Thursday, December 24, 2015

Arithmetic Symmetries, Understanding

Arithmetic Symmetries and Dualities

From the way we usually come at arithmetic, subtraction and division can appear to be subordinate to, and even derived from, addition and multiplication. Why is it not the other way around? And why are positive numbers "natural" and negative numbers not? The answer is straightforward: we bootstrap our way to negative and rational numbers from the whole numbers, the non-negative integers, which, strictly speaking, can only be added and multiplied: we start from numbers intended for counting which itself is then elaborated as addition and multiplication, and end up with negative numbers, and division. 

This would present no problem if mathematics instruction looked back from the results of the bootstrap, and cleaned up the mess. With the full set of integers and rational numbers, there is nothing special about "subtraction of (-M) is the same as addition of (+M)", or "division by M is the same multiplication by the M^(-1)".

For each of addition/subtraction and multiplication/division there are symmetries that allow us to pass back and forth from one member of a pair to the other if that is convenient. This can all be derived by "removal" of the term in an expression for N that is responsible for M "being in" N, thereby mathematizing the concept of "removal":
  • N remove (+M) == N - (+M)
  • is obtained by removing the term + (+M) from
  • N = N + 0 = N + (-M) + (+M), to obtain
  • N - (+M) = N + (-M).  
This shows "subtracting (+M) is the same as adding (-M)". Since we are discussing symmetries it is convenient to use explicit notation for signed values. Interchanging (+M) and (-M) in the above gives

  • N - (-M) = N + (+M),  

the notorious "minus-a-minus is plus-a-plus". Here it is merely one member of a symmetric result: subtracting a positive(negative) is the same as adding a negative(positive), nothing special here. We could have approached this problem as "addition" is the removal of a subtraction operator; there really is nothing special about subtraction (or addition).

We can approach multiplication/division with the same "removal" approach. Now we start with
  • N = N x 1 = N x M(^-1) x M(^1),
and remove the operation "x N(^1)" from it to obtain division:
  • N remove "x M(^1)" == N / M =  N x M(^-1),
With this we have "N divided by M( the inverse of M) is the same as N multiplied by the inverse of M( M)". And again the same result could have started with multiplication as the removal of a division operation.

Trying out the Two-Number-Line

It is always a good idea to confront any process/procedure/scheme
you are trying to understand with the simplest problems you can find. I would like to see how the Two-Number-Line provides the values of the addition expressions:
  • (N + 0) : The relative coordinate system for 0 is the same as the fixed coordinate system for N except its origin is at the far-end of the N-number-segment. The sum can be read off as the coordinate of this origin point.
  • ((-N) + 0) : The same as above except the sum is on the other side of the origin of the fixed coordinate system.
  • (N + (-N)) : In the relative coordinate system for the first problem, the far end of (-N)-number-segment is at the origin of the fixed coordinate system. The sum is the coordinate of this point in the fixed coordinate system, 0. 

and the multiplication problems:

  • (N x 1) : In multiplication the relative and fixed coordinate systems have the same origin point. The far-end of the N-number-segment is the far-end of the unit-segment in the relative coordinate system, ie the point at the far-end of the 1-number-segment in the relative coordinate system has the coordinate N in the fixed coordinate system. This is the value of the product 
  • ((-1) x N) : The point at far end of the (-1)-number-segment in the fixed coordinate system is the end of unit-segment of the relative coordinate system. Now the relative coordinate system is oriented oppositely to the fixed one, and the far-end of the N-number-segment is on the opposite side of the origin and has the coordinate (-N) in the fixed coordinate system. 
It is good to know the value of each of these expressions, because we can then concentrate on how that value arises from the appropriate process on the Two-Number-Line( may be called the TNL in what follows). See the immediately prior post for a detailed discussion of the TNL.

The Difference between Knowing the Answer and Understanding the Problem.

We can see a process on the Two-Number-Line(TNL) that produces a representation of the answer. In playing that back in our minds, we are visualizing a process that evaluates arithmetic expressions; we have something to hang an understanding on, a way to see how the mathematics works. For those of us who use mathematics, this is the sort of understanding we need; we have discussed this before. Arithmetic on a TNL is just one way to see arithmetic. Expect to find others or maybe just pieces of others, and to eventually discover that you are working in a web of understanding. 

There is another reason to think about understanding as a necessary part of the foundation that supports learning. From what I can tell it is generally accepted, unquestioned as far as I can tell, that memorization of the facts of small number arithmetic is an absolute requirement for future success. It may be necessary for executing the algorithms taught to do arithmetic, but that is an academic issue. When and if you come to do calculation, my experience is that you will put it all together for your own purposes with no great anxiety about what you have forgotten from elementary school. You may regret not coming to understand the significance of understanding at an earlier stage, I certainly did.

Let me state this baldly, and wait for enlightenment: it is fraud to state that there is only one route to any of the mathematics any of us is likely to use, and pernicious fraud at that. It somehow suggests mathematics is this dainty, fragile flower that drops its pedals if you approach it with anything less than immaculate preparation.  Relax. Mathematics is robust. break it into pieces, put pieces together, understand what happens. 

Any structure presented to have single points of failure is scary; it is rational to be anxious if you think success requires you retrieve the one fact of arithmetic or the one piece of nomenclature that can solve that problem. Math anxiety is not pathological.

Let me back up a bit on one-digit whole-number arithmetic. I do not think you should rely on a calculator for small jobs. That is silly. I do think small number arithmetic should be thought of as part of our everyday language, not something esoteric that exists only for mathematics. A smart phone has a good dictionary, but you do not look up every word.