Generalizing absolute values of numbers

[lpetrich]

Over at PhysicsForums > General Math > Math Challenge threads, I found one about "value functions" that generalizes absolute values.

For x in some algebraic field F, the generalization |x| is in the nonnegative real numbers R(0+). It satisfies these axioms:

1. x = 0 <-> |x| = 0
2. |x*y| = |x|*|y|
3. |x + y| <= |x| + |y|

A GAV is Archimedean if, for every nonzero a and b in F, there is some positive integer (sum of 1's) n that satisfies |n*a| > |b|.

The trivial GAV is |x| = 1 for nonzero x, and that is non-Archimedean.

|1| = 1 is easy to show. From axiom 2, |x*1| = |x|*|1| = |x|. Thus |1| = 1.


I don't know why that challenge started with a field instead of with a ring.

For a field, the Archimedean condition is |n*a| = |n|*|a|, giving |n| = |b|/|a| = |b/a|

Also for a field, one can easily show that |x^p| = |x|^p for integer p and nonzero x (for a ring, for nonnegative-integer p and all x).

|x^0| = |1| = 1 = |x|^0
|x^(p+1)| = |x^p * x| = |x^p| * |x| = |x|^p * |x| = |x|^(p+1)
Thus the identity holds for nonnegative p.

When x is nonzero and in a field, 1 = |1| = |x*(1/x)| = |x|*|1/x|, thus |1/x| = 1/|x|. Thus, |x^p| = |x|^p for all integer p.

 

[lpetrich]

I'll now consider a finite field: GF(p^n) for prime p and positive integer n. The nonzero elements' multiplicative group is Z(p^n-1), so any one of certain of its elements can generate the group. Let us consider one of them, a. Its GAV satisfies |a|^(p^n-1) = 1, and the only value of |a| that can make that possible is 1. Thus, a finite field only the trivial GAV.

For the real numbers, I start with handling negative ones: |-x| = |-1|*|x|. I next find |-1|. Its square is |(-1)^2| = 1, and the only nonnegative real number with that square is 1. Thus, |-1| = 1.

Now the positive real numbers. I first find what powers of variables can be moved through the GAV. I first consider inverse powers, something possible for positive real numbers. |x^p| = |x|^p with x -> x^(1/p) gives |x| = |x^(1/p)|^p. Thus, |x^(1/p)| = |x|^(1/p). Thus, for general rational power q, |x^q| = |x|^q, and with Cauchy-sequence closure, q can also be real.

Now consider x = e^(log(x)). |x| = |e|^(log(x)), and for |e| = e^c, |x| = e^(c*log(x)) = x^c. Using the third axiom, |x + x| = |2x| = 2^c * x^c <= 2*x^c. Thus, c <= 1.

For the Archimedean property, I note that for c != 1, |x| can get arbitrarily large. To make |n| get arbitrarily large, we must have c > 0. Thus,

• Third-axiom constraint: c <= 1
• Ordinary absolute value: c = 1
• Archimedean: c >= 0
• Trivial (non-Archimedean): c = 0

For complex numbers, one does |w*x| = |w|*|x| for nonnegative real x and unit complex number w. The latter can be approximated arbitrarily closely with nth roots of unity: exp(2pi*i*m/n). || of them to power n gives 1, therefore || of them gives 1, and |w| = 1.

The rest of the proof goes as for real numbers.


For rational numbers, we note that they are ratios of integers. These can be factored into products of positive-integer powers of primes. Thus, dividing one integer by another gives a product of integer powers of primes. Thus,
|x| = |product of primes p^m(p)| = product of primes |p|^m(p)

I can't get further than that.

From the third axiom, the prime GAV's have constraints like |p| <= p, |3| <= |2| + 1, etc.


For this GAV to be Archimedean, this GAV of integers must be capable of being arbitrarily large, and that means that at least one of the |p|'s must be greater than 1.

 

[lpetrich]

Turning from fields to rings, I consider the integers. They turn out to work like rational numbers, dependent on values of |prime numbers|, and with the same constraints on those values. Even the same constraint on being Archimedean.

But integers modulo some prime number are like finite fields, and they have only the trivial GAV. Modulo some composite number, they have no GAV, because they have zero divisors. For integers modulo 6, 2*3 = 0, and thus |2|*|3| = 0. From the definition of a GAV, |2| and |3| must be nonzero, and that is impossible here. So a GAV is only possible for a ring without zero divisors, like the integers or the integers modulo some prime number.

BTW, the ring of integers modulo some prime number is not just a ring, it is a field, GF(p).

 

[Kharakov]

Sometimes I think of absolute value as:

a magnitude preserving rotation to the positive real axis.