Interesting maths? Surely an oxymoron (part 2)

When I last left off we were talking about extending the Natural Numbers to contain values less than zero. And some clever chap had suggested just having the same thing as the natural numbers, just going the opposite direction down the number line. This set of numbers has a name too, it's called the Integers and is defined by I. Using our set notation, we can say the integers are the set of whole numbers {-∞...∞}. So, like the Natural Numbers, the Integers extend off to infinity, however, they also extend off to negative infinity. This means that unlike the Natural Numbers, we can subtract any integer from any other integer. It also means we have to define what happens when you multiply, add and subtract negatives.

It's worth taking an aside here to define a couple of properties our number systems thus far have, which are vitally important to their use. These properties are Distributivity, Associativity and Commutatvity.
Commutativity is what tells us that 3 X 4 = 4 X 3. This might seem trivial, but it's an important property, and it doesn't hold true for all operations in I (or N for that matter); Subtraction is not commutative. 3-4 is not the same as 4-3. More formally;
If a and b are Integers, a + b = b + a, a x b = b x a for all a, b.

Associativity is similar to commutativity. It states that when we perform a series of operations like (3 x 5) x 2, it is equivalent to 3 x (5 x 2) which is equivalent to 3 x 5 x 2. Again multiplication and addition are associative.
More formally
If a, b, c exist in I, a(bc) = (ab)c = abc for all a, b, c.

Distributivity is the final property we have to explain. It's easiest shown by example, as above; 3 x (2 + 5) = 3 x 2 + 3 x 5. i.e we can distribute the multiplication operation to the numbers in the brackets then add the products. So multiplication distributes over addition of integers.

Formally

If a, b, c exist in I, a(b + c) = ab + ac for all a, b, c.

With those properties (which are pretty important) defined, we can talk about integers properly.
If you recall all the way back to the previous entry, we needed to expand our number system to stop subtraction breaking it (the natural numbers were "open" under subtraction), whereas with the Integers, we can take any number from any other, meaning that the Integers are "closed" under subtraction. This gives negative numbers, which, lets face it, are pretty weird and unituitive. Values on nothing? What? But like most unitiuitive things in maths, we can slap some rules on them and use them. We already defined the properites our number system should have, so the negative numbers should obey these. Stepping back a section, we can define each of our natural numbers as 0 + that number (or even more specifically, 0 + 1 a number of times equal to our number). Slightly confusing I know, let's use an example to illustrate;

100 = 0 + 100 = 0 + 1 x 100.

We can do this for the infinity of positive numbers. What does this let us do for our new negative numbers?

Lets let each negative number be equal to 0 minus the value of that number were it positive. Confused? Probably. Illustrative example time!

-3 = 3 subtracted from 0 = 0 - 3 = 0 - 1 x 3.

Hey look, this is basically the same as for our positive numbers with a sign switch. This means that any rules we have for positive numbers should probably apply to our negative numbers!

So lets define our operations using our new negative numbers.

0 + a = a
a + a = 2a
2a - a = a
a - a = 0
=>
0 - a = -a
-a + a = 0
-a + -a = -2a
2a + -a = a

This seems pretty trivial, but we can now use these basic rules to add or subtract any negative number we like. We also know that multiplication is just repeated addition, so we can also multiply negative numbers. Truly we are the masters of our world.

Are we? Really? There's one basic operation we haven't discussed. Division (or if you like, taking the ratio between 2 numbers). Aha, I hear you say, we can divide 4 by 2 and get 2 in both I and N. Yes, we can. In fact we can divide any number by any other number which divides it exactly with no remainder in the integers and natural numbers. Somewhat simpler; we can divide any number that gives us a result that is an integer value, not a fraction. I isn't rich enough to include fractions. It's "open" under division. We need to expand our number systems again for a system that's closed under division.
This number system isn't really interesting enough to get its own post. It's called the Rational Numbers (from ratio between 2 numbers) and it's denoted by Q. It obeys the same rules as, and has the same properties as the integers, except it is closed under division. You can take any integer, and divide it by any other and have a result in Q. It's also worth noting that there is an infinity of fractions between every integer. Yup. That's a lot of fractions.

Formally;
If a, b are integers, there exists a rational number c where c = a/b

So, we can add, subtract, divide and multiply as much as we like. Surely we're done expanding our number systems now. We could be, but before we do, have a look at the length of the hypotenuse of a triangle with 1 unit sides. Quick refresher: . You can assume a, b, c are all integers. I'll be back later when you're done ;)

Interesting maths? Surely an oxymoron (part 1)

To most people, probably. Still, I thought I'd actually blog about some maths on my maths blog (heaven forbid!) and starting with the maths I find interesting would be a good start.


So, what am I going to talk about?


Number systems and how they can be closed or open. I'm not going to rigorously prove things. Partly because I can't and partly because it's not actually that interesting to build up axioms from the bottom up unless you're a proper maths geek.


So, where do we begin? Probably by defining some terms. Firstly a SET is a collection of objects. Sets are denoted by the curly brackets { } for some reason. Stuff inside the brackets is in the set. So the set of all publishers on this blog would be {me}. The set of all even numbers less than 7 would be {2,4,6} and so forth. The symbol ∞ denotes infinity. The => symbol denotes "implies" as in "My umbrella is wet => It is raining".


We can then use this definition of a set to define our first number system: These are the numbers you use every day when you count stuff. They're the most obvious numbers to us, since they clearly exist in the natural world. Thus they get their name; the Natural Numbers. Also called the positive whole numbers. 
They can be (partially) defined as they are the set {0...∞}. Usually this set is labelled by N.
You can add up the natural numbers as much as you like and never reach the end, and you can add any natural number to any other and get a defined result. As multiplication is basically repeated addition, (2x3 is 2+2+2 and so on) you can also multiply any natural number by any other and have a defined result. You can also subtract any number of equal or smaller size from any number. You can't take a larger number from a smaller one though, there is no result in N for this operation. You may be thinking that this sounds a bit complicated and confusing. It looks that way when written down, examples make it simple. For example you can take any number less than or equal to 9 from 9, but can't take 10 from 9, or slightly more formally;
Where a and b are any natural number;
a + b exists in N
a x b exists in N
a - b exists in N if and only if b ≤ a, => a - b ≥ 0

So, the Natural Numbers are not a rich enough number system to contain every operation we might need. This brings us on to our next number system. Somewhere, someone asked "What happens when I take 2 from 1?" It's a fine question. Clearly the answer has to be either zero or some sort of strange less than zero value. If it is a less than zero value what is it? Some historical clever chap thought "why not just extend the natural numbers backwards, and stick a subtraction sign in front of any that are less than zero? This lets us do more stuff and doesn't break anything we know about the natural numbers. Clearly I am a genius. Shower me with riches beyond imagining!" I like to think he (or she) was showered with riches, too.
I'll expand upon this in the second part of this post.

The chain rule

Any time anyone mentions The Chain Rule with reference to Calculus, Fleetwood Mac must be played.

This is actually mostly a WoW post, but that joke has been bothering me for a while.

My mage hit 85 and has been on the heroic and raid grind. Thus follow my observations of magery in Black Rock Descent normal.

Primarily, the trash in BRD is mostly actually fun and interesting. This is a total novelty after ToTC and ICC in wrath, one of which had no trash at all, and the other had multitudes of boring giant undead spiders to AoE down.

The bosses that my guilds casual raid has downed thus far have all been fairly well crafted, interesting fights.
Magmaw can be, however, annoyingly buggy. Blizzards vehicle UI bossfight problem strikes again. Golem council does a novel thing; it's a council fight I don't hate!

I've found so far that Arcane spec is slightly better for both Magmaw and GC, it has more reliable burst for exposed head phases on Magmaw, whereas Fire DPS can tank with a bad run of crit streaks, and Frost doesn't put out quite as much raw DPS. However, if you're not tanking the worms on Magmaw, Frost is really very useful, the ability to reliably root and slow the worms without taking a significant DPS hit is excellent for getting the Parasite Evening achievement. Plus Fire has weaker survivability than Frost or Arcane, so depending on your healer load, that can be an issue.

For Golem Council, I've been finding Arcane pulls ahead entirely on the back of Arcanotrons mana pool dealies keeping mana topped up during the AB4 phase of the arcane rotation. Also worth noting is that when arcanotron puts up barrier, it's worth nuking him and spellstealing the debuff he puts up, it's a DPS boost for all specs.

Maloriak seems to be fairly even across specs. If you're fast with spellstealing Renew Arcane would likely pull ahead in raw single target DPS, but Fire AoE is amazing for killing the adds during Green phases, and the DPS difference is negligible in my experience. The main reason to spec Arcane over any other spec is trivial application of caster slow, which may or may not be important depending on raid comp.

Chimaeron, it probably doesn't matter. Cauterize might provide some wonkiness, but this really isn't a DPS fight, it's all on the healers. We just get to pew pew and get loot.



It's also amazing how quickly heroics become fairly trivial. The 333 to 346 gear jump seems to make a huge difference. Grim Batol is still too damn long though. Also, more wand drops please Blizzard. I finally got mine, but it took ages.

Magery and the pre Cataclysm events.

My mage in WoW finally dinged 80. I've played around with all three specs and formed impressions of all three:

Fire: Undoubtledly the most involved of the three, it requires juggling Living Bomb, FFB and Scorch, watching for Hot Streak procs and timing combustions. The tree has one very, very noticable problem though; pyromaniac requires 3 or more targets so it's useless on boss fights with no adds, but very strong when adds are present. I struggle to get as good overall DPS out of fire spec than frost or arcane, but I have a tendency to clip LB just before it explodes, which is less than ideal.

Arcane: Still a total faceroll of a spec. Arcane blast, repeat. Brings the strongest single target DPS of the 3 specs due to the ease of play and strength of AB. Also has the major quality of life upgrade that is Improved Blink. The main reason I keep this spec at all is for that talent. At 85, with it being accessable from Fire or Frost spec, I can't see me speccing arcane

Frost: Balanced between the DoT watching of fire and facerolling of Arcane, Frost has gone from PVP/Solo only to a fairly competative raiding spec. It plays not dissimilarly to pre 4.0 ret pally, in that you're constantly proc watching. The combination of Icy Veins, Deep Freeze and Cold Snap gives tremendous burst damage too. Although it lacks the amazing quality of life upgrades on Arcane, it has amazing survivability, soloablility and good raid damage, as well as bringing replenishment. This will probably be my secondary spec at 85.

On the pre Cata events:

The elemental bosses are fun, however, the loot tables make them extremely annoying to queue for, as plate gear isn't evenly spread across all three bosses.

1+1=2

Not math related other than the title, but I asked my girlfriend to marry me, and she said yes. Clearly the indoctrination is working.

On heroes

I've been reading two autobiographies lately; Stephen Fry's The Fry Chronicles and Confessions of a Conjurer by Derren Brown.

Although I'd hesitate to call Derren Brown a personal hero of mine, Stephen Fry certainly is, and Brown is something of an inspiration.

The books have made me consider both who my heroes actually are and what I'd actually define as a hero.
So, what do I mean by it? They both inspire me, they're both witty, erudite and knowledgable, and they both seem to delight in sharing that knowledge. Moreover, they both seem to be genuinely nice, kind. individuals. The kind of person who would be a favourite uncle or grandfather.

I think though, it's the passion for knowledge that sets them up as something of heroes of mine. This links into the title of the blog (somewhat loosely, but forgive my indulgences, it's my blog after all) too.
This passion is what essentially got me started on the path to mathland. Some background:

My previous degree required a 4 hour(ish) round trip every day for classes. While the morning trip was mostly spent asleep, the evening one was spend in the papery grasp of a book. I read voraciously during my first and second years, when I would make the trip 4 times a week. Initially I began reading popular science books, but branched out into popular maths (a term which still seems oddly juxtaposed to me). The book which had the most immediate impact on me was Infinite Ascent by David Berlinksi. I've since gone back and reread it, and the prose is pretentious even by my standards (and I used juxtaposed in a sentence), but at the time it, to be somewhat cliched, opened new vistas. It exposed me to the beauty of a subject I was middlingly capable at in school, and mostly ignored in university (despite my mother being a math teacher). This led to books by two people I really would consider my heroes; Professor Ian Stewart, of Warwick University and Professor Marcus Du Sautoy of the Mathematical Institute of Oxford University. Both of them are relentless popularisers of mathematics, and their enthusiasm for the subject really shines out from their books. Prof Stewart in particular was instrumental in me going back into undergraduate education through his books. If not for them I'd probably never have gone into maths at all.

Really, that's who my heroes are; people who inspire me to do different things, to change myself, to be good, great or kind when I could otherwise do nothing.

It's a blog entry!

Mein Gotte!

So, first entry. What should I put in it? I suppose a statement of purpose; what'll I be using this blog for?

It claims to be Adventures in Mathland, so there'll be some math. I'm gonna document the interesting bits and pieces I do at uni that are worth sharing. I also play WoW, so I'll be commenting on my experiences there. I'm in a moderately successful raiding guild based on Kirin Tor US and my main is a Retribution Paladin.

I'll also post any other junk I find interesting or rant worthy.