QUIZ:

Question 1. What is 3 apples plus 2 apples?

Answer 1: 5 apples?

Question 2: What is 3 apples plus 2 oranges?

Answer 2: You can’t add apples and oranges!

Question 3: So, what is 3 apples **times** 2 oranges?

Answer 3: 6 “pears” (pairs)

Example: if the set of apples is { a b c } and the set of oranges is {y z} then the product set in this sense is the set of pairs {ay az by bz cy cz}. The product as usually understood, 3×2=6, is just the count of the resultant set of pairs.

I started this investigation of multiplication some years ago, prompted by curiosity about the physical units of energy and momentum.

In physics, momentum is defined as mass time velocity,
where velocity is distance per unit time, i.e.

p = mv = mx/t

or in units: kilogram meters per second.

Energy is defined as mass times the square fo velocity:
e.g. for mass to energy conversion,
Einstein’s famous equation,

E = mc^{2},

or in more pedestrian situations,

E = mv^{2} = m(x/t)^{2} = m (x^{2})/(t^{2}).

or in units: kilogram square meters per square second.

I have an intuitive and experiential notion of what a square meter is, but what is a square second?

—-

Then thinking in general about multiplying units,
it occurred to me that the product of meters and meters
is not meters but *square* meters.

But I was taught that multiplication is a “closed” operation, that is, the product of any two items from a set it another member of that set. But multiplying items from the set of “lines” (things with length) gives us an item in the set of “surfaces” (things with area).

So how is it that we say the multiplication is “closed”?

I postulate that, in the physical sense of “multiplying” quantities, multiplication is not closed, but is in fact a dimension-increasing operation, forming a set of n-tuples from the arguments, as in the Cartesian or outer product.

I would speculate that physical multiplication would turn out to be a “tensor” product, but for me at least, to verify this will require further investigation.

The beginnings of that investigation have led me to discover David Hestenes’s Geometric Algebra and Calculus, http://en.wikipedia.org/wiki/Geometric_algebra of which perhaps I will have more to say later.

In another vein, I recently discovered that there is a lively discussion going on about whether multiplication is repeated addition. One question was whether “repeated addition” can be rigorously defined. That will be the subject of a future post.