 In the previous video, we talked about some important definitions related to numbers, sets, functions, and operations. And if that felt really fast for you, don't panic. It might seem fast, but it's really going to be okay for us. In this section, it often feels like overload. But like I mentioned in the previous video, chapter one is all about exposure. It's not about deep comprehension yet. That'll come with time, experience, and practice. Really, what we're looking for right now is just exposure. And so the main idea for section 1.2 is the idea of a field. And so a field, first of all, is a non-empty set. So that means there's something inside of this set F. It contains a bunch of elements. It could be a finite set or it could be infinite, depending on the type of numbers we have in play here. Now, when one describes a field, the elements of a field are typically called scalars. This will be in contrast to like a vector quantity, which we'll talk more about next time in section 1.3. So a field is a set that contains numbers, which we call scalars. Now scalars are numbers, but there are different types of numbers. Scalars are only just one type of number. So a scalar will be a number that belongs to a field. Now the field is a set of elements called scalars, but also the field will contain two operations, which we call addition and multiplication. Now this is where things start to get abstract. The symbol we use to describe addition on a field will be a typical plus sign. The symbol we use to describe multiplication will be typical multiplication. We might draw a dot, we might draw an x, we might draw a star, or we might just simply juxtaposition two letters together and there's no symbol whatsoever. That is to represent multiplication. So we use the same notation we use to describe addition and multiplication of real numbers in these field settings. And the reason we're doing that is that the idea of a field is trying to capture what are the important ideas for the real number system. And can we extend that into new realms, new settings? Like if we were a biologist and we discover this brand new island with a brand new species, how would we classify these? It's like, well, that thing has feathers, it's warm-blooded, it flies, that kind of feels like a bird. So maybe we classify it as a bird. Then we look at these other creatures. It's like, well, they seem to have warm blood, they have fur, they give life birth. They give milk to the young. That seems like a mammal. Let's classify that. Then you find a platypus and it's like, what in the world is this creature? It lays eggs, but it has fur, has like a bird, a duck bill, but a beaver tail. It's a secret agent that fights evil scientists. What is this creature? We don't know. And so classification is often, when you find something new, we want to classify it based upon things we've already seen before. And so that's what we're trying to do with a field is we're trying to look for what properties of the real number system can be found on other algebraic species. And so it turns out that the real number system has 10 properties, 10 algebraic properties that we call axioms, which are going to be the definition of a field. And these are properties that the real number system demonstrates that we want to find in other algebraic species. And so these 10 properties are listed on the screen right here. The first one tells us that if we add A plus B, that should be the same thing as B plus A. This is commonly referred to as the commutative property of addition. We also will require that multiplication be commutative. A times B is equal to B times A. Now there are many algebraic settings where the operation is commutative like real number addition, real number multiplication. But as we go deeper into this island and find more creatures like platypuses, or is it plata, plata pi? I don't know. When we find more platypuses, we're going to see that there are some operations which are non commutative. This will be true for matrix multiplication. If you have two matrices A times B, that is not the same thing necessarily as B times A. But for fields, we will require multiplication and addition to B commutative. The second properties are what's commonly referred to as the associative property, which allows us to redo parentheses. If you take A plus B and then you add C, that's the same thing as adding B plus C and then adding to that A, right? So you add the first two, then the third. That's the same thing as the last two, then the first. We require on a field that addition is associative. We also want multiplication to be associative, right? Associative property will hold for addition and multiplication. The third property is going to be about identities. There exists an element which when you add it to any other element, you always get back to the original element. So this element we will typically call zero. It's the additive identity. The additive identity means that A plus zero is equal to zero plus A, which is equal to A. Adding zero doesn't do anything. We also require there is a multiplicative identity, which we will call the number one. One has the property that A times one is the same thing as one times A, which is just A itself. Multiplying by one didn't change the other number. And so we require that for field operations addition and multiplication have identities. The fourth and ninth property will be inverse properties. That is, there exists some element for which when you operate by it, you'll get the identity. So given any element A, there exists an element which we call negative A, which has the property that A plus negative A gives you the additive identity zero. And of course it's commutative, so negative A plus A gives you zero as well. So we can cancel out addition by A by adding negative A. It gives you zero, which the net effect of adding zero is nothing. We require the same thing for multiplication that given any non-zero element. So we take some non-zero element, there has to exist a multiplicative inverse so that A times A inverse is equal to inverse A, which is equal to one. Right? When it comes to additive inverses, we typically use a negative symbol, negative A to describe it. When it comes to multiplicative inverses, we might use this superscript negative one, which this means one over A, the negative or the multiplicative inverse is the reciprocal. Now, when it comes to these axioms, the addition and multiplication are analogous to each other. There is a slight variation when it comes to axiom nine because when it comes to multiplication, we do not require an inverse for zero. We actually forbid it. And the main reason actually is a consequence of the distributive properties that happen down below. If we allowed zero to be invertible and we had the distributive properties, it basically would blow up our algebra. We might talk about that some other time, not so much right now. The last properties are the distributive properties that this is how addition and multiplication interact with each other. If you have A times B plus C, so if you add B and C and then you times by A, that's the same thing as times in A by B and A by C and then adding that together. So we could add first, then multiply the distributive property that says we could then multiply first and then add. So how can we switch the order of operations there? And then we can also just this is the left distributive property. We can also have the right distributive property which says that, and this is our typical distributive property, right? If we add A and B, then times by C, that's the same thing as times by C first and then adding the resulting products. And so these are the axioms of a field and like I said, it might seem a little daunting, a daunting list of 10 axioms. But remember the following, a field is just a number system which we can add, subtract, multiply and divide. So we start off by defining addition and multiplication. But because of the commutative properties and because of the existence of inverses, we actually do have subtraction and division, right? Subtraction is just adding the negative. Division just means you multiply by the reciprocal. So a field is just a number system for which we can add, subtract, multiply and divide using the usual properties of commutivity, associativity and distributive properties, distributive properties. So some examples of fields that we are familiar with. The rational numbers make a field, the rational numbers. By rational numbers, what I mean is, well, this is often denoted as a Q with this extra line in it, Q standing for quotient. We're taking the set of all fractions of the form A plus or A divided by B, where in this setting A and B are integers. The integer set is often denoted using this Z with a double line in it, but we require that B does not equal zero, right? So the integers, let's mention that one here, the integers denoted by this Z right here. The reason we use a Z is actually comes from German, where the German word for number actually starts with a Z. So it's a mnemonic device borrowed from the German here. So it turns out the rational numbers form a field, right? Because if we take rational numbers, we can add, subtract, multiply and divide rational numbers. And this will always produce a rational number, so long as we don't divide by zero. We can't do that. Again, we're not allowed to divide by zero, but we can add, subtract, multiply and divide rational numbers. We always end up with a rational number again. This tells us that the rational numbers is a field. On the other hand, integers do not form a field. And the reason we have a problem there is that when it comes to integers, we can add to integers and get an integer. We can subtract and multiply two integers to give an integer. But when we divide integers, we don't always get an integer, right? If you take one divided by two, this is not an integer anymore. It's a rational number, but it's not an integer. And so the number system of integers is not a field, but the number system of the rational numbers is a field. I kind of mentioned that the axioms of a field are based upon the real numbers, right? The real numbers, this is denoted with an R with a double line. The real numbers would include rational numbers, but also includes irrational numbers like pi for which the decimal expansion is non-repeating. This is a field. And in fact, this is the arc type for all fields. Another field that you probably have seen before would be the field of complex numbers. The field of complex numbers will denote this with a C with an extra line here. And this is the set of numbers of the form a plus bi, where a and b are just arbitrary real numbers. And i has the usual meaning as the square root of negative one. So this is likewise a field. The real numbers is a field. The complex numbers is a field. The rational numbers is a field like we mentioned before. And another set of numbers that you're probably familiar with, if you take the natural numbers, the natural numbers, this will be denoted with a capital N with an extra line here. And this would include the number zero, one, two, three, four, right? Going off towards infinity there. This would just be non-negative whole numbers. I should mention just like the integers, the natural numbers are not a field. For one reason, just like the integers, you can divide to natural numbers and not get a natural number anymore like one half still as an example there. But you also have problems with subtraction. You can't just subtract arbitrary natural numbers, right? If you take, for example, three minus four, that should equal negative one, which is not a natural number. So the natural numbers and integers do not form a field because they aren't closed under all of the operations. But the rational numbers, real numbers and complex numbers do. They do form a field. And so what we're going to see later on in this section is we're actually going to introduce a new type of field that you probably haven't seen before. And it will satisfy those 10 properties that we saw earlier. So stay tuned for that.