Just like we can define multiplication as repeated addition, we can define exponentiation as repeated multiplication. Exponentiation doesn't have the same properties as addition and multiplication, but it does have some worth mentioning, which again we can prove are true based on the definition. It does have closure on the natural numbers, so an is always a natural number if a and n are natural numbers. It is not commutative or associative (in general, an≠na), but it is true that am+n=aman, which feels to me like the distributive law for multiplication and addition. Also, (am)n=amn, which follows from the previous property.
Another property the natural numbers have is that they are ordered. a>b is defined to mean that there exists a natural number n such that a=b+n. This means that for any two numbers, there are three possibilities: they are equal, the first is greater, or the second is greater. The fact that one and only one of these three cases is true is used all the time in proofs. It also means that if a>b and b>c, a must be greater than c. Which means that for any set of natural numbers, it is always possible to order them from smallest to largest.
Next up in blog memes from 2003: pet blogging. I would be all over Friday Cat Blogging, if only I had a cat. Fortunately for me, Friday Cat Blogging has expanded to encompass a wide variety of subjects, such as dogs and cephalopods. So I bring you: Friday Gorets Blogging.
Unfortunately, gorets are famously camera-shy, so I can't offer any photos this week. Maybe next time. Instead, I'll tell you about a recent moment involving my goret, Binkley, that I would post a picture of, if I had one. (Recognizing, of course, that claims of ownership of a goret are dicey at best, and my name for the goret is strictly for my own benefit.)
The other day, Binkley was enjoying a tasty snack of lenticular infundibula. In its enthusiasm for the infundibula, it managed to invert its foon grommet. This was, as you would expect, totally adorable. I thought about trying to grab a photo, but decided it was better to enjoy the moment for what it was. I didn't want to disrupt Binkley's enjoyment of the lenticular infundibula, after all.
Posting to a blogspot blog feels vaguely like I'm living in a 2003 flashback. With that in mind, I'm going to participate in some blog memes from about that time. First up is the Friday Random Ten.
The rule is: create a random setlist in your music player. Post the first ten songs that come up. No shame!
- William Orbit - A Hazy Shade of Random
- Rush - Vapor Trail
- Freezepop - Robotron 2002 (All Your Base Are Belong To Us Remix)
- Netherlands Bach Collegium - Johann Sebastian Bach: Cantata #110, Unser Mund Sei Voll Lachens - Ach Herr, Was Ist Ein Menschenkind
- Pearl Jam - Even Flow
- Hans Fagius - Johann Sebastian Bach: O Lamm Gottes, Unschuldig, BWV 1095
- Deep Purple - Don't Make Me Happy
- Friedrich Gulda - Ludwig van Beethoven: Piano Sonata #10 in G, First Movement
- George Frideric Handel: Israel in Egypt, Part Two
- Leonard Cohen - The Gypsy's Wife
My plan here has been for my posts to follow along with what I have been studying. As I've been going, I've reduced things down to the "important" parts, and left other things out. However, I've realized that before I start on the next major topic (functions and sets), I want to go back and cover some things I skipped the first time.
So, starting over at the beginning, we are currently assuming that the natural numbers exist. (I know that in some contexts, we could start with sets and derive the natural numbers, but we aren't going that far here.) We also assume that addition exists, and that it has three properties. These properties are obvious but they are also powerful, and we need to define them in order to use them.
Property 1 is closure. This means that if x is a natural number and y is a natural number, x+y is a natural number. On the one hand, this is obvious. On the other hand, it means, for example, that there is no greatest natural number. Assume there is a greatest natural number, and call it g. Then g+1 is a natural number, because of closure, and g+1 is greater than g, so g is in fact not the greatest natural number. Therefore there cannot be a greatest natural number. This is a result which may seem insignificant, but it defies real-world experience. I expect to have more to say about it in a future post.
Property 2 is commutativity. This just means that x+y=y+x. This is again obvious but useful, and really annoying in mathematical contexts where it isn't true. I'm just saying, don't take it for granted.
Property 3 is associativity. This means that (x+y)+z=x+(y+z), or more generally that in a series of additions, the additions can be performed in any order. Once again, obvious but useful.
These three properties mean that we can derive multiplication as repeated addition, and prove that multiplication of natural numbers has the same properties of closure, commutativity, and associativity. Multiplication of natural numbers also has one property that addition does not have: the existence of an identity element. n×1=n for every natural number n, so 1 is called the multiplicative identity. Also, we have a new property that combines addition and multiplication, the distributive property, which states that x(y+z)=xy+xz. Like the other multiplication properties, we can prove this is true from the definition of addition.
Next time I will look at exponentiation and inequality.
I have a small follow up on rational numbers and real numbers. We've established that rational numbers have holes, and the real numbers fill those holes. The question is, how big are the holes, and how many of them are there?
It's possible to show that there is a rational number between any two real numbers. The metaphorical proof is straightforward. Imagine the real number line as a sidewalk you are walking on. If you want to find a rational number between the real numbers x and y, imagine a gap in the sidewalk between x and y. If you are taking steps which are a constant rational size, then what happens when you get to the gap depends on the size of your steps. If your steps are big enough, you could step right over the gap and keep going. But if your steps are smaller than the gap size, you must step into the gap. The position of the step in the gap is a rational number, and it is between x and y.
A formal version of the stepping in the gap argument proves that between any two real numbers, you can always find a rational number. Unsurprisingly, a similar argument proves that between any two real numbers, you can always find an irrational number. (An irrational number is just any real number which is not a rational number.)
Here's my interpretation: irrational numbers are holes in the set of rational numbers. Since you can find irrational numbers everywhere, there are lots of holes everywhere. On the other hand, since you can find rational numbers everywhere, that proves that the holes are tiny. It's important that the real numbers fill in the holes in the rational numbers, but depending on what you're doing, you may be able to ignore the holes anyway.