 Tuples and vectors are two very important ideas in higher mathematics. Given some set S, we can form ordered N tuples, an ordered list of N elements, each of which is an element of S. For example, if I take the set of honest politicians, a two-tuple would be an ordered list of two elements, both of which are elements of this set. And so a two-tuple could be... Okay, we need to make sure that S is non-empty. So for example, suppose I have a set of colors, a three-tuple could be red, blue, green. The components of a tuple are typically separated by commas and enclosed in braces, square brackets, angle brackets, or even simple parentheses, though angle brackets are by far the most common. Almost the first thing a mathematician does when encountering a new object is to try and define equality. So if I have two tuples, v1 through vn and u1 through un, they are equal if and only if vi equals ui for all i. That means v1 is equal to u1, v2 is equal to u2, v3 is equal to u3, and so on all the way down the line. As we're basing the equality of the tuples on the basis of the equality of the components, we call this component-wise equality. What this means is that two tuples are equal only if they have the same elements in the same order. So even though these two tuples both include red, blue, and green, they are in a different order, and so they are different tuples. After defining the equality of two tuples, we might try to do something else with them. But this could be a problem. We might not be able to do anything else. So we know these two tuples aren't equal, but which one is larger? And if we can't even decide that, we might have a further problem if we try to do some arithmetic with them. If I hope to be able to do arithmetic with tuples, I need to start with a special type of set. What we need to do is begin with the field. A field is a set f with the following properties. On the elements of f, we'll define two operations, which we'll write as plus and times. It's important to keep in mind that there are only so many symbols, so that even though these are written as plus and times, we should not expect these to behave as our ordinary addition and multiplication. For any three elements a, b, c in our set, the following properties must hold. First, we have to have closure. a times b and a plus b, whatever they are, must be elements of our field. Next, we have to have associativity. If we add three elements, a plus b plus c, it doesn't matter if we find b plus c first and then add it to a, or if we add a plus b first and then add it to c. And a similar property must also hold for our times operation if we multiply three elements. Next, both of our operations have to be commutative. a plus b has to be the same as b plus a, and likewise, a times b must be the same as b times a. We also want to have distributivity. If I multiply a times the sum, b plus c, I want to be able to evaluate that as a times b plus a times c. Next, identity. There are elements which will designate 0 and 1 in our field for which a plus 0 is a, and 1 times a is equal to a. And again, remember, there are only so many symbols, so the 0 and 1 here might have nothing to do with what we usually think about as 0 and 1. Now, closely associated with this idea of an identity is the inverse. There are elements negative a, and for a not equal to 0, a inverse in our field for which negative a plus a is 0, and a inverse times a is equal to 1. So let's try to prove, or possibly disprove, that q, the set of rational numbers using the ordinary operations of addition and multiplication, is a field. It's important to emphasize that you cannot prove a statement by giving examples, but you can use the examples to help guide your thinking. And in this case, it may be helpful to have specific rational numbers in mind as we work our way through the addition and multiplication process. So let's put down our definition of a field, and in order for something to be considered a field, it has to have all of these properties. So let's check them out. So let's take three elements of our set, how about three-fifths, one-fifths, and two-thirds, and the first thing we notice is that three-fifths plus one-fifth is in q. It is a rational number, and if we think about this, this will actually extend to any two things that are rational numbers. Similarly, three-fifths times one-fifth is also a rational number, and again, if we think about this, this is also going to be true for any rational numbers that we multiply together. And so we conclude that our set q is in fact closed under our operations of addition and multiplication. Next, let's check associativity. If I take my three rational numbers, it doesn't seem to make a difference if I add the second and third, and then add the first, or if I add the first and second, and then add the third, I seem to get the same result. That's true for any three rational numbers. Likewise, if I multiply three rational numbers, it doesn't seem to make a difference if I multiply the second and third, and then multiply by the first, or if I multiply the first and second, and then multiply by the third. And again, this seems to be true for any three rational numbers, and so we have associativity. So the next thing to check is commutativity. So again, let's take our rational numbers and see if we get the same thing, no matter which order we add them. And we see that's true for our specific examples, and if we think about it, that's going to be true in general. So addition is commutative, and likewise, our products are also going to be commutative. The next property is distributivity. If I multiply our rational number by the sum of the other two, I can evaluate this as three-fifths times one-fifth plus three-fifths times two-thirds, and this generalizes. What about our identity elements? Since we're using ordinary addition and multiplication, zero and one under ordinary addition and multiplication seem to fit this requirement of being identities. Three-fifths plus zero is, in fact, three-fifths, and importantly, zero is an element of the rational numbers. And this seems to generalize anything plus zero will give us what we started with. Likewise, one times three-fifths is three-fifths, and again, our multiplicative identity one is a rational number, and again, this generalizes one times a equals a. One important thing to emphasize here is that it's not enough that the additive and multiplicative identities exist, they must be part of our set. And the last thing we have to check are the existence of the inverses. We have to confirm that there are elements negative a, and as long as a is not equal to zero, a inverse in our set for which negative a plus a is zero, and the inverse of a times a gives us one. And again, if we use our samples three-fifths, one-fifths, two-thirds, we see that negative three-fifths plus three-fifths is going to be zero, and again, it's not enough that this inverse exists, but it has to be in our set, and negative three-fifths is in fact a rational number. And likewise, we'll try to find the multiplicative inverse of three-fifths, and the multiplicative inverse of three-fifths is five-thirds, and that is a rational number, and more generally, any multiplicative inverse will exist, and also be an element of our set. And since our set q meets all of these requirements, then the set q of rational numbers using ordinary addition and multiplication is a field. And with this in mind, we can now define vectors. A vector is a tuple with n components, each of which comes from a field. If the field is f, we say the vector is in fn. So some of the familiar fields and the associated vectors, first of all, z, the set of integers, is not a field, so we don't really have vectors in z. q, on the other hand, as we've just shown, is a field, so I can take a tuple with n components, all of which are rational numbers, and form a vector in qn. Similarly, r, the set of real numbers, is going to give rise to vectors in rn, and c, the set of complex numbers, will give us vectors in cn.