At times the best way to get something new is to sacrifice something old. In mathematics there are properties that we often take for granted. Dropping them can lead to interesting discoveries.

Non-positive: Ordinarily all one sees in life are natural numbers, the integers to count things with — one cow, two goats, a thousand stars, a trillion trillion atoms, and so forth. But relax that restriction, include negative numbers, and you can solve problems in addition such as X+3=2. (You can also go into debt!)

Non-integer: What fits between the whole numbers? How can equations like X*7=22 be answered? Relax the restriction to integers and you've got fractions, the rational numbers (ratios) like 22/7, 355/113, etc.

Non-rational: Is there room left between all the possible fractions? Yep. A ratio can get arbitrarily close to any number you care to name — but you'll still never quite solve a problem like X*X=2 exactly with rational numbers (see RootsOfCommensurability (26 Jan 2000)). You need the real numbers. They have all the irrationals, including transcendental ones like pi or e that aren't solutions to equations with a finite number of powers of X.

Non-real: Once you've got the real numbers, what could possibly be missing? Well, there are still some formulæ, like X*X=-1, that don't have answers. To solve them, you need complex numbers that have both real and "imaginary" parts. Now, at last, all the numerical equations have solutions. But you can still give up other things and continue on, beyond ordinary numbers.

Non-commutative: Normally A*B = B*A — the order of numbers multiplied together doesn't make a difference. But relax that assumption and you can have strange objects like "quaternions" that describe rotations in space and resolve quantum mechanical paradoxes. (The order of doing rotations matters, as does the order of making measurements on a quantum system.)

Non-associative: It's usual for A*(B*C)=(A*B)*C — the various groupings of numbers being multiplied together doesn't matter. But relax that assumption and you can have strange objects like "octonions", about which I haven't the foggiest understanding. I first heard of octonions ca. 1975, when I was sitting in the back of the room at a high-energy physics seminar and Murray Gell-Mann described them. (He said that he didn't know what they were applicable to, but that he had a hunch they might be good for something.)

When one has thrown away all of the above properties, is there anything left to give up? I don't know. At what point does transitivity go away, for instance? Non-transitive relationships — where, for example, rock beats paper which beats scissors which beats rock — are fascinating and important in economics, politics, military affairs, and of course in games. Could non-commutativity or non-associativity have similar applications?

(see also DenseAndNowhereDense (12 May 1999) and SubbookKeeping (21 June 2001))

TopicScience - 2002-09-07

(correlates: DenseAndNowhereDense, BaseballFootballBasketball, SillySeminarsOf75, ...)