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Okay number field

alright thanks for watching in a previous video we defined what the field is and we said what are as a field and that's what makes it it's more special than its cousins the integers are the natural numbers but here's the thing R is actually much more than a field because you can also compare real numbers let's say to less than or equal to 3 you can say that and in particular that's what makes it our more special than the complex numbers because in the complex numbers we cannot compare them and there's a video on that as well so in particular R is an example what's called an ordered field so definition is if it has a structure so again if F is a field and again it has a structure Leicester equal that satisfies the following and I get a B and C are numbers in that for all a be either one is less than the other or the other one is less than the other so either a is less than equal to B or B is less than or equal to a so that's what's called trichotomy and moreover suppose both cases on a is less equal to b and b is less than equal to a that gives you a equals b and then there's something called transitivity just like for equal a less than equal to b and b less than equal to C implies a less than equal to C and then all of you remember from middle school elementary school so addition doesn't affect inequalities so a less or equal to B implies a plus C that's your equal to B plus C unless what not least same for multiplication except you and you know restrict yourself to positive numbers so a less or equal to B and C is positive implies a see Lester equal to B see also so again those properties hopefully they're obvious to you but but again that's the whole point it's supposed to mimic life you know the real numbers are and just a couple of remarks of course on the examples right or R and Q so again we still haven't told still haven't told you what makes R so so much better than Q but of course some nominee examples are the complex numbers because we cannot order complex numbers but of course m and Z but that's because you're not fields very important and ordered field has to be a field so that's super super important and also by the way so with the far less your equal how do you define greater than equal so note we defined a is greater equal to B if only if P is less than equal to a and not only that is also strictly less than so a strictly less than B means a is less than equal to B and a is not equal similarly for a strictly greater than alright and again just like last time just like for fields for ordered fields we can also have a lot of properties that hopefully seem obvious to you so theorem so first of all if you take negatives then it changes the order so a less than equal to B implies minus a is greater equal to minus B and why is that true it's actually a very neat proof take a is less than equal to B but I had a very complicated element to this so minus a plus minus B so if you add minus a minus B to this equality then what do you get again and we can do this because addition preserves order then well you can cancel that out and you get 0 plus minus B is less or equal to again B plus minus B so 0 plus minus a and therefore minus B is less or equal to minus a how cool is that I didn't know the proof before but it's very well in same with what's called multiplication so as I said multiplication my positive numbers affects the doesn't effect the order but the point is multiplication by negative numbers will affect the order so if a Lester equal to B and C is negative then AC is greater equal to BC and how do you show that yes so the Walkman is simply if C is a positive negative numbered and minus C is a positive number and that's that the property we've just shown so by one C is less or equal to 0 implies minus is greater equal to 0 so it's totally fine to multiply this inequality by that positive number so a less than equal to B implies again minus c/a let's do an equal to minus CB again by this oh five I think but then that's just minus C a is less than equal to minus CB and then you just use a property 1 again so then I can end the fact that minus minus is plus so the CA alright good [Music] well if you multiply two positive numbers you get a positive number DC is greater equal to zero but I believe that's just oh five with a equals one and then you can also show it but obvious but the square of any positive number is positive and that you do basically by cases so first of all note by trichotomy either a is greater or equal to zero or a is less than equal to zero and then you just do it by cases so case one well if it's greater equal to zero then you can just multiply a by a the positive number a so a it's kind of interesting it's kind of met up it so you using this property to prove this property so biopharm and therefore a squared is greater equal to zero and then if a is negative in similar so also in both cases you find that a squared equals one it's positive numbers all right throughout this and then well you can also show that 0 is less than 1 and it essentially follows because 1 is 1 squared so it's a square of a number and therefore you can use for but you would have to show that 0 is not equal to 1 oh that's also interesting um in other words again we have remember we define the inverse a inverse and who tells us that a inverse is still positive so and that's exactly what this is saying so if a is strictly greater than 0 then a inverse is strictly greater than 0 and why is that true because suppose it is strictly greater than 0 but a inverse is less than equal to 0 then what you can do you can multiply this by a so then a inverse a is less than equal to a 0 a a again that's by all five but again the definition of a inverse is this gives you one so then one is less than equal to zero but that contradicts the fact that we've just shown that one is greater than zero so yes contradicts with I believe the stuff we've just shown okay well similarly you can also show that if a is greater than B so seven if a is greater than B is greater than zero then taking multiplicative inverses reverses the order a inverse is less than B inverse and again I'm gonna skip that for now because it's a nice exercise to do at home alright so here's a so we talked about fusing ordered fields right and we said well what makes are so special so different from the integers or the complex numbers is that R is an ordered field but it still doesn't tell us what makes our better than the rational numbers and unfortunately today we won't really find that out because we need one more concept from Section four but let me already tell you a little bit let me also already spoiler dis surprise a little bit but first of all what's nice is you know derived of real numbers are bigger than rational numbers so in other words what that's what's called a selfie so a subfield a subfield a or b is a subset of B that's also a thing for instance again quintessential example the rational numbers are a subfield of the real numbers because you have the real numbers and you have the rational numbers and Arsenal and so it's a subset of the real numbers it's also a field so that's one thing that makes the real numbers kind of special is that like it's bigger than the rational numbers but again that's still not why we care so much about the real numbers here's one we care about this because it has one property that two doesn't really have and that's what's called the least upper bound property so fact there exists actually just wanted an ordered field again first thing that makes our so special it's an order field arm that contains all concert' contains two Q as a subfield so it's bigger than the rational numbers and that also satisfies what's called the least upper bound property and again this we'll talk about in a section for again it's really the main thing that will distinguish the real numbers from the rational numbers but what does that mean heuristic lis well he recently all this means is the rational numbers even though they're or an ordered field they have a gap so really the rational numbers you can picture them as follows so it has a bunch of numbers like minus 1 3 halves I don't know 1750 or something ok but the real number is what's nice is they don't have any gaps so in other words notice the rational numbers is skipping a bunch of numbers like square root of 2 or pi but then the real numbers know they're a continuum they contain of course irrational numbers but also stuff like square root of 2 maybe somewhere like that and then also pi and other numbers and you see there are no rules and no gaps in the real numbers and that's why we like this in analysis cuz in analysis we would like to stuff to have one piece and not be not have any gaps so again hopefully it make you excited a little bit for a section 4 and it concludes our adventure of you know what is a real number so it's really this thing that's super important and I believe there's just one of them those sets that satisfies this alright that's it for today I hope you like this if you wanna see more math please make sure to subscribe to my channel thank you very much