Friday Random Ten!
Okay, I have a lot of Bach, but this is ridiculous.
- Hans Fagius - Johann Sebastian Bach: Prelude in C, BWV 943
- Netherlands Bach Collegium - Johann Sebastian Bach: Cantata #39, Brich Dem Hungrigen Dein Brot - Brich Dem Hungrigen Dein Brot
- The Police - Bombs Away
- Birdsongs of the Mesozoic - Electric Altamira
- Netherlands Bach Collegium - Johann Sebastian Bach: Cantata #19, Es Erhub Sich Ein Streit - Gottlob! Der Drache Liegt
- Pieter-Jan Belder - Johann Sebastian Bach: Suite in A Minor, BWV 818 - Courante
- Rush - Distant Early Warning
- Howard Shore - Ent-draught (from The Two Towers)
- Pieter-Jan Belder - Johann Sebastian Bach: 3-Part Invention #10
- Netherlands Bach Collegium - Johann Sebastian Bach: Cantata #62, Num Komm, Der Heiden Heiland - So Geht Aus Gottes Herrlichkeit und Thron
This next topic makes me nervous, because it's bound to cause confusion, and that confusion will lead to controversy, and I prefer to avoid controversy. However, both the textbook and the lecture notes covered it, so I will too.
We are still looking at extending number systems, and now we're looking at extending the real numbers. Let's go back to the definition of the real numbers: they are a set of numbers that have the least upper bound property, and contain the rational numbers as a subset. So every set with an upper bound has a least upper bound, which is pretty great. Wouldn't it be greater if we could extend the real numbers so that every set has an upper bound?
We can do it. The extended real number system includes all of the real numbers, but it also includes the numbers +∞ and −∞. Positive infinity is defined as a number greater than any real number, so any set of numbers in the extended real numbers has an upper bound. Likewise, negative infinity is less than any real number, and so is always a lower bound of any set in the extended real numbers.
The bad news that this totally breaks addition and multiplication. Any real number added to infinity is still infinity, which easily leads to serious math errors. 1+∞=∞=2+∞, which implies that 1=2, for example. The only way to make things work at all is to make lots of computations involving infinity undefined. For example, ∞−∞, ∞/∞, and 0×∞ are all undefined. (We encountered the undefined value 00 previously. Saying that the result is undefined means that it must take on different values in different contexts, so if you end up encountering that expression, you need to find a different way of solving the problem that avoids it.)
As a consequence, the extended real numbers are no longer a field. There's an exception in one of the multiplication properties for 0, but there aren't any exceptions for infinity. And it's really easy to accidentally compute something using one of the undefined expressions and not notice, so most of the time the extended real numbers aren't worth the effort. We stick with the real numbers, where numbers can move toward infinity, but they can never equal infinity.
Back to building extensions of number systems. The last number system we looked at was the rational numbers. The key point from the digression about fields last time is that the rational numbers have a larger group of properties for addition and multiplication than the natural numbers or the integers.
A few weeks ago, I posted about rational numbers and least upper bounds. That's the next step in extending the rational numbers. As I previously discussed, the square root of 2, for example, is not a rational number, which means that not every subset of the rational numbers has a least upper bound.
The real numbers are defined as the extension of the rational numbers such that every subset of the real numbers has a least upper bound. Actually proving that the real numbers exist is hard (but it can be done), but we can also get away with just assuming that they exist and using them.
The real numbers share all of the properties for addition and multiplication as the rational numbers. (In other words, the real numbers are also a field.) They are also ordered.
We can define rational exponents of positive bases as real numbers. If r is a rational number, then r=m/n, where m and n are integers. Then if a is a real number greater than 0, ar=am/n=(am)1/n. am is a positive real number by closure of multiplication, and if b is a positive real number b1/n is a real number because of the least upper bound property. Therefore ar is a real number for any positive real number a and any rational number r.
We can also define ax for any positive real number a and any real number x. Define a set of numbers of every number ar, where r is any rational number less than x. Then that set has a least upper bound, and we can define ax as equal to that least upper bound.
I've been limiting exponents to positive bases (or 0, for positive exponents only). I will deal with that soon, but next up will be some digressions and rants.
We now have the rational numbers, and we can do addition and multiplication with rational numbers. Addition and multiplication have a bunch of properties on the rational numbers, which I've mentioned as they have come up, but now I'm going to list some of them in particular again.
- Addition has closure: if x and y are rational numbers, x+y is a rational number.
- Addition is commutative: x+y=y+x for all x and y.
- Addition is associative: (x+y)+z=x+(y+z) for all x, y, and z.
- There exists an additive identity 0: x+0=x for all x.
- Every number x has an additive inverse −x: x+(−x)=0 for all x.
- Multiplication has closure: if x and y are rational numbers, xy is a rational number.
- Multiplication is commutative: xy=yx for all x and y.
- Multiplication is associative: (xy)z=x(yz) for all x, y, and z.
- There exists a multiplicative identity 1: 1x=x for all x.
- Every number x except 0 has a multiplicative inverse 1/x: x × 1/x=1 for all x≠0.
- Multiplication is distributive over addition: x(y+z)=xy+xz for all x, y, and z.
In abstract algebra, any set on which addition and multiplication are defined and which has all of these listed properties is called a field. In abstract algebra, the sets don't even have to be sets of numbers, and addition and multiplication can be defined differently, as long as all of these properties exist.
Using the definitions of addition and multiplication that we have been using, the integers are not a field, because the integers do not have multiplicative inverses. However, it's possible to take different definitions of addition and multiplication which result in subsets of the integers being a field.
It's possible to derive a bunch of properties about addition and multiplication directly from the properties that define a field. In the context of the rational numbers, these are all properties that you probably would take for granted anyway, but it's useful to build a framework to show that they are true more generally.
I'm not sure how relevant fields or other concepts from abstract algebra actually are to mathematical analysis, but the textbook called attention to the fact that the rational numbers are a field so I am too.
We're looking at extending the natural numbers, so we can do math that the natural numbers can't do. Last time we looked at addition and expanded the natural numbers to the integers. The useful thing about this extension is that it kept all of the established properties of the natural numbers and added new ones.
Last time ended with the question of negative exponents. This is actually an example of a general case of multiplication problems that can't be solved with the integers. From the properties of exponents, it should be true that ama−m=a(m−m)=1 (as long as a≠0). But it is not generally true that if a and b are integers, there exists an integer x such that ax=b.
The solution is to create the rational numbers as an extension of the integers. The rational numbers keep all the same properties for addition and multiplication as the integers. Multiplication also has the new property that for any rational number x except for 0, there exists a rational number y such that xy=1. (y is the reciprocal or multiplicative inverse of x.) We can define division as multiplication by the reciprocal. Exponentiation is now defined for negative exponents (but not for rational exponents in general) as long as the base is not 0. And the rational numbers are ordered.
The next extension in this sequence is the real numbers, but before I get there, next time I'm going to take a detour through abstract algebra to talk about fields.
Okay, so we have the natural numbers, and we've defined an ordering for them based on inequality. We've also defined addition, multiplication, and exponentiation, which means we can do things with them. This leads to a followup question: what can't we do with them?
Looking at addition first, we have that given any two natural numbers a and b, a+b=n, where n is also always a natural number. But if a+n=b, there is not always an n which makes that statement true. Let me provide a concrete example: if a=2 and b=1, then a+b=3, and 3 is a natural number. But there is no natural number n such that 2+n=1.
This leads to the definition of the integers as an extension of the natural numbers. Given any natural numbers a and b, there is an integer n such that a+n=b. Note that this extension preserves all of the properties of the natural numbers. The integers are ordered. a<b if a+n=b and n is an integer greater than 0. (There's the possibility of circular definition here. n>0 can't simply mean that n=0+n, because that's always true. Saying that an integer n>0 if n is also a natural number avoids this problem.)
Addition is defined on the integers, and has closure and is commutative and associative. In addition, we can now define 0 as the additive identity, giving us another property for addition. Also, it is now true that for any integer a, there exists a number b such that a+b=0. This also means that we can define subtraction as negative addition.
Multiplication keeps all the same properties and is unchanged. Exponentiation is unchanged as long as the exponent is non-negative. a0 is defined as 1 for all integers except 0. 00 is undefined. Basically, depending on how you get there, sometimes 00 should equal 1, and sometimes it should equal 0. If you define it one way or the other, it will end up breaking math. Instead, math says 00 has no meaning, so if you've done a bunch of math and end up writing that down, you should start over with an approach that doesn't end up using it.
We can't define negative exponents yet, but we will soon. In fact, that's a good place to start next time.