The Math Major (Part 2)

Moving on from the last video, let's start with abstract algebra. The basics of this class is the study of groups or group theory. But what is a group? Well a group is a set of elements equipped with an operation like addition or multiplication that satisfies four conditions, but it'll be easier to show you rather than just list the conditions. So let's take the set of integers from negative infinity to infinity, and we're going to use addition as our operation, which again you'll see what I mean. Now we want to know is the set of integers a group with addition as the operation? First off, if we add any two integers, any two numbers in our set, do we get another number in our set? And the answer is yes. Add any two integers and you get another integer. This means the set is closed under addition, and this is one condition that a group must satisfy. Next, is there a number in our set such that if you add any other number in the set you get the same number out? And yes, that number is obviously zero, which is in our set. That number is called an identity element. In our case it's zero, but if we use multiplication as our operation it would be one, because any number times one is itself. Next, for any number in our set, is there another that we can add to it which gives us our identity element or zero from above? And again, yes, it's just the negative of itself in this case, which means every element has an inverse. And if you add multiple numbers together, does how you group them matter? No, it does not. These are the same and equal 19, which means the set is associative. So if a set is closed, has an identity, all elements have an inverse, and it's associative, then the set is called a group. And we've just shown that the set of integers is a group under addition. When I say addition is our operation, notice how we used addition for all of these. If we use multiplication as our operation, then those would all change to multiplication signs, and we'd have to check everything again. Now this may seem weird, why do we learn something so specific? But this is just the beginning, this is like day one. And sets don't even have to include numbers. Let's say you have three objects, which I'll just label as one, two, and three. In how many ways can we change the order of these? Well we could switch the first two, and then keep the third the same, and we can call that move T1, or transposition one. We can also switch the order of the second and third, and call that T2. And we can switch the first and third, and call that T3. But there's also more, we can move them all to the right, and cycle the last one back, and call that C1, for cycle one. And the same thing, but the other way can be cycle two, or C2. Then there's one more, which is the easiest, and that would be to keep them all the same, and we'll call that E. So if we look at the set of all moves you can make, is this a group? And you can actually show that it is. Just like with the integers, if you add two together, you get another integer in our set, which means it's closed. Well if you do like T1 on C2, as in take some setup, perform C2, where you cycle everything to the left, and then perform T1 on that, where you switch the first two, we get 3, 2, 1. But this is the same as taking the original setup, and just performing T3, or switching the first and third if you remember from above. So we can say that T1 on C2 equals T3. And you can show this for any two moves, meaning that since you get something else in the set, the set is closed, just like all the integers added to another one in our set. And I'm not going to, but if you wanted to, you could show the other three criteria as well. This is called a permutation group, and is something you will learn about. And this could go as far as to apply to a Rubik's Cube. You can represent it as the group of possible moves that you can make, and also why if you look up abstract algebra on Wikipedia, the first picture that comes up is a Rubik's Cube. And I'm going to move on in just a sec, but note this class is not about being given a set and just trying to figure out if it's a group. This is just the basics to give you an idea of what a group is. You'll do things like prove there is no formula to solve for the solution to a fifth degree polynomial, as in the zeros. Unlike a second degree polynomial, you can use a quadratic formula. There is not one of these for a fifth degree, which is definitely not obvious. Or you might be asked to prove or disprove that some set is a subgroup of some other group, which is a group within a group. I told you about groups, but there's rings, fields, and more to come. So you see there's a lot of proofs again, and it gets complicated, but it's nothing you need to know now. Moving on, the next big field is topology. Topology is really about the study of shapes, but you're going to study very abstract shapes. Now in normal geometry, you learn about shapes like circles and triangles, and in that class we cared about lengths, area, perimeter, and more. But in topology, we don't care about that stuff. In fact, in topology, the shapes can be thought of as infinitely stretchy. You can morph, bend, and twist them all you want, but you cannot make holes, tear the shape, or fill in already made holes. So in topology, a coffee mug is the same as a donut because you can morph one into the other without making new holes or anything like that. When two shapes are topologically the same, we call them homeomorphic. So a sphere, a cube, or a tetrahedron are all the same in topology and are considered homeomorphic because we can morph them into each other. And one thing you'll do in this class is prove that two things are homeomorphic. In fact, real quick, on a cube, if you count the vertices or the corners, you'll find there are eight, four on the top and four on the bottom. If you count the edges, there are 12, four on top, four in the middle, and four on the bottom that connect the corners. Then if you count the faces, there are, of course, six. If you subtract the first two numbers and then add the last one, you get two. And this is known as the Euler characteristic, or the vertices minus edges plus faces. Then if you do this with the tetrahedron, you'll find there are four vertices, six edges, and four faces. So you do four minus six plus four, and again, the Euler characteristic is two. In fact, if any polyhedron is homeomorphic to a sphere, then its Euler characteristic is two. And you can use this to prove that something is homeomorphic to a torus, a double torus, and so on, which all have different Euler characteristics. But again, this is just the basics. This class is not counting faces and edges of a shape. It's a very rigorous course, and you'll do lots of proofs and analysis, such as proving why Euler's characteristic works. That's something that pure mathematicians have to do. They develop and prove these formulas that we use in math. And in the class, you even get into higher dimensional objects like this Klein bottle, which, by the way, has an Euler characteristic of zero. And you may be wondering why topology is important, but it actually can be used in computer networking, or in quantum field theory and physical cosmology due to the mathematical properties of the universe. But as a math student in this class, you're really just going to do problems in math. And the last class we'll talk about is real analysis. This is really about analyzing calculus concepts in a much more rigorous way than you did in Calc 1. You'll go much more in depth on limits, continuity, differentiability, and so on, which you learn in first year calculus for those who haven't taken it. In Calculus 1, or even before, you learn how to really look at a function and know that it's continuous because there are no breaks or asymptotes or anything like that. But in this class, you're going to learn it on a much deeper level in a way that a mathematician would want to see it. Now this class is called real analysis because it's about real numbers or real valued functions, as opposed to a complex number that has an imaginary component, which is left for a complex analysis. Now you might be thinking that we already know what the real numbers are, like 1, 2, e, pi, and so on. But there's a lot more abstract concepts to come that break everything down into much more detail. It's very hard to show most of the concepts in this class, but one example of a basic proof you might learn So like between 1 7th and 1 6th, which are both rational, just a division of two integers, is there a rational number? Well, this is easy to show that there is. We could take the midpoint of them by adding them together and dividing by two, and we get a rational number, just a ratio of two integers. Well, we can easily show this for any two rational numbers, m over n and p over q, let's say, where all those variables would be integers. If you take the midpoint by adding the two together and then dividing by two, then if you simplify, you would get this expression. And since all the variables are integers, then the top and bottom of this will be integers as well, and thus you get a rational number out. This proof isn't that hard, but then they could ask you to prove that between any two real numbers, a and b, there is also a rational number. So now a and b could be anything that is not complex, like it could be pi, e, the square root of two, or any infinite decimal, and you have to prove there is a rational number in between them. I won't do it here, but that's just a hint of what's to come. These proofs I showed are something you'd learn very early on, and they do not do justice for the class. But I wanted to show you that, yes, we may all know what real numbers are, but there's way more depth than you probably assume. And I'm going to stop there with pure math, even though there's much more. Pure math students will take combinatorics, complex analysis, which can be very proof-based, and so on. But the proofs class plus the other three I discussed are the very famous classes you take that introduce you to higher level math. And the proofs in these classes might not be what you think. If you look at a pure math exam, you would see plenty of math and symbols, but you would also see a lot of sentences. You might see things in the solutions like, suppose that something, like h is a subgroup of g, and x is an element within g, then by some theorem we can show that, and so on and so forth, as you work your way to a solution. But you will have to construct these proofs during exams on your own. Unlike in geometry, the proofs you do in college are not as structured, like step one and step two. It's not just computation either, like the math you've been doing. You have to use logic and reasoning to reach a conclusion. Even experienced mathematicians with PhDs will talk to each other about a new proof they're doing, and ask things like, what do you think about this? Does this make sense to you? And so on. And that's because it's about logic and reasoning. If you've never seen the square root of two proof, then I'll link one below, and if you're new to proofs, you might not have that aha moment your first time seeing it. You might be wondering why that completely proves that it's irrational, and that's because proofs can be like that. But now to sum up all the classes, there is overlap with applied math students and pure math students. Regardless of which path you choose, you will take the calculus series, linear algebra, differential equations, that first proofs class, one abstract algebra course, and a real analysis course. Then as an applied math student, you take classes like vector analysis, numerical analysis, partial differential equations, and more. All the things that have more obvious applications to other fields. Whereas a pure math student might take topology, complex analysis, more abstract algebra, more real analysis, combinatorics, and so on. But this does not account for everything, and it will differ slightly from school to school. Like both these disciplines might take complex analysis, because it does have a lot of real world applications. At some schools it could be required, and others you might be able to choose it as an elective, but it's still a very popular class either way. Overall most of what you see here you can expect in those respective fields, but many higher division electives are up to you. And let's finish with careers. Many pure math students go on to do research in academia, as in they become professors where they teach, but also do research in these pure math fields. This is one of the very few ways to do work in pure math. Because if you want to work for the government or google, and do research in things like cryptography or artificial intelligence, then the math you would do would need to apply to something they are working on, which is applied math. You could use aspects of pure math, but it still must apply to something that they need. And to do research in pure math you do need a PhD. With just a bachelor's you will not be doing much work in pure math. Now if you do just get a bachelor's in applied math or pure math, it can lead to many fields, but usually not as an actual mathematician ironically. You need further education for that, but you can get jobs in software development, which is really common, or in finance, you might qualify for some engineering positions, and so on. If you're thinking that you just want a bachelor's and then a job right out of school, but aren't set on any field, you could do math, but maybe consider computer science or engineering and a minor in math to give you more direct exposure to those fields. With a master's or PhD in applied math, then you could work in academia and research, but you can also do research at private companies or the government. The largest employer of mathematicians, at least in the US, is actually the NSA or National Security Agency. Mathematicians here work in order to analyze intelligence data, do signal analysis, break cryptographic systems, and more. Defense companies can hire mathematicians for things such as missile guidance algorithms, and like I said, finance companies need mathematicians for lots of data analysis. Then companies like Google, IBM, or Microsoft have research divisions which can hire mathematicians as well. Although research positions often require a PhD, and these jobs aren't quite as abundant as being a software developer for Google, let's say. Then some more specific routes you can go include operations research, where you do mathematical modeling and statistical analysis to maximize profit and performance, and or minimize risk and cost for a business. You can be an actuary, which is actually a big field where you analyze statistics and calculate insurance risks and premiums for things like life and car insurance. There's of course teaching, and there's plenty more. So many of these fields involve math, but also other things like statistics, programming, and so on, which is why I said it's common for applied math majors to get a minor, because it's not all just doing math. But I'm going to end there. I hope this gave you a good idea of what you can expect as a math major, and what it could lead to. If you like this video, don't forget to comment, like, and subscribe, and I'll see you all next time.