# 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: ° ° °

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