Math Insight: Multiplication and Multiplicative Identity

By Xah Lee. Date:

Today i had a insight: multiplication is a concept that depends on defining a unit. This is in contrast to the concept of addition.

i'm not sure this insight will be of any significance to others. Here's how it came about:

i was writing about dot product of vectors. As we know, it has this curious and nice property that when the product is 0, it means the two vectors are orthogonal. Now, we can say that this is by design built into the dot product or otherwise… but anyhow, the thing is that dot product is not as intuitive or basic a idea as, say, compared to vector addition.

So, in thinking about any fundamental math significance of the dot product, and its geometric interpretation, i thought about simple product of two numbers.

The setup is this: when we multiply two numbers a and b together, and the result is c. Suppose a ≻ b. What can we say about c greater than a or less? or b? What can we say at all about their order relation with given order relation of a and b? Now we know, if a multiplier is less than one, then the original number shrinks, and also the other way. Now, this is getting odd, because the relative size of the product seems to depend on the unit quantity, but as we know, that a real line is really elastic. There is no absolute rule that says where is 1. It's all relational. So, now we seems to have a paradox.

Put in another way, how can we translate the fact that the relative size of the product depends a specific number 1, into a pure geometry concept that doesn't really have 1.

The final outcome of this, is the realization that multiplication is essentially a idea based on a unit. Now this is in contrast to addition. Addition can be defined geometrically with the real line. From algebraic point of view, this insight can be phrased thus: in a ordered field, the magnitude of a product of 2 element relative to the multiplying elements, critically depends on the magnitude of the unit element. The thing to note here is that we need a ordered field. In typical intro algebra texts (i.e. algebra with no notion of ordering), definition of multiplication does not require a definition of identity.

In geometric inversion, where the inversion circle seems to serve as the multiplication of two distances. But, there the radius of the inversion circle essentially is the unit.