous one so that's the plan next let's do some preparation I advise you watch my previous videos on numbers counting in arithmetic but don't worry if you haven't I'll summarize them as we go we're going to need plenty of mathematical tools here and the preparation will take place in three parts the first part involving some axioms foreign so not only will we fill in the gaps between our numbers we'll also be filling gaps in our axiomatic Foundation now in my previous video I had the axioms of pairing subsets Union extensionality and I also had regularity here we'll add two more axioms to complete the foundation the first is the Axiom of infinity this just says that the set of all natural numbers exists so all these natural numbers that we've made we can just put them into one whole set now it's worth noting here that we can't assume that this set exists using our original rules I mean we could keep taking the successor of natural numbers to create new natural numbers and then we could keep trying to use pairing and Union to group them all together but we can't do that an infinite number of times eventually we have to stop the process this Axiom is required in order for us to assert that an infinite set exists now is this a bit of a leap sure but hopefully it's a comfortable one and we need it if we want to discuss infinitues for me it is a comfortable leap because even though this is an infinite set if I pluck any element from this set I can create that element through a finite application of the successor operator anyway I hope you'll take the leap with me because if not there's little point in continuing so see ya anyway the next Axiom is one that sounds more epic than it actually is power sets it's like a set going Super Saiyan or something but not precisely a Super Saiyan set is one with cardinality greater than 9000. now a power set of a set is a set containing all of its subsets take this set here for example that contains four elements a b c and d a subset of this set is a set containing just some of its elements so if we gather up all of its subsets we have all of the Singleton subsets all of the Two element subsets all of the three element subsets we consider the set itself to be a subset and we consider the empty set to be a subset so the set containing all of these is the power set of this set the set of all of its subsets foreign let's talk about ordered pairs let's say I have two sets A and B what we need to be able to do quite often is we need to be able to construct the set of all ordered pairs where the first element X's come from a and the second element y has come from B how do we construct this entire set from our axiomatic Foundation First of all remember that the ordered pair X Y is actually shorthand for the set containing the set containing X and one and the set containing y and two the elements 1 and 2 are just tagging which element is first in the pair and which element is second in the pair and like I did in my previous video don't worry about the fact that one and two might be members of these sets we're just going to assume for now that they're not if they were we could just use different tags for which is denoting the first and which is denoting the second element in the pair now the set of all ordered pairs such that the first element is in a and the second element is in B is called the Cartesian product of A and B and we write it using the usual product symbol like that so it's defined to be this set here and how do I construct that well if I wanted to construct this with finite sets then sure I could use a combination of subsets and pairing and Union to construct all the pairs and then bundle them all together into a set that contains all of them but not for infinite sense if a and b are infinitely large I can't sit there and apply the subsets and pairing rule an infinite number of times to construct them all we need to do it in a more intelligent way instead let's reframe things instead of a bottom up process constructing all the possible Pairs and then bundling them all together into a set let's construct an overall Universe where all such things like this exist and then we'll use the subsets rule to extract them so let me show you what I mean what we want to be able to do is apply these subsets Axiom to apply the subsets Axiom we need to be extracting the elements from a set so we want to construct an overall Universe where all possible things that look vaguely like this exist and then we'll extract the ones that we need so what is this universe here now fair warning the next couple of minutes are a little bit messy but stick with me because at the end of the process we'll be able to forget about most of the construction anyway we'll start by taking any old X in a and Y in B now I claim that both X and the number one are members of the following set a union the set containing one well it must be because X belongs in here one belongs in here and the union is just all of these things bunched together and since both of these elements are in here then that means the set containing X and one is a subset of a union one so then it must be the case that this set here belongs to the power set of this set because the power set of this is the set of all possible subsets and here we have a subset so it must belong to that power set now we can follow the exact same chain of reasoning to conclude that the set containing y and two will be in the power set of B Union the set containing two okay so this one is in here and this one is in here that means they both belong in the union of these two things I know this is getting crazy but just bear with me so both of these sets belong to this big Union of power sets now if both of them belong to this giant thing here that means the set containing both of them is a subset of this but now here's the final part if this whole thing is a subset of this that means it is a member of the power set of this because the power set of this contains all possible subsets and this is a subset now this whole thing here contains loads of stuff that look nothing like this but the point is we can build it using just our original axioms and we put it there use the subsets Axiom to just extract the things that we're actually interested in now I want to reiterate here now that we've been through that entire process and we know that we can construct this set of all other pairs from now on we don't even need to think about this we know it's possible to do it so just for shorthand from now on we'll just do a cross B the Cartesian product of A and B is just the set of all pairs X Y such that X is in a and Y is in B and remember when I said this notation looks like coordinates Cartesian coordinates so coordinates in the plane like that they are formed from the set of all ordered pairs of real numbers whatever a real number is we're getting there I promise Now we move on to an area in which my expertise cannot be questioned relations specifically the set theoretic binary relationship describing a subset of the Cartesian products what do you think I meant so far we've thought of the contents of a set of these individual packets we haven't really thought much about how to organize or how to compare these elements that's what relations allow us to do to see how the elements relate to one another so that we can get used to relations let's just deal with numbers colloquially I'm going to take the set containing the numbers one to five and we'll drop all of our axiomatic stuff and just deal with numbers as we're used to for now now all a relation is is something that takes two elements of the set and returns a value of true or false depending on what the relation is you're already used to some relations let me show you equality is an example of a relation because we could take two elements and depending on what these elements are it returns a value of true or false this one happens to return true when the elements are identical so here we have a value of true whereas here we have a value of false let's look at a different relation so we've got equality but we've also got uh a less than so this is now returning a value of true notice that in some relations it matters which order the elements are placed so here we swap the elements and now we return a value of false now relations are really flexible so let's look at a few different examples just so we can get used to them so is not equal to is an example of a relation this Returns the value of true if these elements are not the same we could also have divides so this returns a value of true if the second element is a multiple of the first so this is false as 2 isn't a multiple of three but for example this returns a value of true because 2 does divide four four is a multiple of two and the last one that we're going to look at is one that I'm going to call the parity check relation so this Returns the value of true if these numbers have the same parity that's if they are odd or even so here we have true whereas here we would have false as always it's always worth asking how are we actually constructing these in the language of our sets because the only things that we know exist are our axioms and so far the natural numbers well here's how we do it we do it a lot like we do with functions so we let's call the set a this set that I've created the set containing the numbers one to five now the relation is actually a set and it's a set of ordered pairs that encodes the relation so here is the less than relation but encoded as pairs this will return a value of true if and only if the pair is a member of the relation set if I make a different inquiry like two and one because the ordered pair two one is not a member of the set this returns a value of false usually we don't think of relations as sets but we just encode them in this way the real way we usually think about relations is in this kind of context where we are getting a true or false reading depending on what the relation is now steal yourself because I need to teach you three new words to help us discuss relations unfortunately there's no way around this we're just gonna have to power through the first word is reflexive now a relation is reflexive whenever every element is related to itself so for example equality is reflexive because all numbers are related to themselves because all numbers equal themselves for example less than is not a reflexive relation because it is not true that 2 is less than two and before we move on I encourage you to pause the video here and decide whether it divides the parity check and is not equal to are reflexive you should hopefully get that divides and the parity check are reflexive but is not equal to is not reflexive secondly remember when I said that in some relations the order matters so here we have true but here we have false if the order doesn't matter we call the relation symmetric so less than is not symmetric because I've just swapped them and now I've got false whereas for example equality is an example of a symmetric relation because whenever this is true then this is true and again I want you to pause the video here and try and decide whether it divides the parity check and is not equal to our symmetric you should hopefully get that divides is not symmetric the parity check is symmetric and is not equal to is symmetric finally let's discuss when relations are what we call transitive so this is when if we have a is related to B and we have that b is related to C then automatically a is related to C if a relation has that property we call it transitive I like to think of it as we can cut out the middleman so whenever we have a chain of related elements we can cut out the middle one and say that the outside members of the chain are indeed related and less than is an example of a transitive relation because if a is less than B and B is less than C then surely a is less than C but for example is not equal to is not transitive because for example 2 is related to 3 because they're not the same three is related to two because they're not the same but that doesn't necessarily imply that two is related to two we've cut out the middleman but suddenly we've got a falsehood so this is an example of relation that isn't transitive again pause the video here decide whether equality divides and the parity check are transitive you should get that in fact they all are isn't that nice so if you've done your homework you should have something like this so this is describing whether each of the five relations we're using in this example are reflexive symmetric and transitive and something very special happens when they have all three of these properties so we've got for example equality and the parity check they have all three of these properties a relation that has all three of these properties we call an equivalence relation and something magic happens when we have an equivalent relation let me show you let's look at the parity check relation in more detail because this had all three of those properties it's what we call an equivalence relation so what I'm going to do is I'm going to pick an element like one for example and I'm going to build the set of all of its relatives all of the elements that one is related to so the priority check relation is true if the numbers have the same parity so one is odd two is even so they're not related it is related to three because three is also odd and it is related to five because five is is also odd and this is now the collection of all the relatives of one and now I repeat this process for all the elements of the sets but notice that if I choose to build the set of relatives of three I'm going to generate this exact same set here and it kind of has to be this way imagine I'm constructing the set of relatives of three and I find something new that 3 is related to but because this is a transitive relation because one is related to three and three is related to this by transitivity one is also related to this so it should have been in the set of relatives of One to begin with so these end up as these closed bubbles of relatives all the relatives of five are in here and we can't find any new ones so let's choose a different element then so for example let's build a set of relatives of two two is even so it's not related to any of these but it is related to four and notice what we've done we start to put the set of numbers from one to five and by constructing the sets of relatives we've broken the setup into subsets with no overlap and everyone being accounted for this is called a partition of the original set and any equivalence relation induces a partition so if you have any equivalence relation by building the collections of relatives you end up with these bubbles of elements that don't overlap and that accounts for every element and that is going to be absolutely essential going forward it's the most important New Concept in this video it's the key to understanding how we construct all the more complex numbers like the integers the rationals the reals and the complex numbers because it's so important I'm going to just do one more example just to make absolutely clear that we know what we're doing so this time I'm going to define a relation not on numbers but on pairs of numbers so I've got this set the set containing 0 1 and 2. I've taken the Cartesian product of that with itself generating all pairs of numbers a b such that A and B are both either zero one and two so it's this nine element set here and the relation I'm going to Define is these two pairs are related if and only if they have the same sum so if we add up the numbers in the pair and I get the same sum over here then they are related so for example 1 1 and 2 0 are related because one plus one is the same as two plus zero they both add up to two whereas one one and two one are not related because one one makes two two and one make three they don't have the same sum I claim that this is an equivalence relation and it's pretty easy to check everything must be related to itself because if you add up the elements it's the same sum as if you add up the same elements it's a symmetric relation because if uh if two sums are the same then they must be the same in reverse order and it's transitive because if two pairs have the same sum and two other pairs of the same sum then they all all three must add up to the same thing so the two outside members of the chain must have the same sum you can do it in a more formal process but you can do that in your own time if you if you want to now because it's an equivalence relation it induces a partition so this is a splitting of the set in a way that everything is accounted for and there is no overlap so let's see what happens so zero zero that is in a set all by itself it doesn't have any other relatives because it's the only pair that adds up to zero similarly with two two it's the only pair that adds up to four so it also doesn't have any relatives two one and one two they add up to three and similarly zero one and one zero they both add up to one and now this collection here in the middle all three of these add up to two so they are in a set of relatives all by themselves so you can see what's happened here we've partitioned the original set according to whichever two pairs have the same sum we have partitioned the set I know there was a lot to take in so just a quick summary a relation that's a way of comparing two elements of a set a relation which is reflexive symmetric and transitive it's called an equivalence relation and an equivalence relation induces a partition of the original set that is a breaking apart into subsets with no overlap and everything being accounted for I know that was a lot in one go but trust me it will be worth it because we are finally ready to embark on our Quest now would be a good time to take a quick break if you need to but come straight back because it all about to go down very quickly since my last video I broke 10 000 subscribers thank you so much for supporting the channel I'm so glad that so many of you out there are willing to support me because I've got loads of ideas for upcoming videos and if you're excited to see those please make sure you subscribe also below you'll find a link to my subreddit where I'm gathering questions if I get enough good questions I'll make a 10K q a video it should be a less scripted and light-hearted bit of fun anyway let's get to it what is a negative number and why do we need them well in my last video we defined addition and multiplication on the natural numbers but we couldn't Define subtraction because the answer to some subtractions is not a natural number so like 3 subtract 5 the answer to that isn't a natural number it tries to escape from the set so we need a larger set of numbers which encompasses all such answers so that's one way we can think about negative numbers it's the answer to a subtraction where the first number is less than the second now if we're talking about less than I'd better actually Define what less than is so so far I've got the set of natural numbers we've got 0 1 2 3 Etc and the way I Define them is as follows a natural number is the set containing all the previous ones so in this way it's actually quite easy to Define less than I'm going to say that a a number natural number a is less than b exactly when a is contained in B and that feels a bit weird but it's just a way of encoding our intuition of what less than means so for example 2 is less than three because two is a member of the three set and two is a member of all the sets that are further down the list so two is going to be less than each of these the collection of these numbers I call n for natural and as I said we can think about negatives as the result of subtractions when the first number is less than the second number since we're dealing with subtractions which is something we do on a pair of numbers we're actually going to start not with the set of Naturals but with the Cartesian product of n with itself I.E the set of all ordered pairs of natural numbers let me show you what I mean so we've got the set of all ordered pairs of natural numbers and I'm going to define a relation on this set of pairs I'm going to say that two pairs are related if they have the same subtraction so for example uh 2 1 and 3 2 are related because 2 subtract 1 is equal to three subtract two now I can already hear you screaming we haven't defined subtraction on the natural numbers yet and you are absolutely right to be screaming that so instead what we can do is we'll rephrase things and instead of framing things around subtraction we can frame them around addition I want you to imagine we're going to add D to both sides and add B to both sides of this and we'll get an identical stipulation that involves only addition so now we can say that two pairs a b and c d are related if a plus d is equal to C plus b now that gets around the subtraction problem but in your head I want you to be thinking intuitively that this relation is all about subtraction now I claim that this is actually an equivalent relation and it's fairly straightforward to prove so I'm not going to go through the all the details right now you can do it in your own term if you want to and I'll leave some notes in the description if you want to check the proof but in brief an element is related to itself because its own subtraction is equal to its own subtraction uh if two elements have the same subtraction then they have the same subtraction in a different order you need the the fact that addition is commutative in order to actually prove that which we proved in the last video and if two elements are equal and two elements are equal their subtractions are equal I mean then uh it must be the case that those outside elements have the same subtraction as well so it's transitive um so since it's reflective symmetric and transitive it is an equivalence relation again I'm not going to go through the full details we're just going to take it as given that it is an equivalent relation and what do we know about equivalence relations they induce partitions so we'll start with zero zero and I'm going to construct the set of all of its relatives well zero subtract 0 is 0 and you'll notice that all the elements on this main diagonal all have a subtraction of zero so the set of relatives of zero is going to look something like this where we strip away everything else and just have this whole main diagonal here now let's look at one zero one subtract zero is one and two subtract one is one so again we follow this pattern where all the pairs on this diagonal are related to one zero and you'll notice the same for all the elements going this way and they're gonna go on forever because this is an infinite set now on this side we're going to get the same pattern zero subtract one has the same answer as one subtract two if you're uncomfortable with the fact that we're using subtraction you can use the addition definition so for a b and c d a plus d equals c plus b is the stipulation for the relation I the outside elements have to add up to the same thing as the inside elements and we see that here 0 plus 2 is the same as one plus one so these are related and you'll see it's also related to this one we get the same pattern forming over here where the set of relatives of zero one is this whole diagonal here the set of relatives of zero two is this diagonal and so on up in this direction and we can see if we extend this in both directions we'll eventually absorb all of the pairs so this is a partition everything is accounted for and there is no overlap and here is the magic so I'm going to give a new name to this collection of relatives the set of all relatives that is uh related to zero zero I'm going to give it a new name I'm going to call it integer zero and the set of all pairs related to one zero I will call one the set of all pairs related to 2 0 I will call all too the set of all pairs related to 3-0 and so on in that direction in general the set of all pairs related to n zero I will call integer n and in this direction I'm going to give a new name to the set of all pairs related to zero one I'm going to call it integer negative one I'm just just using a symbol we might not have seen it before but I'm just using it here I'm just going to call it negative one the set of all pairs related to zero two I will call the integer negative two and negative three and so on up this chain here so in general the set of all pairs related to zero n I'm going to call the integer negative n and there we have it a definition of all the integers positive negative and zero and like I said at the start of the video we've upgraded the numbers our new definition of integer relies upon the definition of natural numbers because the natural numbers still in here but the definition of an integer includes the natural numbers it's like upgrading the operating system of the numbers it relies on the old architecture but it does everything that the old system did and more because it also contains the negatives now if we bundle all of these integers into a set we give it a special name we call it Zed for integer actually it's Z for zalen which is the German for numbers why German I'm not entirely sure I couldn't find the answer apparently the first recorded use of Zed for the set of all integers is due to Nicholas bobaki the story of bobaki is so cool that it's deserving of its own video but anyway we've got these new numbers to play with let me show you how we add subtract and multiply using these new numbers we'll start with addition and before I do I just need to explain this notation so these square brackets that just represents the collection of all relatives of this particular element here so remember how we defined the integer one was the collection of all the relatives of one zero so that's what these square brackets mean now to add these if I take these two integers so one integer being the set of relatives of a b and one being the set of relatives of CD then the answer is the set of relatives of a plus c b plus d b just add the components let's look at an example let's say I want to do 3 plus negative 5 well what three actually is is the set of all relatives of three zero and negative five is the set of all relatives of zero five and then this definition of addition up here just says okay you do three plus zero to get three zero plus five to get five and the answer is the set of all relatives of three five and this you will note because 3 subtract 5 is the same as zero subtract two this set of relatives is equal to this set of relatives and this is what we defined to be negative two so as we expect three plus negative five is negative two let's do multiplication next which is a bit more difficult but it makes sense once you think about it now multiplication is a bit more complicated a b times c d is equal to the set of all pairs related to this whole mess I will explain where this whole mess comes from but first let's just look at an example so again we'll start with uh three and negative five and just remember we are expecting an answer of negative 15 here but let's go through the whole process so three that's the set of pairs related to three zero and negative five is the set of pairs related to zero five now this looks a bit complicated but just think of a and b as three and zero and C and D is zero and five and we'll feed this into the multiplication definition so it's saying a times C which is zero plus b d which is zero uh so we get zero here and then over here we have a d which is 15. uh plus BC which is zero and so there we have it this is indeed what we are calling the integer negative 15. it works now I want to make clear these Representatives here are arbitrary so this is the set of things related to one zero but I could have defined this as the set of things related to 2 1 or 3 2 or anything related to one zero it's going to define the same set and this definition of multiplication will work regardless of which representative we choose so for example if I'd chosen a different representative here like uh like seven four for example so that is related to three zero like our canonical representative but I'm going to show that this definition of multiplication still works and a different one for negative five so for example I don't know three eight it's gonna be more difficult to do but it is going to work and I'll label these as a b c and d again so you can follow what's going on now this multiplication definition says we do AC which here is 21 and then BD which is 32 21 and 32 that makes 53 then it says a d which here is 56 plus BC which is 12 so this gives 68 and now notice that this has a subtraction of negative 15. this is the same set of Representatives as those related to 0 15 which once again is the integer we are calling negative 15. so it works regardless of the representatives we choose so what is this doing here why is this working well I'll show you where this comes from and it'll become really clear so where this comes from is the fact that this set of relatives a b is really encoding the notion of a subtract B that's where these uh relations came from it's this notion of subtraction and then CD is the uh subtraction C subtract D and what happens if we multiply these together well we know how to expand brackets or multiply out brackets you might call it if we multiply them out and then extract a negative sign what we get is this we'll get AC plus BD subtract the sum of a d plus BC I'm skipping steps here you can do it in your own term if you want to and that is exactly how we specified the answer to the multiplication this is the subtraction AC plus BD subtract a d plus BC which is what we see here so that's why it works we can even Define some subtraction something that we were unable to do with the natural numbers and I'll leave it to you as an exercise to demonstrate why we use this definition of subtraction so a couple of remarks before we move on so the first is that these operations are dependent upon the operations we defined for the natural numbers you'll notice the inside each of these are addition and multiplication of natural numbers but never subtraction and because addition and multiplication on natural numbers are well defined we did that in the last video these definitions are reliant upon that but we can create new definitions of these operations on this new system of numbers again we're kind of overwriting what we've done but it's dependent on the previous work and we're able to Define subtraction on integers using only addition of natural numbers the second point is that isn't this awfully complicated I know when I first just saw this I thought do we really need an entire infinite set for each of the integers why not for example just have -2 be the ordered pair zero two well the reason for that is because actually all the operations that we have here are much easier if the entire collection of relatives of zero two is defined to be negative two so we saw in the example of three plus minus five that the answer was this pair here three five and if we use this kind of naive definition of the integers this isn't one of our integers but because this belongs to the set of all pairs related to zero two we know that it Bears the name minus two so when we do three plus minus five we get a member of one of these sets so we say that that is the answer to them the addition so it looks more complicated but actually when you you think about it it's the cleanest way of doing it so that's addition subtraction multiplication but what we don't have is division and there's a good reason for that because if you take any two integers and if you add or subtract or multiply them you get another integer but that's not the case for division so for example 2 divided by 3 is not an integer the operation of division is trying to leak out into the gaps between the integers so the next step let's fill those gaps [Music] so we're about to start talking about rational numbers but before we do I just want to reiterate that you know how I always say that we don't think of natural numbers as those sets well it's true for integers as well we don't permanently think of them as these infinitely large collections of pairs of natural numbers we've assigned the labels like the integer three and the integer negative five to those collections so from now on we can just think of Zed the set of integers as the sets of all of those collections of pairs of natural numbers but we're just going to use the labels that we've assigned them so from now on we think of Zed as the set containing 0 1 2 3 Etc minus one minus two minus three Etc going on infinitely in both directions we know that hiding underneath that symbol one is actually the infinite set of pairs of natural numbers that are related by subtraction as we set up in the previous section but from from now on we don't really need to think about that we can just use the shorthand that we've created and we've proved that this shorthand works with addition multiplication and subtraction and we can even go ahead and prove that they have all of the nice properties like commutativity associativity distributivity and the cancellation law and all of that good stuff I'm not going to prove them in this video because this video is more about how we Define the larger classes of numbers anyway on to the rational numbers now the way we set up the integers was to think about all the subtractions of natural numbers it's going to be a very very similar process except this time we're going to think of divisions of integers and as before a division is what takes place with a pair of numbers so we need to start from all pairs of integers so we take the Cartesian product of Zed with itself to generate all pairs of integers and we're actually just going to take a subset of this so we're only going to take the ones which have the property where Q the second entry is not zero and the reason for that is because we're thinking of these pairs as divisions and we know we can't divide by zero so we're just gonna specify that Q can't be zero at the start of this whole process you'll see why shortly so this is the set which we're going to deal with and use to define the rational numbers so this time the relation is going to be two pairs are related to each other if they divide to produce the same answer so if P divided by Q is equal to r divided by S and once again I can already hear you screaming we haven't actually defined division on the integers we only have addition multiplication and subtraction but we can manipulate this in order to make the same stipulation but instead of involving division it's going to involve multiplication so let's do that so I want you to imagine we're going to multiply both sides by q and multiply both sides by is so you might know this is cross multiplying so we get that PS must be equal to RQ for two pairs to be related now we're doing this to avoid the problem that we haven't established division in the integers but as far as our intuition goes I want you to think about these pairs being related if their divisions are the same and you know the drill by now this relation is in fact an equivalence relation I'm not going to go through the full details now but in brief it should be fairly clear that the division here will be equal to its own division so it's reflexive if two pairs have the same division then they'll have the same division in a different order so it's symmetric and if these two pairs have the same division and a third pair had the same division then all three pairs would have the same division so it's transitive the full details I'll leave in the description if you're interested but from net from here we'll just go ahead and assume that this is indeed an equivalent relation and what do we know about equivalence relations they induce partitions I apologize in advance for how terrible this looks it looked better in my head and as soon as I started drawing I thought this is going to be a way too busy for a diagram but I'll try and uh let's just get through it all right so we've got um we've got all the pairs of integers where the second entry is not zero increasing this way is the first entry so in columns all the zeros in the first entry all the ones all the twos in the threes in the first entry and so on and then negative in this direction in the rows it's the second entry that's changing now this goes on infinitely in all directions because we have the integers from the negative to the positives in this direction and the integers from the negatives to the positives in this direction so it's not quite as clean a diagram as in the last case but let's look at the partitions so I'll try to do this diagram as best I can but it's not quite as neat as it was with the pairs of natural numbers so we're thinking about division so this entire Central column here that all represents zero because zero divided by anything is zero so we end up with this partition here that kind of trims away all of these pairs uh like that and then when the entries are the same we have one one two two Etc that represents when we are doing one number divided by itself which is equals one so they're all going to be the same so we end up kind of peeling off these but we also have it down here in the negative direction as well so something like that and I've done them in the same color because this is one set one subset of the partition it's just broken apart like this because we've got this awkward region there in the middle that's kind of separating things out but the blue represents one single set and then we have the uh the other examples of uh of equal divisions so for example one two that's like one divided by two which we know is a half and that's the same as if we do minus one divided by minus two we would also have one up here which is two divided by four so the partitions they kind of uh in green and I've already used green stupid uh we let's do purple so we have something that looks like that it collects that one and then it goes up at a sharper angle than the blue one and similarly down here we're gonna go down and it's going to go down like this so what you want to imagine is these kind of lines of increasing gradient and eventually everything will get uh absorbed uh so what else have we got well one divided by -1 or a number divided by its negative or it's negative divided by its positive they will equal minus one so they're all going to be the same so let's uh let's draw these in now something I want to call your attention to is these ones were the second entry is one that represents two divided by one which we know is the number two so we're gonna have a partition that looks like this it would absorb uh the number up there which would be uh four divided by two that also has a division of two uh similarly down here as well we also uh absorb these ones like this and so on now you kind of just have to use your imagination here if we keep drawing these bubbles we'll eventually absorb all of these now here's where that the magic happens so instead of writing these as ordered pairs I'm going to assign new names to them so this blue one here for example I'm going to call this the the rational number one over one this purple one here for example instead of writing it one comma two I'm just going to write it as 1 over 2 and so on for the others like this one up here I will call this minus one over one and so on hopefully you can see by renaming these we are generating the rational numbers a lot like we did with generating the integers from the natural numbers we just assigned new notation to those collections those infinite collections of pairs of natural numbers we're doing the same thing here we're just assigning new names to these infinite collections of ordered pairs of integers let's get rid of this diagram and tidy things up a little bit because I hate it so a lot like we did when we defined the integers the fraction p over Q is the set of all ordered pairs that are related to the ordered pair PQ where the relation is if they have the same division and we can go from here and actually Define the four operations that we are used to addition and subtraction multiplication and division on these fractions so here we have the four operations and notice that in each case in the definition we are only relying on addition subtraction and multiplication of integers never division so in here in addition we've got multiplication addition and here subtraction of integers which we know are well defined and have all the nice properties like we mentioned in the last section this line here is not really division of integers it's just separating out the first entry and the second entry of the ordered pair now there is an incredibly weird and important property of the setup of all rational numbers but before I get there I just want to explain a few additional things so you'll notice that if we have a fraction that looks like this so a integer over one then that represents the integer P so p over one we're just going to call that P so it's like I said at the start of the video we're layering numbers on top of the previous ones something else that we're going to specify is just a matter of standardization that we will never have a negative denominator so let's say for example I have something like P over negative Q well that's fine that's that's is a rational number but we're just going to restate that as minus p over Q so for that reason we can always specify that the denominators are positive if the denominator was negative we could just change the signs of both numerator and denominator so the the denominator was positive and one last thing we need to set up with these rational numbers before we talk about that special property is the fact that we can specify when one rational number is greater than another so here we have two fractions and from here we're going to apply this standardization we're always going to assume that q and s are positive so we say that P over Q is less than R over s exactly when PS is less than QR and here all we're doing is multiplication of integers and comparing two integer values to determine which of the rational values is larger so with these things in mind I want to talk about density and this is a property of the set of rational numbers and none of the sets we've examined so far have this property so what do I mean by density well that just means that if we have any two rational numbers we can always find a rational number in between them so if you imagine a kind of number line here we've got a rational number there and a rational number there then there always is another rational number in between them that isn't the case for any of the number systems we've looked at before because clearly the integers and the natural numbers they have gaps but if you zoom in on the rational numbers even if these numbers are very very close to each other there's still always another one in between them and that's true for this rational number and this rational number we'd be able to find another number in between those and in between those and so on forever so no matter how much you zoom in on the number line of rationals you can still always find a rational number in the Gap and the proof of this is actually fairly straightforward let's do it okay so the goal here is that if we have two rational numbers and they are different that means that one of them must be less than the other then we can find another rational number which squeezes in between them now what I'm going to do is I'm going to add these fractions together and then divide them by two so in other words I'm going to take the arithmetic mean of both of these numbers okay once we add them and divide by 2 we get this fraction here and my claim is that this works in other words that this is in fact greater than this one yet less than this one so let's first prove that it's greater than p over Q so as always with proofs we start with what we know we know that PS is less than QR this is given to us here in the fact that P over Q is less than Qs and now what I'm going to do is I'm going to add PS to both sides of this inequality my motivation for doing that is because I want something that looks like this PS plus QR involved in this inequality down here because we're trying to prove facts about this fraction here now I'm going to multiply everything here by q and the reason for doing that is because I want Qs involved where this 2qs is here I've got 2ps but I want to get Q involved so let's do that and remember I can do that because Q is not allowed to be zero in order for this to be a rational number so I am allowed to multiply through by a q so let's do that okay and I've used the associative property here to just to rewrite this section so that it's really clear what's going on it's 2qs times p is less than this entire bracket times Q because now we are done this satisfies this inequality definition here this is meaning that P over Q is less than PS plus QR over 2 Qs so we've proved the first half of the result that this is greater than p over q and to prove that it's less than R over s is it a very very similar argument that I'm not going to bother going through here you can do it at home as an exercise if you want to uh it's it's almost identical just slightly different so it's it should be fairly easy to work out but the point is we can always find a rational number between any two rational numbers and that is going to be an essential property going forward so that's the rational numbers and all of their nice properties now if we bundled together all of those rationals into a single set we give it a special a special a special name we call it Q for rational actually it's Q for quotient for German reasons and why the word rational well it's easy to think that rational derives from the word ratio since we are taking the ratio of two integers actually it's the other way around the word ratio derives from rational a rational number being one that can be expressed as a division between two integers and rational comes in contrast to irrational which derives from the ancient Greek alogos which means inexpressible to the ancient Greeks the irrational numbers Were Somehow wrong incommensurable and they knew that they were out there the ancient Greeks after all were familiar with Pythagoras's Theorem which is named after Pythagoras although it's not believed to have been devised by him originally the simplest right angle triangle you can think of with legs of length one has a hypotenuse of length the square root of 2 which is easily proven to be not rational so even though the rational numbers kind of form this nice line where even if you zoom in on them really close you can still find another rational number it's still patchy there are still holes there are gaps in our number line that need to be filled in [Music] okay fair warning this next stage of the journey is the most complicated but stick with it because it's amazing now let's say this red line here represents all the rational numbers and We Know It kind of fills in the gaps between the integers but it's in some way patchy it has holes in it which are these irrational numbers so this runs from negative to positive and somewhere on this number line is an irrational number like for example the square root of two how do we describe where the square root of 2 is given that all we know so far are the rational numbers well the square root of 2 this is the number X such that x squared is equal to two and we know that there must be a bunch of rational numbers below it and we know that there must be a bunch of rational numbers above it and we can describe those rational numbers as follows so if you are lower than the square root of 2 then that means means either you are negative or you have a square that's less than two so this is this set over here and if you are greater than the square root of 2 then you must be positive and you must have a square that's greater than two and that's this set over here and these sets are very very very very close to each other they actually kind of collide where the square root of 2 is but there's this tiny tiny infinitesimally small gap between them and that point there is what we think of as the square root of two so having all of the rational numbers below and above we can kind of Zone in on exactly where the square root of 2 is and it's almost like we can cut this number line exactly where the square root of 2 is and that's where it will be that's how we will describe it and this cut has a special name we call it a dedicinned cut named after the mathematician who dedicates his kind name let's just be a little bit more precise a delicate is a splitting of the rational numbers into two sets A and B and it has to follow a few rules in order to be a genuine dedicated cut so firstly A and B between them have to cover the entire set of rational numbers so for any rational number number it's in a or it's in B uh neither a nor B are allowed to be empty so they both must contain something and here are the two crucial rules that make this a dedicated cut firstly it's the a is what we call closed downwards that means that it just goes on forever in that direction so precisely if you pick any member of a if you pick any other rational number that's less than this one then it must also be a member of a so it just means that a it starts somewhere and then goes on forever in the negative Direction and as a consequence because between these two sets they cover the entire set of rationals B is closed upwards it goes on forever in that direction and finally here's the really brain melting one a must not contain a highest element and let me just explain what I mean in a bit more detail there so what I mean by that is that there is no member of a which is greater than everything else in a but how can that be because a eventually stops and B takes over to carry the rest of the rational numbers so surely there must be some number in here that's bigger than everything else in a but it is possible to construct sets with no greatest element and in fact that is really the defining property of a dedicated cut so I'm going to spend a bit of time proving that this is true we can construct these and why it's so important so let's say I pick a number R and I build the set of all rational numbers that are strictly less than R then I claim that this has no greatest element and the way we do that is to say imagine if it did have a greatest element let's call it m for maximal so let's assume that this exists so m is in a and m is greater than any a in big a well it must be the case that m is less than R we know that because this is the set of all things less than R but because m is a rational number and R is a rational we know that there is some rational number X which that X is greater than M but less than R this is the density property that we proved at the end of the last section but because X is less than R this tells us that in fact X must be a member of a but since it's greater than M which was supposed to be the biggest element of a this is a contradiction no such M could possibly exist so what's basically happened here is we've said okay imagine we find this m here that's bigger than everything else well actually because of the density of the rationals we can always find some number here which is in between what we thought was the maximum one and R and now if we think if that one is the maximum one we can find another one that's bigger and we can keep finding ones in that smaller and smaller Gap so a in this case contains no greatest element and here is the magic because this is the set of all rational numbers less than r that is how we Define the real number R we do it using these dedicated cuts so this real number R it sits somewhere on the number line how do we Define it well we take all of the rational numbers less than it and we take all of the rational numbers greater than it and this cuts the set of rational numbers a deadly can cut exactly at the point where R is so the real number R is defined to be the set containing A and B the delicate cut that defines a and b describes where it is using two um sets of rational numbers and we can even redefine the rational numbers in this way so in the case where R is a rational number let's say two-thirds well we'll take all of the rational numbers less than two-thirds and all of the rational numbers greater than two-thirds this is a dedicine cut that describes where the where the number two thirds is so this is a way of redefining the rationals in terms of delicate Cuts but the main point is that we can describe exactly where the gaps in this number system are so like with the square root of 2 2 the square root of 2 is defined to be the pair of sets of rational numbers such that we take all the rational numbers greater than the square root of 2 and all the rational numbers less than the square root of 2 and this is the dedicating cut that describes the square root of 2 exactly where it is I like to think of these as imagine this set here is like a big arrow that points along the set of rational numbers and points at the square root of two it never actually gets there but describing all of the rational numbers less than the square root of 2 it tells us exactly where the square root of 2 is it's like a big arrow that points to it it's like in a sense the square root of 2 is the very next number after this set of rational numbers here and we can describe all of the irrational numbers in this way for example any old thing like the fifth root of 17 you know this is an irrational number but we know that it cuts the line of rational numbers at some point and we can take the set of all rational numbers Q such that Q to the power of 5 is less than 17 and then we take all the rational numbers Q such that Q has a q to the power 5 greater than 17. this is a dedicated cut we call this one a we call this one b and the set containing A and B which we sometimes write like this we call the fifth root of 17 and that is a way of defining the irrational numbers in terms of the rationals you might be wondering about numbers like pi and it's going to be on the scope of this video but there is a way of describing Pi using dead again cuts so for example there are infinite series which describe Pi you know infinite series of rational numbers and you can do that with dedicine cuts you know you take the sum that's less than the sum that's greater and in between them is exactly where Pi lives uh maybe include some notes in the description option if you want to check that out okay so the next thing is that the real numbers have a very very special property and it's the property that really sets it apart from all the number systems that have come before uh now before I get there I just need to talk about when uh one real number is less than another real number so let's say I've got a dedicine cut a b which describes a real number and a dedicated CD that describes a another real number so in this case this one is less than this one but how can I say that this one is less than this one in terms of the cuts well just focus on sets A and C for the time being because these sets are closed downwards C goes on forever in that direction and a goes on forever in that direction we can see that a will actually be fully contained in C because C starts here and goes that way and a only starts here and that is how we Define when real numbers are less than each other so here I've got the the notation that we use for dedicated Cuts with this bar it separates into two sets A and B we say that again cut a b is less than CD exactly when a is completely contained within C and that's how we compare real numbers and this is going to be essential going forward in discussing that really really special property of real numbers and just what is that special property well it's the notion that the real line is a single unbroken Continuum there are no gaps and this is the first number system that we've come across with that property it's really the defining aspect of the real numbers the integers have very clear gaps between them the rational numbers is like this foamy expanse between the integers but it still has these Pockets missing these little gaps the irrational numbers but in filling those in there are now no more numbers on this line it is a Continuum and it's just amazing now how do we prove that the real numbers are this unbroken Continuum well the proofs involved here are very sticky and very technical so I'm not going to give all the details here but I will give you the gist of how it works and I know I keep saying that in this video I keep saying I'm not going to give you all the details and the reason for that is because I don't want the videos to get too bugged down with too many technical details these videos aren't intended as a replacement for a formal education they're just here to spark your interest in your curiosity and if you wanted to study this on your own you've got some intuitive feel for what's going on and you can fall back on that intuition when required anyway I'll give you the gist of how this works so let's say we have two real numbers here described by these dedicated cuts and we'll just focus on the lower part of each of these cuts and I want you to imagine that these two dedic in cuts are very very very close together and imagine that we were worried that there is actually a hole in our real number line that there is some number right in between the set a and the set C what we will do is demonstrate that in fact there is something in that Gap that's how we will prove that this is in fact an unbroken line so let's do it okay so the idea here we've got these real numbers a and C described by their delicate cuts the lower sets of the real number a is Big A the lower set of the real number c is Big C just like this uh setup that I've got here on this diagram and we are worried that there is nothing in between them so how do we show that in fact there is always something that lives in there well in this case here we've got the a is less than C and that is if and only if this set here is contained in that one that's that's how we described less than in terms of the delicate Cuts now if a is contained in C notice that we can't have the a is the same set as C otherwise these real numbers would be the same number so in fact I'm going to draw a little symbol here which means that they are not equal so they're not the same set yet a is contained in C but if they're not the same that must mean that c contains something that a doesn't so there must be an element of C that is not contained in a so let's call that Q so we've got this element Q which is in C but not in a now you might be thinking that we are done here because after all we found some q that lives inside C that is not in a so it must be in this Gap here and this is where the diagram can be a bit misleading because what if Q is in fact a what if the only thing that is not in a but is in C is this real number a then we've not actually found anything in the Gap so we just need to be very careful here here about how we do this and what we're going to do is we're going to show that we're going to go through the cases of whether a and C are rational or irrational we can always find something in this Gap so we'll start by assuming that they are both irrational so in this case if both of these are irrational numbers since Q is a member of C which is a subset of the rationals Q is a rational number so Q cannot be equal to a so Q must genuinely be between a and C so where they're both irrational we can find a number in that Gap now let's move on to the case where a is rational but C is irrational so in the case that a is rational and C is irrational well we know that Q lives somewhere in here so either Q is actually equal to this a or it's in the Gap if it's in the Gap then we're done we've found a number in that Gap but if Q is in fact equal to a then we're still done because C is a dedicated cut which means it does not contain a greatest element which means there is a number that's greater than Q that's still inside C so we have found something in the Gap now we'll reverse the roles here imagine that a is a rational and C is rational well this case is rather like the first case we can't have the Q is equal to C because Q is contained in the set Big C and we can't have that Q equals a because Q is a rational number so Q must live inside the Gap so the only remaining case is what happens if both A and C are rational well then it doesn't really matter about this whole queue business because we've got a rational number here we've got a rational number here and we know that from the density of the rationals there is always a rational in between them we can find something in that Gap now I know the actual proof is more Technical and I'm being quite hand wavy here so so the property you're looking for if you want to research this on your own is called completeness so the completeness of the reels and uh that's encapsulated in the greatest lower bound property so if you want to go ahead and self-study that's what you're looking for and as always with this definition of real numbers we are able to Define addition subtraction multiplication division and prove that they have all of the nice properties using this dedicated definition of real numbers it's a bit more Awkward than the last cases but we can do it and if we take all of the dedic in cuts of the rational numbers and bundle them together into this one giant set we call it R for the German real which means real now you might be wondering if the real numbers really are this continuous unbroken line then what room is there for more numbers well let's just remember how the other numbers were motivated so we couldn't always subtract in the natural numbers we needed the integers we couldn't always divide in the integers we needed the rationals we couldn't always take square roots in the rationals we needed the reals like the square root of two there's an irrational number and we still can't take all square roots because the way we set this up a negative multiplied by itself is positive and a positive number multiplied by itself is positive what is a number like this it doesn't exist anywhere on this line but it deserves to exist just like the others deserve to exist and this is where we get to complex numbers now despite the name complex this is actually the simplest step out of all of them so like we did with the integers and the rationals we're going to start with the set of all ordered pairs X Y of real numbers so these exist in the Cartesian product of the reels with itself and that's it they are the complex numbers done okay we're not quite done we don't need any relations or anything like that the only thing that makes these the complex numbers is how we Define multiplication we do it in a way that isn't quite obvious let me show you so if we take two of these pairs of real numbers x y and u v then we Define the multiplication as follows it's the ordered pair X U subtract yv and then the second entry is XV plus y u that's what makes these complex numbers and I'll show you why because remember we were motivated to do this because we didn't have a square root of negative one and why do we set it up like this well we want to encode the idea that we can square to make a negative so what we do is we just come up with a new symbol let's call it I and we'll make it such that I squared is equal to negative one and the way we usually write a complex number is as follows X Plus i y and this over here would be U Plus I V and now we use the fact that this has this property of I squared being negative one to Define this multiplication here because look if I take X Plus i y and multiply it by U plus IV then what I get here will be x y and then I'll get I squared IV which because I squared is -1 is just yv and then here we will collect together an x v and a y u lot of I so this is going to be i x v plus y u but in the language of ordered pairs this is the first entry of the pair and this is the second entry of the pair now we've used the notion that I squared equals negative one to kind of reverse engineer this multiplication but it really does work so the number I in the ordered pair form is this 0 1 because it's zero plus one lot of I and if we multiply this by itself well this is telling us to do the first two things times together so that's zero and then we subtract the final two things times together so one times one so zero subtract one is negative one and then in the second entry we take the two outside things so zero times one which is zero and then we add the product of the two inside things so y times U which is one times zero which is zero so in other words this is I times I and this written in the usual way we write complex numbers is minus one plus zero I so in other words minus one so this really does encode the notion that we can square to make a negative and that's really all there is for complex numbers it's just the Cartesian product of the reals with itself with a specified system of multiplication and the set of all such things we of course call C for complex not complex is in complicated but complex as in consisting of multiple Parts which we usually call the real part and the imaginary part the natural question to ask now is are there any more numbers well there is a sense in which the complex numbers complete the number system so we created more and more interesting numbers out of necessity subtraction letters to create the integers division letters to create the rationals rooting positive numbers led us to create the reals and routing negative numbers led us to create the complex numbers and that's kind of it even if you did something like the square root of I that is just another complex number there's no real operation you can do with the complex numbers that require the existence of higher level numbers so there's really a sense in which the complex numbers finish off the number system this is known as the fundamental theorem of algebra that is if you take any polynomial with complex coefficients the solutions lie in the complex numbers there are other systems like the quaternions and as to whether they are numbers it's kind of open to interpretation for me the complex numbers round off and complete the number system um but I'm sure that will make some people angry so to summarize a complex number like two plus three I is actually the ordered pair two three of real numbers where the real number two is actually the dedic input of all rational numbers less than two and greater than two where each rational number is a collection of ordered pairs of integers related by division where each integer is a collection of ordered pairs of natural numbers related by subtraction where each natural number is constructed by recursively taking the successor of the empty set easy thank you so much for watching this concludes my four part series of building numbers up from this mathematical Foundation if you like this I've got loads of ideas for upcoming videos so please make sure you subscribe so you don't miss them a huge thank you to my supporters on patreon your support is genuinely being quite life-changing my current teaching job is only part-time and I deliberately chose that in order to give myself time to try and make more videos and your support has given me that flexibility to try and get my channel off the ground so from the bottom of my heart thank you this video's patreon question comes from Ian thank you so much for your support you've been so generous and he asks are there any mathematicians or logicians who inspire me either personally or professionally yes too many really I could probably make a whole video about it but for now I'll choose John Conway so he was a genius you know he was prolific in many areas of mathematics but he was particularly instrumental in the classification of finite simple groups and I'm a group theorist my PhD was in group Theory and we had this this giant book that we would appeal to and you know there's this big ring-bound A3 tattered you know dog-eared book with like yellowing pages and a faded front cover and it was called the atlas of finite simple groups and he was the lead author of that and he and many others were really um you know really instrumental in making group Theory the subject that it is today and I'm really grateful for that but more than that he's said to have had a really playful style to mathematics so he would toy with mathematics in play and entertain games and squeeze Theory even out of the silliest of ideas that no one else would entertain and I I just love that approach also on a personal note so it may surprise you to learn that I'm a bit of an introvert um but so is he but when he moved to a new area in his late teens there's this great story about him where he realized no one in this new place knows who I am so I could completely reinvent myself and be the most extroverted charismatic person ever and he did and he became known as the world's most charismatic mathematician anyway that was a very long and boring answer to that question and I apologize this has been another proof under another roof until next time [Music] [Music] thank you

Channel: Another Roof

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Id: IzUw53h12wU

Length: 80min 16sec (4816 seconds)

Published: Mon Sep 26 2022