Laws of Algebra

Introduction

Just as one can gain a better appreciation of one's own culture by traveling to exotic lands with different customs, in mathematics one can gain a better perspective on the everyday laws of algebra, by exploring exotic number systems where the same laws may or may not hold -- and where even when they do, one's "cultural intuition" about how the numbers work as a result of those laws may be wrong.

Distributive:	x(y+z) = xy+xz, (x+y)z = xz+yz
Associative:	(x+y)+z = x+(y+z)	(xy)z = x(yz)
Commutative:	x+y = y+x	xy = yx
Identity:	x+0 = 0+x = x	x1 = 1x = x
Inverse:	x+(-x) = (-x)+x = 0	x(x-1) = (x-1) = 1

These laws are color-coded according to which type of number system must obey them (from least to most stringent):

Commutative Group:	°
Commutative Ring:	°	°
Field:	°	°	°

The Woodrow Wilson Leadership Program in Mathematics lpt@www.woodrow.org
The Woodrow Wilson National Fellowship Foundation webmaster@woodrow.org
CN 5281, Princeton NJ 08543-5281 Tel:(609)452-7007 Fax:(609)452-0066