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I guess most people are pretty familiar with the natural numbers {0,1,2,3,...}, otherwise known collectively as N, the integers {...,-2,-1,0,1,2,...} called Z (from Ger. Zahl), the rationals (the fractions like 1/3 and -27/11) called Q (from Ger. Quotient) and the reals (because numbers like e and pi can't be written as fractions) called simply R. Of course quite a few of you will be pretty au fait with the complex numbers, C, too. Start with the real numbers and throw in the number i with the property i2=-1. Some of you might debate all night over whether they exist but the rest of us will just get on and use them in applications such as electronic engineering, digital signal processing and computer graphics. Q, R and C are all fields - basically this means you can add, subtract, multiply and divide them in the familiar way and in other ways are very familiar too. And if you haven't seen it before, this formula due to Euler ought to blow your mind:
eit=cos(t)+isin(t).

Talking of graphics, quaternions are pretty popular too. You've seen how to make the complex numbers by adding in a new number i so why not try our luck again and throw in j whose square is -1. The catch now is: what is ij? Well let's call it k. In fact if we add the rules ij=ji=k, jk=-kj=i, j2=k2=-1, and a few more, I'm sure you're getting the idea by now, we end up with H, the quaternions (damn! notice Q's already taken!). Anyway, any quaternion can be represented by 4 real numbers. For example 1+2i-4j+11k is represented by (1,2,-4,11). That means we can embed good old 4 dimensional spacetime in there. In fact that's what Hamilton originally used them for. Well, Einstein hadn't come along yet so actually he was only using the last three components to represent points in space but you can use the first for time too. Turns out the quaternions do a really sweet job of doing geometry which is why graphics people like to use them. If you thought Euler's formula was cool that's nothing compared to what you can do with these babies. In fact their main application is in eliminating all those complicated trig formulas you get when you try to carry out rotations in 3D. Notice that we lost something by using these numbers. Two times three is three times two but i times j isn't j times i. We've sacrificed what's known in the game as commutativity.

Well there's a bit of a pattern going on. Start with R, move on to C and then Q. The quaternions were 4-dimensional. What about 3-dimensional? Or 5-dimensional? Well you can try making these things (exercise!) but you find they're not all that interesting. They just aren't well behaved. That is until you come to 8 dimensions. Then you find you can make the octonions, otherwise known as the Cayley numbers and labelled O. Still, you have to make more sacrifices. When you multiply a by b and c it matters whether you do ab or bc first. No longer are we guaranteed things like (2.3).5=2.(3.5). We have lost associativity. Still, there's fun to be had with octonions. You can discover the delights of spinors and triality, find that there is a well defined cross product in 7 dimensions and if you've enough stamina you'll see they have neat connections with string theory.

How long can we keep this up for? Well if you really insist you can make the 16-dimensional sedenions but they're so unruly that I just wouldn't bother if I were you!

But you might have noticed. All of these things were built from the real numbers. The whole point here is to get away from them. What other alternatives do we have? Well you could play with the integers modulo N. Just do normal math but every time you get an answer bigger than or equal to N divide by N and keep only the remainder. For example modulo 9, 8+7=6 and 3*3=...hmmm...0. That zero isn't too good, we don't want to end up with zero when we multiply perfectly respectable numbers. That's easy to fix: choose N to be prime. (Think about it!). Now we can even divide these numbers. In fact Z/NZ (that's their other name BTW) forms a field, just like R and C. That means you can add, subtract, multiply and divide by anything non-zero. Still, numbers that only go up to N-1. Sounds a bit useless. Au contraire! They're the at the heart of algorithms like RSA encryption.

But we're still not being alternative enough. After all, numbers modulo N are still, well, numbers. Can we get a bit more radical? Try this for size: you're used to decimal expansions that go on forever like 1/7=0.14285714285... Maybe we can make digits go infinitely far to the left of the decimal point instead. Consider a number like ...99999999. Looks pretty big. But try adding 1. Well 9+1=0, carry 1. Write down the zero. Now 9+1=0, carry 1. Hmmm...looks like the answer is going to be ...000000 with that 1 getting carried out to infinity and its final cashing out getting postponed forever. So what we have is ...9999999=-1. That might look familiar to some of you. I'll give you a clue: this is called tens complement arithmetic. Or at least I've just called it that. So it seems that arithmetic with infinite sequences of digits might just work out. The catch is that you can't always divide these things. However, if we work not with base 10, but base p digits it turns out that we can always divide by any non-zero number. These numbers are called the p-adics, or Zp. For each prime p the p-adics form a field. They're very useful for doing number theory and of course that means they're good for cryptography. BTW, going back to the 10-adics again, try calculating directly that ...9999992=1.

But what about infinity? Can't we make that a number? Turns out that there are lots of ways to introduce infinity to a number system. Let's start with the cardinals - numbers used to count how many elements there are in a set. For example {red,green,blue} has 3 elements meaning that the set of primary colors has three elements. Here 3 is being used as a cardinal. But what about this perfectly respectable set {0,1,2,3,...}=N, what is the size of that? Well that's the first transfinite cardinal and it goes by the name of aleph0. (I apologize, I don't know how to do hebrew HTML.) The next biggest cardinal is called aleph1 and you can guess the rest of the pattern. But here's an interesting conundrum: how big is the set of real numbers R? Well it's called c and it's bigger than aleph0. But is it bigger than aleph1? Turns out it's your choice! Whichever way you go the rest of mathematics is happy to accommodate you.

There's more than one way to make an infinity. You can play with the transfinite ordinals too. Go back to the beginning when nobody had invented any mathematics. We had nothing, in other words the empty set {}, which we'll call 0. On the next day we can try making another set - after all we now have something to put in a set, the empty set itself. So define 1={0}. Now we can define 2={0,1} and I'm sure you see the pattern n={0,...,n-1}. These are the ordinals. But why stop there, how about {0,1,2,...}? Well that's the first transfinite ordinal and we call it w (that's meant to be omega, I don't do Greek HTML either). Now we can make {0,1,2,...,w}. That's called w+1. How about {0,1,2,w,w+1,w+2,...}. Well that has a name too, 2w. Want to carry on? Don't let me stop you. But I'd better warn you that you might not be able to get all cardinals this way because I've only shown you how to get the accessible ones.

Maybe you're not interested in big. We could try small. In calculus you probably got fed up of always using the fact that as dx->0, dx2 could be ignored. Why don't you just set dx2 exactly equal to zero and be done with it. Well that's what nonstandard analysis is about. It introduces a new kind of infinitesimal number that has just this property. Useful if you're lazy about proofs.

But maybe mathematics isn't your thing and none of these systems looks like fun to you. Then there is one last thing I can throw at you. Combinatorial games. What do games have to do with numbers? Though invented by Conway (yes, that's the Conway), Don Knuth gave them another name: Surreal Numbers. Here are some surreal numbers: 0, *1, *2, *3... And here's how to add them: *a+*b=*(a xor b). That's the good old xor that's written '^' in C. If you know how to win at Nim then you know why these are good for winning games. The *n are called nimbers and they are just a tiny part of the entire collection of surreal numbers, all of which represent positions in games. What's more, the surreal numbers include Z, Q and R. Hard to believe but real numbers are all positions in games. But that's not all, the surreal numbers include the transfinite ordinals, lots of different kinds of infinitesimals and countless other weird and wonderful numbers. If you want to beat your friends at Hackenbush, Nim, Mogul, and maybe even Go, then these are the numbers for you.

And if I manage to get this story past voting I might just follow it up with one on alternative logic.

--

God created the integers, all else is the work of man.
Kronecker

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The Alternative Numbers | 155 comments (131 topical, 24 editorial, 0 hidden)
Good article (2.20 / 5) (#2)
by DesiredUsername on Mon Sep 23, 2002 at 02:17:07 PM EST

But the Kronecker quote is flamebait. Which is entirely in character for him, but doesn't add much to the article.

Play 囲碁
OK, good point (none / 0) (#12)
by DesiredUsername on Mon Sep 23, 2002 at 02:48:02 PM EST

Let me get the ball rolling: All of mathematics is discovery, not invention.

Play 囲碁
[ Parent ]
Wow. (none / 0) (#74)
by Arkayne on Mon Sep 23, 2002 at 10:33:59 PM EST

The perfect punchline to this rather dry, albeit interesting article would be if you two suddenly fought it out like ninjas.

[ Parent ]
Are you sure (none / 0) (#18)
by krek on Mon Sep 23, 2002 at 02:58:25 PM EST

To say that we are discovering mathematics, while not incorrect, is not exactly precise either. Did we invent the wheel or discover rotation? Did we invent light bulbs or did we discover that running electricity through tungsten produces light?

Is mathematics inherently part of our universe or is it mearly the way we interpret our universe?

I would not be so quick to decide on this one if I were you.

[ Parent ]
I think... (none / 0) (#23)
by Rocky on Mon Sep 23, 2002 at 03:13:37 PM EST

...you're confusing science/math with engineering.

I invent the wheel.  (engineering)

I discover an abtract reason why it's good for the axle to be in the center of the wheel.  I apply this to fans, frisbees, and such. (physics)

If we knew what it was we were doing, it would not be called research, would it?
- Albert Einstein (1879 - 1955)
[ Parent ]

At the time wheel was being invented ... (none / 0) (#102)
by bob6 on Tue Sep 24, 2002 at 09:10:39 AM EST

... nobody did a distinction between engineering, crafts and artwork on one side and science and abstraction on the other. The dichotomy appeared during the 20th century and it is not fully established yet. For examples, take a look at computer science journals and conferences where articles jump from "truth discovering" science to "industry improving" engineering, and back.
In short, Science and Engineering feed one each other in a more complicated way than we would have wanted it. So let's not draw those lines because they only serve to discard arguments.

Cheers.
[ Parent ]
I disagree (none / 0) (#43)
by X3nocide on Mon Sep 23, 2002 at 05:04:54 PM EST

All maths start with basic assumptions, like what addition means. Thats a mental invention. We then discover the implications of this, within the context of that assumption. To pull our discoveries outside of our assumptions and call them discoveries is poor logical form.

pwnguin.net
[ Parent ]
If you assume objective reality, you have numbers (none / 0) (#86)
by abulafia on Tue Sep 24, 2002 at 01:41:29 AM EST

Sorry, but you're wrong, unless you're of the opinion that all reality is subjective.If you are of that opinion, it must be difficult for you to talk to many people outside of certain realms (I'll omit listing them so as to not draw flames). If you agree that objects fall towards the earth in a predictable fashion, can be measured, and that this knowledge can be verified by other people, then you are stuck with all but a few realms of mathematics. Everything is built upon a very few assumptions which I, and most of the rest of the world, find rather difficult to dispute. For that matter, take any other simple observation, such as the water displacement of a body, the amount of heat required to boil water, what have you. Sorry if I tend to come at this froma physics perspective, but that's my inclination and approach to math in general. And economic behaviour, for that matter. Two of my favorite quotes are unfortunately from some rather fringy people, but they're still smart, no matter what your philosophy: "higher power = 213,466,917-1" --Tim May "All crypto is economics" -- Eric Hughes. [to make this make sense to the non-obsessive members of the audience, that higher power is the current Mersenne Prime, and crypto is merely specialized math with a social element.] -j

[ Parent ]
I bet (4.00 / 2) (#5)
by tranx on Mon Sep 23, 2002 at 02:29:56 PM EST

When you wrote

Fed up of seeing the same numbers over and over again?

You were thinking of the Vset: {-1, 0, +1, +1 FP}

Can anybody come out with some interesting math properties of V?


"World War III is a guerrilla information war, with no division between military and civilian participation." -- Marshall McLuhan

Vset is a group (4.50 / 2) (#38)
by phliar on Mon Sep 23, 2002 at 04:35:22 PM EST

Vset: {-1, 0, +1, +1 FP}

Can anybody come out with some interesting math properties of V?

This set with addition forms a group.... let's call "+1 FP" f. With
1 + 1 = f
f + 1 = -1
f - 1 = 1
f + f = 0
I think we have closure. (Somebody check my math!)

Wait a minute... write the set as {0, 1, -1, f}. This group is isomorphic to {1, i-i, -1} with multiplication.

Voilà!


Faster, faster, until the thrill of...
[ Parent ]

That's really twisted.[n/t] (none / 0) (#84)
by abulafia on Tue Sep 24, 2002 at 01:26:06 AM EST



[ Parent ]
Hmmm.. (5.00 / 1) (#91)
by Kwil on Tue Sep 24, 2002 at 04:10:57 AM EST

From what I know of the Vset system:

f + f = f
f + 1 = 1
f + 0 = f
f + -1 = 0
1 + 1 = 1
1 + 0 = 1
1 + -1 = 0
0 + 0 = 0
0 + -1 = -1
-1 + -1 = -1

And while commutative, the system is not associative.

How you get +1 FP + 1(section only) = -1 (dump it) I'm not sure.

That Jesus Christ guy is getting some terrible lag... it took him 3 days to respawn! -NJ CoolBreeze


[ Parent ]
Interesting properties (5.00 / 1) (#55)
by godix on Mon Sep 23, 2002 at 06:32:41 PM EST

95 +1's added together equals at least one, although not neccesarily just one, comment saying 'Not another Iraq story!'


Don't mind the plummeting noise, mojo always makes that sound after I post.


[ Parent ]
It's just a generator set (4.00 / 1) (#100)
by roiem on Tue Sep 24, 2002 at 08:44:22 AM EST

For a much larger (infinite, in fact) set: the set of all possible voting results of stories.
A voting result is an ordered pair of numbers (s, f), where s is the score of the story, the number of positive votes minus the number of negative votes, and f is what I'll call the fp-score of the story, the number of +1FP votes minus the number of +1 votes. If s reaches 95 and f is positive, the story gets to the front page. If f is negative, it gets to the section page. (I don't know what happens if f is zero). If s reaches 20, the story is dumped.
Thus, we have
  • -1 = (-1, 0)
  •  0 = (0, 0)
  • +1 = (1, -1)
  • +1FP = (1, 1)
with the normal addition and subtraction operations, (a1, b1) +/- (a2, b2) = (a1 +/- a2, b1 +/- b2).
90% of all projects out there are basically glorified interfaces to relational databases.
[
Parent ]
Numbers (3.66 / 3) (#9)
by Noam Chompsky on Mon Sep 23, 2002 at 02:41:18 PM EST

are disappointed words.

---
"They are in love. Fuck the war."

Suddenly, I want bananas. (nt) (none / 0) (#17)
by graal on Mon Sep 23, 2002 at 02:55:52 PM EST


--
For Thou hast commanded, and so it is, that every
inordinate affection should be its own punishment.
-- St. Augustine (Confessions, i)
[ Parent ]

Stay away from toilets and giant crabs though. (none / 0) (#109)
by Gumpzilla on Tue Sep 24, 2002 at 11:56:28 AM EST

Can't really recommend the smegma stew, either.

[ Parent ]
Great. Nightmares now. (nt) (none / 0) (#147)
by graal on Fri Sep 27, 2002 at 05:03:57 PM EST


--
For Thou hast commanded, and so it is, that every
inordinate affection should be its own punishment.
-- St. Augustine (Confessions, i)
[ Parent ]

No no no!!! (none / 0) (#64)
by CodeWright on Mon Sep 23, 2002 at 08:41:43 PM EST

The split sentence

is the medham signature! Keep it together man! Buck up!

--
"Humanity's combination of reckless stupidity and disrespect for the mistakes of others is, I think, what makes us great." --Parent ]
You were saying something nice about me, (none / 0) (#68)
by Noam Chompsky on Mon Sep 23, 2002 at 09:12:30 PM EST

I can feel it.

---
"They are in love. Fuck the war."
[ Parent ]

I was (none / 0) (#120)
by CodeWright on Tue Sep 24, 2002 at 04:51:41 PM EST

Sometimes the comments of your various personae are why I read this site...

...and sometimes they are why I stop reading this site.

I know you and Ralph hate the casus of small-minded hobgoblins, but can it be so bad given the alternative?

--
"Humanity's combination of reckless stupidity and disrespect for the mistakes of others is, I think, what makes us great." --Parent ]
This is my only active account. (none / 0) (#126)
by Noam Chompsky on Tue Sep 24, 2002 at 07:08:55 PM EST

It really is.

---
"They are in love. Fuck the war."
[ Parent ]

Yes (none / 0) (#138)
by CodeWright on Wed Sep 25, 2002 at 10:04:53 AM EST

As long as the definition of "my" refers only to your currently active personality splinter. :)

--
"Humanity's combination of reckless stupidity and disrespect for the mistakes of others is, I think, what makes us great." --Parent ]
Spacetime Numbers (5.00 / 1) (#15)
by Seth Finkelstein on Mon Sep 23, 2002 at 02:52:09 PM EST

There's also spacetime numbers a.k.a. hyperbolic numbers.


-- Seth Finkelstein

Great article... (3.00 / 1) (#19)
by Logan on Mon Sep 23, 2002 at 03:00:39 PM EST

But not a single link to Planet Math!

Logan

i did! i did! (none / 0) (#34)
by Xcyther on Mon Sep 23, 2002 at 04:16:24 PM EST

second paragraph - imaginary number 'i'. view source - ctrl-f - planetmath - enter :)

_________________________________________
"Insydious" -- It's not as bad as you think

[ Parent ]
owie (3.20 / 5) (#26)
by sykmind on Mon Sep 23, 2002 at 03:30:30 PM EST

my brain hurts....make it stop

Agreed. (none / 0) (#73)
by Arkayne on Mon Sep 23, 2002 at 10:31:30 PM EST

Heh.. I know what you mean. I have zero experience with all this but figured I was up for the challange of trying to read the entire article.

You know that scene from Scanners where the guy's head explodes? So me right now.

[ Parent ]

Damn You! (2.16 / 18) (#31)
by PullNoPunches on Mon Sep 23, 2002 at 03:57:05 PM EST

Now I must wank!

------------------------

Although generally safe, turmeric in large doses may cause gastrointestinal problems or even ulcers. -- Reader's Digest (UK)

Rate this up (5.00 / 1) (#99)
by Stick on Tue Sep 24, 2002 at 08:30:09 AM EST

Highly amusing :-)


---
Stick, thine posts bring light to mine eyes, tingles to my loins. Yea, each moment I sit, my monitor before me, waiting, yearning, needing your prose to make the moment complete. - Joh3n
[ Parent ]
Hey, Thanks (5.00 / 1) (#113)
by PullNoPunches on Tue Sep 24, 2002 at 01:14:28 PM EST

Now I must....

Get back to work.

------------------------

Although generally safe, turmeric in large doses may cause gastrointestinal problems or even ulcers. -- Reader's Digest (UK)
[ Parent ]

Erratum #1 (5.00 / 4) (#36)
by My Alternative Account on Mon Sep 23, 2002 at 04:23:13 PM EST

In the quaternions ij=-ji=k.

ah thanks (none / 0) (#60)
by PotatoError on Mon Sep 23, 2002 at 07:54:05 PM EST

I spent half an hour trying to figure that out.

when he wrote "ij=ji=k" and then later "but i times j isn't j times i" it really confused me.

Guess I should have read comments as well :P

good article tho

[ Parent ]

Natural numbers? (none / 0) (#37)
by yamla on Mon Sep 23, 2002 at 04:25:02 PM EST

I thought natural numbers started at 1 and went up by one each time. The set of whole numbers, then, was the set of natural numbers with the number 0 added. Am I wrong?

I take it (none / 0) (#79)
by skim123 on Mon Sep 23, 2002 at 11:33:20 PM EST

I think it's probably better to leave out the zero myself and I wish that was what I had done

You are not a computer scientist? :-)

Money is in some respects like fire; it is a very excellent servant but a terrible master.
PT Barnum


[ Parent ]
The least natural number (none / 0) (#41)
by Three Pi Mesons on Mon Sep 23, 2002 at 04:51:57 PM EST

Some people think that the "natural numbers" ought to be {1, 2, 3, ...}, and some think they should be {0, 1, 2, ...}. There's no real consensus as to which choice is correct - it's something you can argue about forever without convincing anyone.

The set of whole numbers is not the same as the naturals including zero: negative numbers are "whole", but not natural.

:: "Every problem in the world can be fixed with either flowers, or duct tape, or both." - illuzion
[ Parent ]

Whole numbers include negative? (none / 0) (#56)
by yamla on Mon Sep 23, 2002 at 06:42:35 PM EST

Where I was brought up (England, I think... but it could have been in Canadian schools), whole numbers did not include negatives.  Thus, we had:

Natural numbers: 1, 2, 3, ...
Whole numbers: 0, 1, 2, 3, ...
Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...

I had never heard of anyone claiming (and I'm not saying you are wrong) that whole numbers included negatives.

[ Parent ]

Whole numbers... (none / 0) (#72)
by nustajeb on Mon Sep 23, 2002 at 10:02:51 PM EST

http://mathworld.wolfram.com/WholeNumber.html
http://www.sparknotes.com/math/prealgebra/wholenumbers/summary.html

When learning math, as a child in the U.S., I also learned that the set of whole numbers was the set  containing 0 and the natural numbers. However outside of grade school, natural numbers seem to be used quite frequently as synonymous with the whole numbers. Later still you find writing that uses the forms mentioned in Math World above, so it seems to be a matter of where you look.

[ Parent ]

Negatives (none / 0) (#98)
by Three Pi Mesons on Tue Sep 24, 2002 at 07:41:55 AM EST

Well, I always thought of negative numbers as being "whole", in the sense that they're "completely" integers, rather than integers-plus-a-bit.

:: "Every problem in the world can be fixed with either flowers, or duct tape, or both." - illuzion
[ Parent ]
ten's/two's complement is different. (4.00 / 1) (#39)
by nex on Mon Sep 23, 2002 at 04:40:15 PM EST

incredibly interesting article, interesting links. however, i doubt that "...9999999" is a number and i'm sure that it certainly doesn't equal 1. of course computers use strings of figures that look like that to represent negative numbers, but they have a very clearly defined end to the left.

I wouldn't touch the 10-adics (none / 0) (#44)
by Humuhumunukunukuapuaa on Mon Sep 23, 2002 at 05:10:46 PM EST

The 2-adics are pretty cool though and I have a I have a diary on them. The 10-adics are perfectly good for multiplying, adding and subtracting (and ...99999 is definitely -1) but there are some divisions that you can't do with the 10-adic numbers. I've not heard of tens complement arithmetic before but I just did a quick web search on the term and it seems to me that 10-adic arithmetic is another way of thinking about this. With the 10-adics you can ignore the instruction to "disregard the 1" because it ultimately gets carried out to infinity.
--
(&()*&^#@!!&_($&)!&$(*#$(!$&_(!$*&&!$@
[ Parent ]
What's a number? (none / 0) (#46)
by phliar on Mon Sep 23, 2002 at 05:22:19 PM EST

... however, i doubt that "...9999999" is a number
This brings up the interesting philosophical point: what are the attributes we want from an object for us to call it a number?

Forget the fact that we use the digit "9" and we're using the ellipsis in the way we normally use it to the right of the decimal point. If we change the notation slightly -- say we represent the object above as A9 and instead of the equals sign, we use a tilde "~" to represent an equivalence relation that holds. Then we can say

A9 ~ -1
If we can talk about these objects and say that "addition" is
A2 "+" A5 ~ A7
i.e. we define enough rules that you can always "add", and maybe perform other operations that we name "multiplication", "division" etc., and the expected properties hold like the distributive law etc. -- then would you accept that these are "numbers"?


Faster, faster, until the thrill of...
[ Parent ]

'tis too. (none / 0) (#77)
by mindstrm on Mon Sep 23, 2002 at 11:12:26 PM EST

You get into infinities, infinities of infinities, aleph, etcetera... I don't have my little book handy, and it's way too fuzzy.

You can do math in infinities as well.

If you have a number represented by an infinite string of nines... obviously if you add 1 to it, you end up with an infinity of zeroes, and a carried 1 that we don't know what to do with.

[ Parent ]

re. to all above: ...999 does NOT equal -1, mkay? (none / 0) (#119)
by nex on Tue Sep 24, 2002 at 03:22:20 PM EST

just because someone doesn't know what to to with the carry, it's not true that you get something that equals 0. imagine an infinite string of nines. the number this represents in base 10 must be some special kind of infinity. now if you add one, the result is exactly the number you had before, incremented by a value of exactly one. you could represent these values in entirely different notations and you wouldn't get a string of 0s at all after adding them. you would immediately see that you've still got some incredibly big number, some special kind of infinity.

2's / 10's / n's complement is something entirely different. it's about writing/calculating with/storing negative numbers, not by storing a bit representing the sign, but by shifting the range of available numbers down by half that range, so you get (approximately) equal chunks of positive and negative numbers. that shifting is done in a clever way that ensures that a word of 0s still has the value 0 and that microchips can still perform additions, subtractions and other arithmetic operations in the usual way.

so, on the one hand, you've got an infinite string of nines, which is defined to mean the sum of all 9 * 10^n (nine times ten to the power of n), where n runs from 0 to infinity. on the other hand, you've got a finite string of nines, which is defined to mean -1. it equals -1, because it's defined to mean -1. but it's not the same as the incredibly huge sum described above. it's something entirely different.

if i failed to get my point across, feel free to ask.

.. .. .. .. .. .. .. ..
:: My user page just told me I'm not currently a paid member,
:: but I could become one. Sure, of course I'd like to be paid!
[ Parent ]

What do you mean "does NOT"? (5.00 / 1) (#123)
by phliar on Tue Sep 24, 2002 at 06:42:11 PM EST

If you don't know what a p-adic is, you can't go around saying something does or does not equal something else. You are in the position of saying "a negative number cannot have a square root, so obviously i2 CANNOT be -1." Here's a good introduction to p-adics (in PostScript).

Perhaps you are getting too hung up on the place-value notation and carry and all that. Think of them as mnemonics. Think of a p-adic as an infinite sequence of integers 0 <= ai < p. Operators we name "addition," "multiplication," etc. can be formally defined on these objects, and these operators are consistent, useful, and follow the properties we normally expect from addition, multiplication etc. on integers.

Does that make it better?


Faster, faster, until the thrill of...
[ Parent ]

5.00 for the interesting link (none / 0) (#128)
by nex on Tue Sep 24, 2002 at 07:41:23 PM EST

thanks for the interesting link. however, i was referring more to the original article, which implies that "...999" (the number!) equals -1. i didn't say a negative number couldn't have a square root. of course, when you say, this p-adic equals that p-adic, you're perfectly right. the article is also perfectly right when stating that ...9999992==1, since it's made clear that these mnemonics denominate p-adics. but the implication i mentioned before says you could do that with your ordinary integer numbers, which is wrong.

i hope we're of the same opinion now :-)

[ Parent ]

Well okay.. (none / 0) (#150)
by mindstrm on Fri Oct 11, 2002 at 12:42:37 AM EST

So what is 9999999 repeating equal to in integer math?


[ Parent ]
makes sense (none / 0) (#129)
by nex on Tue Sep 24, 2002 at 07:44:52 PM EST

yup, that way it makes sense perfectly, i agree.
(see my reply to the other reply to my reply g)

[ Parent ]
I'll stick to non-base-ten, myself (none / 0) (#48)
by rickward on Mon Sep 23, 2002 at 05:25:01 PM EST

Did anyone else have to read The Magic Numerals of Ali Khayyam in school?

"Crack don't smoke itself." —Traditional

you forgot (none / 0) (#53)
by thirstyfish on Mon Sep 23, 2002 at 06:24:18 PM EST

C, C#, D, Eb, E, F, F#, G, Ab, A, Bb, B

Hmm... (none / 0) (#58)
by RofGilead on Mon Sep 23, 2002 at 07:08:50 PM EST

Western music is really more of a base 7 system.

Each scale consists of the I-VII notes.

So, its really just integer math.

-= RofGilead =-

---
Remember, you're unique, just like everyone else. -BlueOregon
[ Parent ]

Hahahaha! (none / 0) (#111)
by x31eq on Tue Sep 24, 2002 at 12:27:58 PM EST

Well, it's both. The equal tempered system is a new arrival. The most popular tunings historically are meantones, and they need two numbers to describe a note. Staff notation is a legacy of this (and the older Pythagorean) -- C# is distinct from Db.

You can look at the general case if you like. But the simple result is that any interval in staff notation can be expressed by both the number of steps on a 7 and 12 note scale, and you need both for a complete description.



[ Parent ]
Homogenous Matrices vs. Quaternions (4.50 / 2) (#54)
by Blarney on Mon Sep 23, 2002 at 06:26:16 PM EST

Quaternions are really neat creations - you can do lots of really nice mathematics. However, as many computer graphics people simply use them to rotate a point in 3-space around a specified direction by a specified angle (works out well with the 4-entry quaternions), it might be good to mention homogenous matrices as well.

Homogenous matrices can perform any operation that quaternions can - basically rotations and scaling - and can combine rotations by multiplication just as quaternions can. However, homogenous matrices can also represent shearing and translation, and can combine these operations by multiplication as well. Finally, homogenous matrices may be generalized to higher dimensions far more easily than quaternions can be. This greater generality comes at the price of using 16 numbers for a 3-space point instead of the 4-number simplicity of a quaternion... but is good enough for many common graphics APIs, including both DirectX and OpenGL.

Homogenous matrices work just like ordinary matrices ... but instead of feeding your 3-vector to a 3x3 matrix and multiplying, you start with a 4-vector with the first 3 entries being your 3-vector and the last entry being a "1". Multiply by a 4x4 matrix instead of a 3x3, and finally convert your resulting 4-vector back to a 3-vector by dividing each of the first 3 entries by the last entry and then discarding the last entry. If you got a "0" in the last place, then your 4x4 matrix doesn't represent a valid transformation ... it scales to infinity. Just be sure not to do that, and it'll be lots of fun.

Another reason. (4.00 / 1) (#90)
by Slothrop on Tue Sep 24, 2002 at 04:09:36 AM EST

I think that the most compelling reason to use quaternions, other than storage space considerations (4 words v 16 or 9) is that they can be smoothly interpolated between one another, producing intermediate rotations.  This makes them ideal for using in games and animation, since you can smooth any animation expressed via quaternions to an arbitrary level of detail, or use them to blend several together.  

Of course, other representations can be used to interpolate, but all of them seem to have different, negative qualities that make them problematic when interpolating.  Gimbal lock, jerkiness, infinity problems, etc.
__________________________
Provide, provide!
[ Parent ]

uhm (2.00 / 1) (#61)
by kha0z on Mon Sep 23, 2002 at 08:10:28 PM EST

infinities make my head hurt... :) i'll stick to finites for the time being.
--kha0z
That's what Kronecker said (5.00 / 1) (#62)
by the on Mon Sep 23, 2002 at 08:24:15 PM EST

But he found he had to give up the real numbers as a result.

--
The Definite Article
[ Parent ]
j whose square is -1 ? (2.66 / 3) (#63)
by chbm on Mon Sep 23, 2002 at 08:33:49 PM EST

So what's the diference between j and i ?

-- if you don't agree reply don't moderate --
Okay.. (none / 0) (#76)
by mindstrm on Mon Sep 23, 2002 at 11:09:03 PM EST

So both i and j square to -1.

What does "different quaternions" mean?

-i and i square to -1, yes.

Do -j and j square to -1 as well?

Are i & j not the same thing then?

[ Parent ]

They act the same in those equations (none / 0) (#82)
by PurpleBob on Tue Sep 24, 2002 at 01:13:56 AM EST

The parent poster was giving an example of how two numbers that give the same result in an equation can still be different numbers. Finding another example of how they give the same result does not prove anything.

An example where they are different:

i + -i = 0
j + -i = j-i which does not equal 0.

You could ask "But why isn't j-i 0? With my assumption you'd be subtracting a number from itself and then you'd get 0." You could indeed assume that. Then i=j=k, and you end up getting the complex numbers instead of the quaternions, and losing the special properties of the quaternions.

[ Parent ]

Thanks. (none / 0) (#151)
by mindstrm on Fri Oct 11, 2002 at 12:54:12 AM EST

Thanks to all who replied. I get it, sort of.

I'm still stuck thinking standard algebra I guess.

[ Parent ]

Example (4.00 / 1) (#103)
by roiem on Tue Sep 24, 2002 at 09:36:03 AM EST

2 squared is 4, and also (-2) squared is 4, but that doesn't mean that 2=-2, just that they share a property. You just need to let go of the notion that every number has exactly two square roots. Every number has exactly two complex square roots, but more than that (eight? not sure) quaternion roots, and even more (16?) octonion roots.
90% of all projects out there are basically glorified interfaces to relational databases.
[
Parent ]
Precision (none / 0) (#149)
by dtclausen on Sun Oct 06, 2002 at 12:46:44 AM EST

It's important to note that ALL of the systems mentioned in in the article can be defined precisely starting from the simple axioms of set theory... for example, assuming the reals are already defined, you can define the complex numbers as the set of ordered pairs (a,b) (a way to define the ordered pair (a,b) in terms of sets is {a, {a,b}}, which implies the fundamental property: (a,b)=(c,d) if and only if a=c, b=d, which is the only property we want/need) where a and b are real numbers. You *define* addition on complex numbers as (a,b)+(c,d) = (a+c,b+d) and multiplication as (a,b).(c,d)=(a.c - b.d, a.d + b.c), where the . on the right hand side is multiplication of real numbers. There's no need to worry about whether i exists: it's simply (0,1) and was precisely defined. Similar constructions exist for quaternions, as I mentioned, and from them it's obvious that j is different from i, and all of your questions should melt away.

[ Parent ]
Explanation please (none / 0) (#154)
by hesk on Sat Nov 02, 2002 at 07:45:04 PM EST

a way to define the ordered pair (a,b) in terms of sets is {a, {a,b}}

I remember seeing this in my logic class two years ago, and I'm totally lost on this. What's the rationale behind this? I understand, that (a,b) != (b,a), because {a,{a,b}} != {b, {b,a}}, but that surely isn't all, or is it?

--
Sticking to the rules (red lights etc.) doesn't improve your safety, relyi
[
Parent ]

Engineering (none / 0) (#121)
by paugq on Tue Sep 24, 2002 at 05:05:22 PM EST

Normally engineers use "j" instead of "i", because insensity is also represented "i". If you use "i" both for "intensity" and for the imaginary unit, you get a mess.

[ Parent ]
i is for current, that's why EE's use j (n/t) (none / 0) (#132)
by Big Sexxy Joe on Tue Sep 24, 2002 at 10:00:07 PM EST



I'm like Jesus, only better.
Democracy Now! - your daily, uncensored, corporate-free grassroots news hour
[ Parent ]
How can we talk about R (4.00 / 2) (#66)
by dcodea on Mon Sep 23, 2002 at 08:47:39 PM EST

Without mentioning Dedekind cuts? Remember, what you think it a real number is really just the set of all rational numbers less than the number you think you're thinking of.

Who Dares Wins

hexadecimal! (2.33 / 3) (#67)
by Fen on Mon Sep 23, 2002 at 09:05:00 PM EST

If you keep using decimal, then you have no hope at all.
--Self.
Oh no (none / 0) (#83)
by PurpleBob on Tue Sep 24, 2002 at 01:22:37 AM EST

This sounds like "thinkit", who has drifted between various newsgroups and mailing lists, searching for any use of decimal, and then invariably replying with a rant about how hexadecimal is vastly superior and decimal users will go extinct like the dinosaurs. One instance was on, I believe, a health newsgroup, where he wanted people to stop talking about things in metric and use his newly-invented hexadecimal metric system. He latched onto the Lojban language mailing list for a while too, telling newbies the lie that all numbers in Lojban are hexadecimal.

thinkit, is that you? Get help.

[ Parent ]

Wow! (4.80 / 5) (#75)
by glog on Mon Sep 23, 2002 at 10:51:38 PM EST

You have a gift for writing informative, entertaining content. This has got to be one of the most refreshing math articles that I have ever read. And I mean it. I am no weenie when it comes to math but it's just that there are one too many professors and books that make it all seem so cut and dry. I love the way you mix in the plain old vanilla stuff with tasty chunks about applications, bits of history, and a touch of humor.

I would love to see more articles written by you about this or other topics. In addition, if you haven't already written a book consider doing so.

Nonstandard Analysis / Hyperreals (4.75 / 4) (#80)
by Repton on Tue Sep 24, 2002 at 12:23:49 AM EST

I'll say a bit more about nonstandard analysis, because I think it's interesting.

You can make the real numbers out of the rational numbers by considering sequences: You can represent any real number as a converging sequence of rational numbers. For example, the sequence for pi would be:

3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ...

(a converging sequence is a sequence of numbers that gets closer and closer to some number ... in this case, these numbers, which are rational because they have a finite decimal expansion, are getting closer and closer to pi)

(strictly speaking, this is not the sequence for pi; rather, it is one of many)

If you try the same trick with real numbers, and you also decide not to care if the sequences converge or not, you get the hyperreal numbers, R*.

The hyperreals add two interesting things to the normal number line: infinitessimal numbers and unlimited (or infinite) numbers. They are basically what you think they are: a positive infinitessimal is smaller than every positive real number, but larger than zero. A positive unlimited number is larger than every real number.

This gets interesting if you want to do calculus. Any "normal" sequence of real numbers can be "extended" to a sequence of hyperreal numbers. And likewise, any normal real-valued function (eg, sin()) can be extended to a hyperreal-valued function.

So if you want to find the limit of a sequence at infinity, all you do is look at the Nth term, where N is some unlimited hyperreal number. (in a sense: since we have infinite numbers on our number line, if we want to find out what a sequence does after infinitely many terms, you just go to the infinity-th term and look at it...)

And if you want to take the limit of a function f(x) as x tends to zero, you can just evaluate that function at an infinetissimal number.

The upshot of this is that you can prove a lot of calculus results without using limits (and genuinely new results have been proven in this way).


--
Repton.
They say that only an experienced wizard can do the tengu shuffle..

Irrational sequences (none / 0) (#89)
by treefrog on Tue Sep 24, 2002 at 04:05:31 AM EST

OK, question time :-)

We all know that PI is an irrational number, i.e. there is no rational number that expresses the ratio of the circumference of a circle to its diameter. In fact, I believe that in base 10 PI has a lot of other interesting properties involving the randomness of the decimal sequence.

Do these properties hold in other bases?

Scratching his head.... treefrog


Twin fin swallowtail fish. You don't see many of those these days - rare as gold dust Customs officer to Treefrog
[ Parent ]

Don't confuse... (none / 0) (#101)
by Rocky on Tue Sep 24, 2002 at 08:55:10 AM EST

the representation of a number with its intrinsic value.

If we knew what it was we were doing, it would not be called research, would it?
- Albert Einstein (1879 - 1955)
[ Parent ]
He/she wasn't (4.00 / 1) (#104)
by roiem on Tue Sep 24, 2002 at 09:44:46 AM EST

Your parent was asking about certain properties of the decimal representation of pi, and whether these were also properties of the binary, octal, etc. representations of pi. It's obvious (to me, at least) that the properties of thr number pi (It's irrational, it's the ratio between the circumference and the diameter, etc.) hold no matter how you represent the number.
90% of all projects out there are basically glorified interfaces to relational databases.
[
Parent ]
quaternions (2.50 / 2) (#85)
by Big Sexxy Joe on Tue Sep 24, 2002 at 01:30:35 AM EST

How many roots does an equation have under quaternions? (i.e. Does the fundemantal theorem of algebra apply?) Does Euler's formula apply for both i and j?

Great article, by the way.



I'm like Jesus, only better.
Democracy Now! - your daily, uncensored, corporate-free grassroots news hour

Damn it (4.50 / 4) (#87)
by kholmes on Tue Sep 24, 2002 at 01:53:32 AM EST

My brain exploded. Look what you did.

Why do you think people stop taking math after high school? This is dangerous stuff.

If you treat people as most people treat things and treat things as most people treat people, you might be a Randian.

Never mind Tylenol; pass the Quaaludes! (none / 0) (#107)
by Mr Incorrigible on Tue Sep 24, 2002 at 11:12:48 AM EST

My head hurts; I'm too friggin' ignorant in this area... *mew*

Damned interesting article, though. I'm going to have to go digging through some old calculus texts to grok it; I never bothered with any maths heavier than trigonometry in high school. After high school, most of my education came from RTFM and I was a wee bit more interested in programming and philosphy than in maths. Besides, I think that for most bread-and-butter programming (the stuff code monkeys like me do to pay the rent) a rock-solid grounding in formal logic and algebra are more important than the really heavy stuff.

All the same, a little more knowledge never killed anybody. At least, not after the power of the Roman Catholic Church was shattered during the Reformation and later, the Enlightenment

--
I know I'm a cheeky bastard. My lady tells me so.


[ Parent ]
Kewl. Thanks. (none / 0) (#117)
by Mr Incorrigible on Tue Sep 24, 2002 at 02:04:15 PM EST

I'll have a look at these when I have a weekend to be slack. I hope some of 'em run under FreeBSD, though. Thanks!

--
I know I'm a cheeky bastard. My lady tells me so.


[ Parent ]
Math and high school (none / 0) (#118)
by phliar on Tue Sep 24, 2002 at 02:53:17 PM EST

My brain exploded.
Yeah -- ain't it cool?!

I think the problem with math education in school (and this holds in all societies, I think, not just the US) -- is that we make it dry and tool-based: this is how to solve problems involving buckets being filled by pipes. And the students ask: how often in real life am I going to calculate how long the bucket will take to fill? We need to teach that math is cool the same way music and literature are cool. (And as a bonus we can calculate how long it will take the bucket to fill.)

"Good -- he did not have enough imagination to become a mathematician." -- Hilbert, after being told a student had quit math to become a poet.

Faster, faster, until the thrill of...
[ Parent ]

Complex Numbers (4.00 / 2) (#88)
by YellowNumber5 on Tue Sep 24, 2002 at 01:59:15 AM EST

Bah, I Don't beleive in these complex numbers, I don't even think they are Real. Sorry, couldn't resist - Yellow #5

believe this... (none / 0) (#155)
by spectatorion on Mon Nov 11, 2002 at 08:38:19 PM EST

In a senior-level quantum mechanics class at my university (where things like this should not happen), when being told of the mathematical prerequisites for the course, which include a basic knowledge of complex numbers and their properties, the gentleman sitting next to me turned to me and said, "You know, I don't believe in the complex numbers." Honest.

[ Parent ]
Off-topic, for further reference (4.00 / 1) (#92)
by metropacem on Tue Sep 24, 2002 at 05:21:48 AM EST

This file contains info about various mathematical and diverse HTML entities, so if you want to display aleph, you'd write &#8501; [ℵ].

I hope this is helpful. :)

--
[shiki soku ze kuu]
Unfortunately, No (none / 0) (#134)
by MyrddinE on Wed Sep 25, 2002 at 02:39:28 AM EST

Unless an aleph is a square box. :-)

I'm probably missing some extended font support.

[ Parent ]

Here's an interesting set (4.00 / 2) (#93)
by salsaman on Tue Sep 24, 2002 at 07:07:08 AM EST

The set of binary numbers which starts with a 1, and does not contain 11. The first few of these are:
1
10
100
101
1000
1001
1010
10000
10001
10010
10100
10101
100000
100001
100010
100100
100101
101000
101001
101010

These numbers are interesting for two reasons. Firstly, they can be used for compression.

Suppose we make:
1 = e
10 = t
100 = i
101 = a
1000 = n
1010 = s

etc.

We can now compress text by sending these codes with the digits reversed, each code followed by a 1. For example, the word "tins" would be:

01 1 001 1 0001 1 0101 (1*)
 t     i      n      s

(*We can omit the last 1 if we know the string length.)

Just for fun, try and decode this:

0001111101101

The other interesting fact about these numbers is that they have a relation to a Fibbonaci series.

We can show this by counting how many such numbers there are for a given length.

Looking at the list above, we can count for each length:

Length (L)            Number of codes (N)
1                            1
2                            1
3                            2
4                            3
5                            5
6                            8

It is possible to prove quite easily that L,N do form a Fibbonaci series; this excercise is left to the reader :-)

Is not that... (none / 0) (#125)
by gmuslera on Tue Sep 24, 2002 at 07:05:45 PM EST

... Huffmann compression? Understanding how is generated (binary tree, left 0, rigth 1, etc etc) could make things clear.

[ Parent ]
It's a type of Huffman code (none / 0) (#127)
by salsaman on Tue Sep 24, 2002 at 07:16:15 PM EST

but in general Huffman codes allow '11', and allow numbers to start with a '0'.

[ Parent ]
Inconsistency (none / 0) (#94)
by dark on Tue Sep 24, 2002 at 07:08:15 AM EST

"i times j isn't j times i", but also, "ij=ji=k". Which is it?

Typo ? (none / 0) (#95)
by salsaman on Tue Sep 24, 2002 at 07:09:29 AM EST

I think he meant j times k isn't k times j.

[ Parent ]
Or maybe not (none / 0) (#96)
by salsaman on Tue Sep 24, 2002 at 07:13:46 AM EST

See this comment

[ Parent ]
Ok, thanks. (none / 0) (#97)
by dark on Tue Sep 24, 2002 at 07:24:02 AM EST

Weird, I did read the comments before posting, but I missed both references to this mistake.

[ Parent ]
Ralph & Earl (2.00 / 1) (#108)
by Lenny on Tue Sep 24, 2002 at 11:36:16 AM EST

The universal language makes the universe noxious.


"Hate the USA? Boycott everything American. Particularly its websites..."
-Me
A new definition for MLP (none / 0) (#114)
by kallisti on Tue Sep 24, 2002 at 01:28:51 PM EST

Your link to the p-adics goes to John Baez's explanation of the octonions. Where did you intend the link to go to? I didn't quite understand the p-adics from the description given and the Treasure Trove entry is rather obtuse.

Great Math Link Propagation article.

math section (4.50 / 2) (#130)
by cronian on Tue Sep 24, 2002 at 08:27:21 PM EST

With all the math articles that are being posted these days, it seems like it would make sense to make a section of the site just for math articles. What does everyone think?

We perfect it; Congress kills it; They make it; We Import it; It must be anti-Americanism
I like it (5.00 / 1) (#133)
by dirvish on Tue Sep 24, 2002 at 10:35:02 PM EST

Yeah, then those articles would be easier to avoid..

Technical Certification Blog, Anti Spam Blog
[ Parent ]
Love to see an article on non-standard logics (4.00 / 1) (#135)
by CleverNickname on Wed Sep 25, 2002 at 02:47:48 AM EST

Please be sure to cover: intuitionistic logics, different modal logics including the more popular ones like temporal CTL/LTL. I'd also be up for more esoteric stuff.

Nice article BTW! Any chance of seeing something that covers abstract algebra?

is this right? (2.00 / 1) (#136)
by fringd on Wed Sep 25, 2002 at 07:22:05 AM EST

so why not try our luck again and throw in j whose square is -1. The catch now is: what is ij? Well let's call it k. In fact if we add the rules ij=ji=k, jk=-kj=i, j^2=k^2=-1,

this tweaks me. if jk = -jk, then shouldn't j = -j and j=0? same with k? also, if j^2=k^2=-1, and k=ji, then shouldn't k^2=j^2 x i^2 = (-1)(-1) = 1. doesn't this yield -1=k^2=1 !?

maybe there's a typo, or maybe it's just so weird that i'm not groking it. corrections or explanations would be great :)

wait, i get it. (none / 0) (#137)
by fringd on Wed Sep 25, 2002 at 07:56:20 AM EST

ok i've read the erratum.

ij=-ji=k. ok so k^2 != (j^2)(i^2), k^2 doesn't even equal (i^2)(j^2), k^2 = (ij)(ij) = i(ji)j = i(-1ij)j = [i think i can move the -1] ... -1(ii)(jj) = -1(-1)(-1) = -1 which grocks... wacky.

this whole mess really wacks one's concept of what numbers are. it's easy for a guy to imagine that a number is something that describes the count of something, but i guess it more describes the amount of something, or eventually i guess it just describes something in someway.

i think i'm just starting to grok the role of a number as an abstract descriptor of any type, but it's taking a second. i'll have to take more number theory classes.

anybody have a good generalized definition or way to think of what a 'number' really is? if it can be anything, what makes a number worth using? does it even have to relate to anything concrete, or even describe something?

[ Parent ]

Descriptors, commutativity, and quantum mechanics (4.50 / 2) (#142)
by Dr. Zowie on Wed Sep 25, 2002 at 07:48:34 PM EST

Thanks, MAA, for a really cool exposition.

Numbers are really good at describing, well, anything at all -- be it geometrical (like the number line) or weird (like, er, rotations in 5-space) or even logical (like, er, mathematical proofs). Godel's theorem (which among geeks is so cliche and trite that we forget how powerful and interesting it is) is the famous proposition that every interesting logical system must have unprovable tautologies -- universal truths that cannot be proven within the logical system. "Interesting" here means that the system is powerful enough to contain statements about itself. He proved it, in part, by pointing out (in another revolutionary argument that is now so trite it seems hokey) that it's possible to represent any statement in any logical system with a number; and it's possible to construct mathematical operations that correspond to the laws of derivation for any logical system.

On another topic, non-commutativity has a lot of really interesting applications in the real world. For example, you can use matrices (another weird, multidimensional form of number) to represent rotations in three dimensions. There are special (but simple) rules for multiplying matrices. If you have two matrices that represent rotations, you can find out the result of making both rotations by multiplying the two matrices together. But it matters which order you do rotations (for example, turn your head 90 degrees right, then 90 degrees up. Now you're looking straight up, with a crick in your neck. If instead you turn your head 90 degrees up then 90 degrees right, you end up looking to the right with your head bent sideways). So of course matrices have to break commutativity too, to match that behavior.

Another really deep, strange, and interesting application of non-commutativity comes with quantum mechanics. It's so interesting that I'll take a few minutes to go off on a tangent and describe it.

There are two very interesting formulations of classical mechanics that use energy, rather than forces, to represent the system (Remember, in classical mechanics you're trying to describe the behavior of, basically, systems of springs, levers, ropes, wheels, brakes, and such -- macroscopic objects). Those two formulations are are Hamiltonian and Lagrangian mechanics, and they're taught in upper-division college physics curricula everywhere. In both systems you write down all the possible free parameters of the system as "generalized dimensions" that aren't necessarily spatial -- they could represent, for example, how far you have bent your little finger, or how electrically charged you've become, or whatever -- so long as each one describes some aspect of the system.

So far, so good. Lagrange mechanics is really neat, because it turns the subtle art of figuring out where things will move into a mechanical process: just write down all possible parameters of the system, figure out how the potential and kinetic energy of the system depends on each one, and plug in a simple formula -- and out pops the answer for how the system behaves.

But that's not the cool bit. Enter a mathematician named Emma Noether, from around the beginning of the 20th century. She made an incredible, brilliant discovery that tied together two bits of physical theory that most folks don't even think about. Allow me to digress for a couple more paragraphs, to set the stage.

One important concept to any aspiring physicist is the conserved quantity. You've probably heard of the conservation of energy, in the sense that there's something called "energy" that is associated with every physical system. You can compute how much there is, and then come back later and account for it. The energy isn't created or destroyed, only moved around the system. Likewise, Newton's second law ("for every action there's an equal and opposite reaction") can instead be expressed as a conservation of momentum. There are perhaps half a dozen of these important conserved quantities that you learn in first-year physics.

There's another aspect of physical law that people wonder at, which is invariance, and many people have spent time trying to figure out how you can tweak physical systems and leave the physics intact. By that, I mean that the experiments come out the same. For example, it doesn't matter what time you do an experiment -- we expect that physics experiments will get the same outcome whether they're done Monday morning or Friday afternoon, or 100 years in the future. Likewise, we don't expect the experiments to change just because we move the apparatus across the room or around the world. (Unless, of course, the experiment depends on the stuff around it -- you have to move the whole apparatus, or at least the bits that matter -- so it's no fair pointing out, for example, that moving a boat from the ocean to to Death Valley at sea level will make it fall 300 feet to the ground.)

Noether's great discovery is that symmetries in physical law (like invariance under motion or time of day or angle) are intimately connected to conserved quantities. In face, she constructed a mechanical apparatus -- an algorithm -- for converting physical symmetries into conserved quantities, and vice versa. You can start with a physical symmetry like invariance under displacement in time, plug it into the Noetherian apparatus, and get out a conservation law (in this case, conservation of energy). Plug in displacement in position, and you get out momentum. Plug in change in angle, and you get out angular momentum. Every symmetry has an associated conservation law, and vice versa. Wow.

even curiouser, if you multiply together any conserved quantity and its associated symmetric variable (like energy and time, or momentum and position), you get out something with units of "action", energy * time. That's the same units as Planck's Constant, the fundamental quantum mechanical constant.

In fact, if you start with ordinary Lagrangian classical mechanics, and then take away commutativity (in a limited way), you get back out quantum mechanics for free! That's very strange. The rule change is that you keep commutativity (ab=ba) for most things, but that any quantity and its associated ``Noetherian J-form'' (the thing that's conserved if physics is invariant under changes in your quantity) fail to commute. In particular, you just have to set (for example) (momentum) (position) = (position)(momentum) - i h/ 2 / \pi, where 'i' is sqrt(-1) and h is a measurable physical constant with units of action (6.6e-34 J s). When you've done that, all of a sudden you can use exactly the same theorems and techniques to solve your problems, except that now you get quantum mechanics instead of classical mechanics.

That's bizarre, but it shows how important non-commutativity can be!

Yow -- that was a lot longer than I intended.

[ Parent ]

D'Ni Number Systems (3.00 / 1) (#140)
by exZERO on Wed Sep 25, 2002 at 04:40:52 PM EST

If you're interested, check out the site www.DniDesk.com to learn about the D'ni mathematical system, from the "MYST" series of games.  It's base 25, and is very detailed considering its only used for 3 books and 3 games.  The symbols used for one through 5 are based loosely on "normal" numerical symbols

Shorah
<<Zero_out>>

Not bad for a cheesy end-game pun... (3.00 / 1) (#141)
by Dr. Zowie on Wed Sep 25, 2002 at 07:09:58 PM EST

Remember, D'ni started as "Duny", a very bad pun on "done-ee" in the original Myst game, because when you got there you were done.

Myst and Star Wars both have extensive, nay enormous amounts of post-facto backstory. The difference is that the Myst guys did it right :-)

[ Parent ]

Inventing or discovering mathematics? (3.50 / 2) (#144)
by trident on Thu Sep 26, 2002 at 02:58:17 PM EST

First, thanks for a great article! There is just one thing...
Go back to the beginning when nobody had invented any mathematics.
I always jump whenever I see people talking about "inventing" mathematics. Is mathematics really something you can invent, or is it a fundamental property of the world that we, mankind, are here to discover? Is mathematics a human or divine invention?

This is a controversy that sometimes pop up, and there are a lot of people who are infuriated whenever someone is talking about "inventing" mathematics. The idea is that since mathematics is universal - cardinal numbers are cardinal numbers anywhere in the universe. Such numbers also "existed" prior to mankind's discovery of them; the were not "created" by man. Civilizations on the other end of the universe, if such civilizations exist, would have found just the same cardinal numbers as we have.

On the other hand, if we had not ever thought about the notion of cardinal numbers, would they really have "existed"? Perhaps we "invent" mathematics by "creating" the questions that are answered by our mathematical theorems?

It is quite interesting once you start thinking about it.

Kant vs Hume (none / 0) (#145)
by kholmes on Fri Sep 27, 2002 at 02:19:36 AM EST

Thats a very deep argument that has plagued some of our greatest thinkers.

The rest of us just ignore it and go on with our lives :)

If you treat people as most people treat things and treat things as most people treat people, you might be a Randian.
[ Parent ]

In Conway's book "On Numbers and Games". (none / 0) (#146)
by the on Fri Sep 27, 2002 at 01:21:03 PM EST

...he plays with the idea of the integers being created on various days. On the first (or rather zeroth day) we have the empty set. On day 1 we have 0 and 1. On day two we have 0, 1 and 2. On day omega we have the number omega. This captures the sense that to make an ordinal we need all of the previous ones. It's more interesting if you look at the surreal numbers because on day 1 you find there are quite a few numbers you can make, not just the number 1 (for example you can also make *1 and -1). There is a function on the sureal numbers saying which day they were each created on.
Civilizations on the other end of the universe, if such civilizations exist, would have found just the same cardinal numbers as we have.
You're simply assuming that. I suspect it's true but remember that some cultures on earth don't have numbers beyond about 3 or 4 (apparently).

--
The Definite Article
[ Parent ]
p-adic integers vs. p-adic numbers (5.00 / 2) (#148)
by dtclausen on Sat Oct 05, 2002 at 12:43:18 PM EST

You denoted the p-adics by Z_p ("Z sub p"), but in fact that is the common notation for the p-adic integers, which do not form a field, only a ring. The notation for the p-adics, or the p-adic numbers, is Q_p.

Hehe (2.00 / 1) (#153)
by tetsuwan on Sat Nov 02, 2002 at 05:45:56 PM EST

I inclining towards thinking that the integers are giving to us by experience of the outside world, that is they're not there a priori,

Njal's Saga: Just like Romeo & Juliet without the romance

The Alternative Numbers | 155 comments (131 topical, 24 editorial, 0 hidden)
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