okay I am starting the chapter summaries now for Pure year 2 and hopefully these should be really good for some revision and just to kind of summarize a playlist so this first one we're looking at is proof by contradiction and I've got it chapter 1A which is um part of the algebraic methods chapter really because the first part of this chapter and the second part are very different and I think it makes sense to have them separated so for proof by contradiction there's really only a few things you need to know for this it's really just doing a lot of practice and getting into the habit of how these go so what you do for a proof by contradiction is you start off by assuming that the opposite is true this is the the kind of rather broad part that I've got here and then you just do some maths you'll see what I mean by that with some examples and then what you do when you're doing the maths is you find the part that doesn't work or doesn't make sense and because that part doesn't make sense then the thing we assumed at the beginning cannot be correct which means we can say that the original statement was therefore true so we always finish off by writing down our conclusion hence by contradiction blah blah blah blah blah you must have that that conclusion statement to be able to get that Mark so it says for the first example prove by contradiction that there exist no integers X and Y such that 15x + 20 Y is equal to 1 okay well we better start off and assume that there do exist some integers that this is true so it's going to be a lot of writing in this video so I'm going to say assume you can just say assume but I always like to say assume for contradiction to kind of make it clear that I'm going to to do appr proof by contradiction that there do exist that there do exist integers X and Y such that 15x + 20 y = 1 so we're just going to pretend that this thing that we know is not true is true and we're going to try and show that something doesn't make sense now number two says to do some maths well when I look at this so when I try and think what math can I possibly do with this I look on this left hand side and my brain is instantly going oh there's something to do with the five times table here so we can either factor out of five we could divide by five I know my instinct is probably to factor out of five if I factor out of five I would have a 3x + 4 y = 1 okay and now I'm probably going to divide by that five so that I get that 3x + 4 Y is equal to a 5 and I can't really think of any other maths that I can do here so I guess I need to now try and find the part that doesn't work or doesn't make sense well we've just assumed that there are integers X and Y that exist well if X and Y are integers and they're being multiplied by an integer it's never going to be able to equal something that is not an integer so here is the thing that doesn't work or doesn't make sense if X and Y are integers we're never going to be able to get something that equals a fifth so I'm going to just write that down and I'm going to write down a conclusion statement so I'll say as X and Y are integers 3x + 4 Y is also an integer but cannot equal 1 as it is not an integer I'll say that it is not an integer so we have a contradiction so I can say hence we have a contradiction we have the thing that went wrong hence we have a contradiction so much writing in this video and therefore there exist no integers X and Y and I'm just going to put here Etc and then just finish off the conclusion you need to finish that off I'm not going to waste your time in this video writing out that sentence again okay we got a different one so now says prove by contradiction that there are no positive integers A and B with a odd such that a + 4 b equal 4un a so we're going to start off with our assume part we are going to assume for contradiction and there's so much writing like I keep saying assume for contradiction there are positive integers A and B with a odd such that a + 4 b equal 4un a now we get to the fun part of doing some maths well look at this what maths could I possibly do to this well there's a square root sign here let's try and get rid of that square root sign it kind of feels like the only thing we could possibly do so what I'm going to do is square both sides now if I Square both sides I get that a + 4 B is equal to when I Square the four I get 16 and when I Square the root AB I get the ab like this oh that should have the square sign so the left hand side is a s we would then get a 4 A and A 4 a that's 8 A+ 16 b^ 2 and that is equal to 16 a well this kind of looks like a quadratic so I'm going to take this and I'm going to subtract it from this side so that I get a s 8 a - 16 a is - 8 a + 16 b^ 2 is equal to zero now this thing that we've got here we know that this thing is the same as this the only difference now is that I've got a minus here so that means I should be able to quite easily factorize this one that I've got I should be able to factorize it really quickly now if I factorize this I would get an A minus 4 b^ 2 is equal to Zer now that's a really hard step to spot but really there's not much math that we can actually do in this situation other than squaring it and kind of playing around with making equal zero this is why people find this topic hard because it's very much what should I do next now that I've got to this stage I can see that this part must be equal to zero so I get that a is equal to 4 B which is the same thing as 2 * 2 B and so I'm going to come up here for this part we can now tell that a is even because it's B multiplied by 4 or 2 B multiplied by 2 but we have assumed that there are integers where a is odd so we have a contradiction so I'm going to say hence a is even and we have a contradiction because we said it was odd we have a contradiction so I'm going to say hence and I'm going to say Etc make sure that you write out the conclusion in full for this because there's always a mark for getting that last part with that complete con conclusion sentence now I know what you're going to say you're going to be like how would I ever notice this well the doome maths part is the thing you're going to start getting used to square root sign let's Square both sides brackets let's expand them okay it's kind of like a quadratic quadratics equal zero can I factorize it it's hard I get it it's it's a hard thing to do and I'm going to look at three examples to do with irrationality now irrationality kind of has probably been mentioned a few times but this is probably one of the most important uses of something being rational or irrational now let's talk about something being rational to begin with if a number is rational it can be written in the form A over B where A and B are integers with no common factors and I'll explain a bit more about these no common factors afterwards well clearly three is a rational number we could write it as 3 over 1 - 4 over 5 is the same as - 9 /2 and 2/3 I mean it's yeah 0.6 recurring that's a rational number because we can write it as 2 over3 like this now examples of things that are irrational some irrational ones that are useful to know obviously Pi E I don't know even like Pi + 1 that's an irrational number all of those are examples of things that are irrational oh and of course the square root of two lots of thirds are irrational so we're going to look at this one it says prove by contradiction that if x is rational and Y is rational then X Plus Y is irrational so the thing we're going to try and flip around here is this last part of the sentence we're going to prove by contradiction that if x is rational and Y is rational we're going to assume that X Plus Y is rational so let's assume and you don't have to say this a little bit but I like to say assume for contradictions so it's kind of making it clear what I'm about to do that x + y is rational that's the part we flipped the meaning on its head instead of being irrational we've said it's rational so we can say X + Y = A over B that's what we can say for rational numbers we've also said that X is rational and X is going to be well I'm not going to do A over B because I've already used those letters so I'm going to say x is equal to C over D and you can either write this in words or with symbols but I'm going to say where a b c and d are all members of the integers if you don't want to write it using this language you've not seen it before this just means are members of The Zed represents the integers that we've got here so we've now said this one is rational and this one is rational people often say well how represent an irrational number we can't represent an irrational number so we're not going to say oh y equals something that is irrational we can't do that so we're only going to play with the ones that actually can be written as rational numbers now what's the math that I can do here well I guess I could try and find out what Y is because I know that this x I could substitute in here so if I take that X and substitute it in I get that c over D + y equal A over B so Y is A over B minus C over D now this looks like it's going to be rational but I really want to make it explicitly clear that this is going to be a rational number by making it one single fraction so I'll do my cross multiplying I'll have my a D minus my BC over b d so I can now just do a little bit of explanation I can say um a D minus BC is an integer as is b d hence Y is rational because we've got an integer divided by an integer but we've said in the question we were saying that Y is irrational so this is a contradiction this is a contradiction because originally y was irrational so I'm going to say hence and then I'm going to write Etc conclusion never write what I'm writing here in a test this is me just saving some time in this video okay this is probably one of the most famous proofs prove by contradiction that the square root of two is irrational so what I'm going to do is I going to assume the square < TK of two is rational you notice how I didn't write by contradiction you don't have to it's perfectly fine to do it this way as well so if I've assumed it's rational I'm going to say that it's A over B now I'm going to do some maths the doing some maths here is let's get rid of that square root sign so I get 2 is equal to a^ 2 over b^2 so this means that a^ 2 is two lots of b^ 2 now what does this tell us so a s is even now you could do a proof for this as well you don't need to do this but if a s is even then a must be even as well so a squ is even whoops a squ is even and so is a now if a is even let's let a be equal to something so a could be equal to 2 K so we now go back from this line that we've got here we have that a 2 which is 2 k^ 2 is equal to 2 b^ 2 so that's going to be a 4 k^ 2 is 2 b^2 so we then get that b^2 is equal to 2 k^2 which means that a oh that b s is even and hence B is even so b^ 2 is even and so is B now I'm going to come up to this part and say um I guess I should have said here where A and B are integers with no common factors you'll see why I need that are integers with no common factors that was one of the things we talked about with irrational numbers up here okay so going back to this we've said that b is even and and so is B so I can say as a and b are both even they have a common factor so we have a contradiction because that was one of the things that we put in the assumption that A and B are integers with no common factors so I'm going to explain why that even means anything so I'm going to say hence Etc conclusion so let's just think about this right if we said that they had a common factor if they both had a common they both even so we can divide them both by two well then that means and this is a part you don't need to write down the square root of two could be I don't know instead of it being A over B it could be written as C over D and then we would go through this process again and we would say oh both C and D are even so we could cancel them by a factor of two each time and we get that the square root of two is now equal to e over F but e over f are also going to be even and so we would end up with numbers that you can continually divide the top and bottom of the fraction by two because they would always be even numbers and that doesn't make sense there is no fraction there is no number exists that you can infinitely cancel it down by dividing the numerator and the denominator by two eventually you're always going to get to a fraction in its lowest term so the fact that the A and B are always going to be even numbers in this case means that we would eternally be able to cancel this fraction down which doesn't make any sense that's the issue in this part but even if you just don't have to explain that you can just show that they both have a common factor and then I like this one here we're going to prove by contradiction that log base 2 of 7 is irrational so I will just assume log base 2 of 7 is rational so log base 2 of 7 is equal to A over B and I will say that A and B are integers and have no common factors it's not going to actually be un needed in this one the fact that they have no common factors but it's worth always using this so now let's do some maths let's rewrite this statement so the base is two and the power is A over B and that is equal to seven well the math kind of feels like I want to do something with this this is a over B if I do both sides to the power of B I get 2^ of a = 7 to the power of B and now we need to think to our does this make any sense can we ever have where A and B are integers can two to the power of something be equal to seven to the power of something and the only way that this could work is if a and b were equal to zero so we this is doesn't make any sense so we can just say this cannot be true unless a = b = 0 the reason why is because if you think about it you remember when you did like um prime factor decomposition where you could say something like 12 is equal to 2^ 2 * 3 when you do stuff like this 12 is equal to 2 * 6 which is 2 * 3 any number is going to be built up of different prime numbers and in this case something that's built up of twos can never be equal to something that is built up of sevens it's pretty easy for you to see why this cannot be true and if you do need to explain it we could say the left hand side is always even the right hand side is always odd lots of different ways of explaining this so I will say hence we have a contradiction and I'm going to say hence blah blah blah Etc conclusion now this probably is enough to to say this cannot be true but I think we should probably say why unless a equal b equals z and I'll say this is because let's just add in because the left hand side is even and the right hand side is odd but it might not always be even and odd it could be things like I don't know a different example could be if we had like 2 to the^ of a being equal to 5 to the power of B you just need to explain that two to the power of something can never be equal to 5 to the power of something unless those powers are zero in this case we had even and odd but that case it wouldn't be because um 5 squ actually no five is always even an odd as well so anyway I'm digressing now going on to different parts so um if you're watching this video in 2024 in sort of the winter of 2024 maybe January 2025 um do check out my bison matths award it's a scholarship that is available to five students there is a video that's Linked In the description um if it's past this maybe I'm also going to be running a scholarship in the future so you can check that out as well so if you do want to use um any of these questions that I've got here or this annotated PDF I've also linked that in the description too so good luck with your studies and I hope to see you in another video soon