Tuesday, November 29, 2011

Confusing Mathematics Definitions

What is it with mathematical defintions? Are they deliberately trying to be obtuse? Consider Rosen's definition of a primitive root:
A "primitive root" modulo a prime p is an integer r in Zp such that every nonzero element of Zp is a power of r.
Wikipedia was no clearer
a primitive root modulo n is any number g with the property that any number coprime to n is congruent to a power of g modulo n. In other words, g is a generator of the multiplicative group of integers modulo n. That is, for every integer a coprime to n, there is an integer k such that gk ≡ a (mod n). Such k is called the index or discrete logarithm of a to the base g modulo n.
When after I had unpacked these definitions and looked at several examples my idea of a clear definition would be:
Given a prime p, a “primitive root modulo p” is an integer r such that all of the integers from 1 to p-1 appear in the set Zp. The set Zp consists of powers of r modulo p.
Of course, none of these definitions really make much sense without an example:

Example: Since all the integers from 1 to 10 appear in Z11 when r=2, 2 is a primitive root of 11. Powers of 2 modulo 11: 2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 5, 2^5 = 10, 2^6 = 9, 2^7 = 7, 2^8 = 3, 2^10 = 1.

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