Re: [Phys-l] Types of scalars

[David Bowman <David_Bowman@georgetowncollege.edu>]

Regarding John D's example:

> ...
> Supplying units is formally equivalent to supplying basis vectors.  
> Technically you don't have scalars at all once you start doing this.
> Example  5 apples plus 3 oranges is a vector in a two dimensional
> space.

The example sum is not a vector in a two dimensional vector space if ordinary arithmetic is used along with the restriction that the apple number and orange number each be nonnegative integers.  This is because a vector space requires that each of the component scalars be elements of a field (in the algebraic sense).  But the nonnegative integers do not form a field under ordinary arithmetic.  This is because they don't have additive or multiplicative inverses.  So either we need to allow our apple numbers and orange numbers to have negative and fractional values *or* we need to define a kind of arithmetic for which just nonnegative integers form a field (in the algebraic sense).  Unfortunately, even though I can imagine a fractional number of apples or oranges, I can't imagine a negative number of them.  But we can make certain finite sets of nonnegative integers form a field if we limit the maximum number of oranges or apples that are allowed in our 'vector' space.  We can do our arithmetic mod(n) where n is some prime number bigger than 5.  It has to be at least bigger than 5 because we already have 5 apples in our example, and modular integer arithmetic with n less than or equal to 5 doesn't have integers that big.  We need n to be a prime number because only modular arithmetic mod a prime number has a full compliment of the multiplicative inverses needed for them to form a field.  If, for example, we did our arithmetic mod(7) then we can have apple and orange numbers of 0,1,2,3,4,5,and 6.  In this case 5 apples plus 3 oranges would be a vector in a two dimensional vector space with the nonnegative integers mod(7) being the field over which the vector space is built.  But it would be kind of weird having 4 apples minus 6 apples being 5 apples, and having 6 times 5 oranges being 2 oranges (mod(7)).  But at least in such a weird situation we would have a legitimate vector space on our hands.

David Bowman