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Prove zip field

hey there so in our previous episode we mentioned several sets of numbers and discuss the abstract notion of a group and we said that some of these sets namely the integers with respect to addition and the rationals with respect to addition and the rationals without zero with respect to multiplication are groups and we looked at it carefully and saw what the properties of being a group are and how they are satisfied by these examples okay and we ended with with a statement that rational numbers the the fractions the quotients are in fact more than just a group with respect to these two operations they're what's called a field so what I want to do now is first write the very very mouthful it's not just going to be a mouthful it's going to be a board full of the definition of what it means to be a field and as we write it we're gonna kind of keep the rational numbers in mind and and and understand that all these properties are indeed satisfied by the rational numbers so let's get going so here's the definition definition note that I'm reducing the font because I'm really gonna fill this board definition a set oops so a set is now just a collection a set F I'm gonna call it f for field a set F with two operations which I'm going to denote one by plus and the other by dot I'm just too lazy to write X all the times so these are the two operations and you should think rationals with plus and times with addition and multiplication so a set with two operations is called is called a field that's what I'm defining if the following hold the following eleven axioms eleven properties that have to be satisfied so the first property property number one for every a and B which belong to F can you read this for any a and B which belong to F for any two elements a plus B also belongs to F so when you take the first operation which we're gonna call addition okay add two elements the sum is also an element okay do you agree that when you add two fractions when you add two quotients of integers two rational numbers the result I hope everybody hears and here knows how to add fractions okay and and you may say hahahaha of course we know but just just take a look and verify that you know right we don't just add the numerators in the denominator separately right we need to do a common denominator make sure you know how to do this okay so for any two elements the sum is also an element what would this property be called what is the name of this property right closure closure and this is closure for the plus operation okay the second property still relates to the plus operation is the associativity for every a B and C which belong to F for every three elements a plus B plus C equals a plus B plus C right so property number two is the associative property for addition okay for the addition operation property number three note that all these properties so far are precisely the same properties for being a group okay property number three there exists so property number three is going to be the additive identity identity for addition and now we're not just gonna call it e we're just gonna call it zero okay there exists an element zero in F satisfying that such that a plus zero equals a for every a that belongs to F again tell me if you can read this if you feel comfortable with reading this there are hardly any words in English right it's all meth there exists remember this symbol there exists a zero element in identity for the operation of addition what does it mean to be an identity for the operation of addition that when you add zero to any element you just retrieve that same element it doesn't affect addition okay property number four which is the property of having additive inverses additive inverses namely for every for every remember the symbol for every exists for every a in F there exists minus a in F that's the inverse for addition what does it mean to be a dick an inverse such that a plus minus a is just that identity element is zero good everybody okay so these four properties together read if you think for a moment let's look at the next board which I left from the previous clip these four operations are precisely the four operations which say that a field is in particular an additive group a group with respect to the operation plus do you agree does everybody see that it's precisely these four properties good ok let's go back to this board and I'm gonna add here a fifth property of addition note that all these properties are still only properties of addition and this property is saying that it's not just the group it's a commutative group with respect to addition namely for every a and B that belong to F for any two elements a plus B equals B plus a do you agree okay so this is just one half of it this is just the properties that say that F a field is just an additive commutative group but it's more than that and now there are gonna be five more properties that say that these same things hold for multiplication with the little modification that the inverses exist for any element except zero so let's write it down I'm gonna write them a bit more quickly so six for every a and B that belong to F a times B also belongs to F this is closure with respect to multiplication seven associativity with respect to multiplication for every a B and C that belong to F a times B times C equals a times B times C that's associativity of multiplication eight the existence of a multiplicative identity there exists a 1 in F an element which we're going to denote by 1 such that a times 1 equals a for any a in F that's what it means to have a multiplicative identity property number 9 this is the property that's going to be a bit modified so for any a not equal to 0 for any element except this 0 element from property 3 except the additive identity there exists an element which we're going to denote by a to the power minus 1 in if such that a times a to the minus one equals one so every element has a multiplicative inverse except zero except zero okay so this means that a field is a group also with respect to multiplication these are the four axioms as being a group except that you need to throw out the zero element okay and then property number ten is gonna be the commutativity property of the product of the multiplication for every a and B in F a times B equals B times a so a field is also a commutative multiplicative group without the zero okay and then finally there's an eleventh property which I'm gonna write down here because it it relates to both of these operations and the eleventh property we call distributivity the distributive law for any for any a B and C that belong to F a times B plus C equals a times B plus a times C this is the distributive law with which binds these two operations which otherwise would be kind of independent right otherwise you'd have a structure that that or a set that has two very separate structures that don't feel each other but this says that in fact these structures are living together there's a relation between the structures okay okay so you agree that this is really a mouthful and what you have to do now is think carefully about all of these with respect to these integers let's go back one board for a second here's here's the sets of numb here are the sets of numbers that we've already encountered so Q over here was the integer sorry was the rationals or the quotients so you have to think does Q in fact satisfy all these eleven properties going back here you have to look at the properties one by one and think suppose we take an element I'm looking at property number nine for example suppose we take an element in Q let's say four over five it's not zero good does it have a multiplicative inverse is there an element such that four over five times that element equals one right of course five over four is the inverse of four over five the multiplicative inverse okay and you have to think about this otherwise you you see one big blur right this is really very scary okay but it's not you just have to read carefully get used to these two symbols which show up all the time exists and for any and this belong to sign and decipher the the abstract notion here and see that in fact it's really just stripping cue stripping all these properties of addition and multiplication in cube okay so let's write that let's write that I think I by now I can erase the the definition of a group in this course we're gonna spend a lot of time with with fields fields are going to be the structure that we the structures that we're going to consider many times usually they're gonna be the real numbers and the complex numbers not just rational numbers we're not gonna say too much about groups okay the group was just an easier thing to pick up it had only four properties rather than eleven that's why I started off with groups okay so example q with plus and multiplication is a field verify verify that you believe all these eleven axioms do it it's good practice okay so the question is are there numbers are there numbers that you already know that are not rational and right you may have heard the word irrational which kind of hints that there are and what are these numbers give me an example of an irrational number right pi is a good example I want it I want to discuss another one so here's the the real axis that kind of all the numbers that you know and suppose zero is here and there is a number somewhere with a little bit here which is called the square root of two okay and Pythagoras you may have heard of Pythagoras right you know the Pythagorean theorem in a right triangle right that when you take two sides the two sides adjacent to the right angle you take the square of one plus the square of the other you get the square of the hypotenuse everybody knows that right so in the time of Pythagoras this is about 2500 years ago in the sixth century BC very long ago in ancient Greece Pythagoras believed that the only numbers were rational numbers in fact they Pythagoras and his students were actually either that they made it into a religion okay they worshipped the numbers the rational numbers they thought that nature is governed by these numbers okay and they found very interesting relations between numbers and architecture and geometry and music and and very very interesting things but for them that the numbers numbers were these rational numbers okay what you can do with integers and they knew that they knew the the Pythagorean theorem of course their leader was Pythagoras and if you take let's say a right triangle with sides of length one that's very easy to do measure one or take a square a 1 by 1 square and chop it in half using the diagonal so here's a right triangle with sides that are equal to 1 so this is 1 and this is 1 so by the the theorem of Pythagoras the length of the hypotenuse is going to be a number which squared equals 1 plus 1 so that number we're going to call root 2 so there's no doubt that this number exists it's some real number you can measure it on on the ground or on on the board or whatever okay but then came along one of the students of Pythagoras his name was he passes he passes and he said well this number cannot be written down as the quotient of two integers it's not a rational number the proof is very easy the same proof that he passes gave you can find it in Wikipedia for example I'm not gonna do it here because it's kind of standard here in the Technion that it's done in the calculus course so I'm gonna leave it to them but he proved it he exhibited in a way that's really one cannot dispute that this number is not the quotient of two integers you cannot find two integers whose quotient is gonna square to be two okay and you know what they did they were baffled and then they were very disappointed and that I mean they're their religion was was worshipping these these numbers they took the guy in exiled him and another another theory says that they took the guy and drowned him they just couldn't cope but that doesn't change there are numbers which are not rational which are not in queue and and those so all the numbers that we know form another set of numbers which we call the real numbers let me write that write that down so this little picture proves or or or proves that that the number root 2 is a very real number and hypothesis proved that root 2 does not belong to the rationals ok and then therefore there is a fourth set of number denoted by r with a double-leg called the real numbers or the reals and they're called real because then there's going to be a fifth set of number which people really couldn't believe and they called them imaginary those are the complex numbers which will follow but the real numbers and I'm gonna cheat a bit that a formal construction of the real numbers is way beyond what we can do in this course it's very very delicate I'm just gonna talk to your intuition and say the real numbers are the set remember the set of all numbers X such that it's a point on this real line so X is anything between infinity and minus infinity okay this is really cheating but for us it's gonna suffice okay so the real numbers are any real number you can think of okay in particular root 2 and PI and E if you know e from the exponent e to the X and all these numbers are real numbers root 3 root 6 some of them may be rational and others are not a discussion of what is real and what is not what is irrational so not belonging to Q means it's irrational not a ratio the word rational comes from the word ratio it's not a ratio it's not a quotient of two integers and a discussion of which numbers are in which are not is a bit more complicated and we're not gonna do it let's just add a few more get getting used to our symbol so pi the number pi the the pi from the circle right the way the circumference of a circle relates to its radius so pi belongs to r pi is a real number but pi does not belong to Q so that again requires proof that pi is irrational pi is a real number but not a rational number another way of denoting this I'm gonna introduce it is that pi belongs to R minus Q so this back tilted slash here is minus of sets so pi belongs to the reals without the rationals so pi is irrational okay is is this clear to our without the rationals okay so all these are various ways of denoting so is so is root 2 by the way you can substitute grew to here okay so what one can what one can verify and we're gonna work with it a lot so some remarks remarks 1 R is a field it satisfies all the 11 axioms and you should think about it you should think of things that you know but think about them in order for them to to kind of to feel to feel good with them to feel that you understand what this statement means so you have to go back to the definition of a field which is whoa and think is it true look at this board please is it true that for any two numbers or any three numbers when you add them a plus B plus C real numbers two plus five plus 17 is it the same as root 2 plus patwa 17 and you know that it is okay these are things that you know but think about them see that you actually believe them that you actually understand these statements in concrete examples in a in in in the context of a concrete structure like the the rationals are the reals or the complex numbers which we're gonna learn later okay so R is a field that's one statement another remark that I want to make is that if you look at a set which contains only one element a set with only one element in that element we're gonna call zero okay a set with only zero in it and you look at those 11 axioms you're gonna see that this set with one element is a field a very boring and dull one but it's a field okay so 0 satisfies all 11 axioms but it's so dull that usually we don't consider 0 to be a field ok so usually the convention is that when we discuss a field there have to be at least two elements playing the roles of 0 and 1 and they cannot be the same but usually the convention is usually we require require oh boy we require that in a field zero and one are not the same element okay so that would force a field to have at least two elements zero and one and and if you take the set including zero in one that is is a field you can check all these eleven axioms okay it is a field so so there there is an entire branch of mathematics that deals with fields okay but we're really just gonna touch them we're not gonna we're not gonna go very deep into the theory of fields it's it's it's it's not uh it's a big theory a big theory I just want to mention there are more fields the first one is what we called C the complex numbers and these I'm leaving now is a promo for for the next lectures so we're gonna discuss the complex numbers later and in particular we're gonna see that they're a field okay but we first have to define them and define how we add complex numbers and how we multiply complex numbers and then gradually we're gonna see that all these eleven axioms holds for addition multiplication of these mysterious imaginary numbers okay and in fact there are fields there are more but I want to mention that there are fields which usually do know by denote by Z sub n Z sub n are finite fields namely fields which have only finitely many elements so the field which has only zero in one and the regular operations but you have to be careful there so for example if you have only zero in one what is one plus one it can't be two because two is not an element anymore so you declare one plus one to be zero okay Wow yeah so so finite fields with with the addition and the multiplication done in kind of a circular way where you go back to where you started it's called addition modulo N or multiplication modulo n this is also a topic that requires a longer discussion if you want to really understand what's going on there we're not it's not in the scope of this course I'm just mentioning it okay so there are these finite fields and and these Z ends are only fields when n is prime okay but there are there are fields there are finite fields with not necessarily prime a prime number of elements but but that's really beyond us for now okay so so these are examples so for example Z 5 is the field that contains the element 0 1 2 3 & 4 where you add modulo 5 so for example 2 plus 4 is not 6 it's 1 because 2 plus 3 is 0 so 2 plus 4 is 1 & 2 plus 5 is 2 ok Wow hmm right very confusing so if you're interested in this you can look up finite fields or Z ends and in the internet and read about them it's very cool very cool and and relates to many many things you know okay so for example your if you have a watch on your hand and not a digital one but an old-fashioned one with you know the watch real wha it it really adds according to z12 right because when you take the hour to and you add 11 you don't get to 13 you get to 1 again right so that's addition modulo 12 okay anyway that's all I want to mention about these so the last thing I want to do is see that there are more things that are valid for fields many many many many many more little kind of properties that are satisfied and the reason that we don't list them down in these 11 axioms is because we don't need to they follow from the axioms we can prove them knowing just the axioms and that's what I want to do in the next clip kind of show you how we formally prove things in a very abstract setting okay again it's something you have to get used to but at first it looks whoa it looks like we're proving things that are what do you need to prove here for example I'm going to prove that that for any number a times zero equals zero that's not one of the axioms okay let's look at the axioms again there's no property that says that if you take a and you multiply it multiply by the additive identity a times zero not a plus zero a times zero but a times zero is always zero okay if you think of real numbers you know that right 7 times zero is zero five times zero is zero pi times zero is zero right but it's not an axiom here how do you prove it how do you show that it necessarily happens even in this abstract setting so that's what I want to do next