Thoughts on Arithmetic-III

Arithmetic Part III


Counting numbers are a subset of the integers which are a subset of the rational numbers.
The Rationals are a subset of the Reals are a subset of the Complex Numbers. The algebraic operations are extended at each step to the new numbers. While the algebraic operations were originally defined to fill a real, everyday sort of need, as they are extended to these larger sets of numbers they lose a real world intuition which is replaced by mathematical rules intuition. It is the laws of exponents that have to be kept intact, not intuition using herds of sheep. “It’s the laws, Baby.”
2*3 means to multiply 2 times itself 3 times = 2 x 2 x 2. The intuition is intact. But 2*-3 doesn’t mean to multiply 2 times itself a minus three times. Physical intuition is lost. 2*-3 is defined as 1 / 2*3. 2*-3 is defined in terms of previously defined things, 2*3 and division.
The basic rule in exponents is that a*n x a*m = a*(n+m). This means that I want
2*3x2*-4 = 2*(-4 + 3) = 2*-1 = 1 / 2
and so it is for this reason that I define a*-n = 1/a*n.

Now I have exponents defined for all the integers...sort of.

I have not tried for logical correctness or completeness and perhaps I should have said more. As a matter of fact it is a decision for the teacher to make: How much should I say?
But something has to be said. Magician manuals say that when presenting a trick, you tell the audience what you are going to do. (I’m going to make a coin disappear), do it (I make the coin disappear) and then tell the audience what you have done. (“See, I’ve made the coin disappear.”)
The same technique might be used in class.
Dick Askey was an instructor at the same university where I was going to graduate school. At the beginning of each semester Dick would ask me what courses I was taking and then he would tell me what was going to be covered in those courses. He would tell me the flow of the theorems and how one followed from another, he would tell me the ideas behind the proofs. I would then know what the course was about and what the point was; all that was left was seeing the actual, formal, complete proofs. It didn’t really take him all that long, I usually took three courses, and it was very, very helpful. Topics didn’t come out of nowhere to whiz by before I knew what the point was. I liken it to driving down the freeway at 85 mph at night with the dims on. Off ramps come and go before you can read the green sign.
Of course I was prepared to assimilate the information Askey passed onto me. And a teacher must prepare his students to be able to accept general mathematical ideas.


I have tried to give the flow of my approach to arithmetic. I am a big believer in “flow”. I see the counting numbers, N, as the font from which all good things flow. N spreads its glory to the integers, I; + , x and exponents are carried along.
The integers pass the glory on to the fractions, Q; +,÷, x and exponents are carried along; division is born.
By “carried along” I mean that, say, exponents have to be defined for rational numbers in terms of how they are defined on the integers. That is, a*1/n = n th root of ‘a’ if a>0.

The idea is that the objects that are manipulated need a context; I try to make the original context the more or less observable world for both the objects and the manipulations.
But the context has to change to a mathematical context. If you keep everything in the Real World you lose the power of mathematics. When you move into the Ideal World of mathematics it is to some degree like moving to Point Barrow. It is a cold, unforgiving world. Life is not lived sloppily there.
The context here is the obedience to rules. The rules of exponents still have real world significance in special cases, say compound interest, but that is a small part of the mathematical significance of exponentials; and the real world becomes a third world. (e * iπ = -1, give me a break.)
I think the difficulty in making this change of context is considerable. It is jumping off a cliff into the sea of abstraction...with your eyes closed. Don Juan (per Castaneda) says that to become a man of knowledge there are four enemies you must defeat and the first enemy is fear. When you first start to learn something, I mean really learn it, to understand it, it is nothing like you had imagined. The familiar hand holds and crutches are gone; this doesn’t seem to be what you wanted to learn. And you meet your first enemy, fear. Carlos asks Don Juan how to defeat this enemy and Don Juan says you don’t really defeat it. You take one step after another in the face of your enemy until that first step when your enemy retreats. Then you see that you can learn anything. Some people retreat from the enemy. (And drop the class, I thought.)
I thought this was an excellent metaphor for the student’s first meeting with mathematical abstraction.
So, you might say, some people are just born with the ability to abstract. I saw a 14 year old Chinese kid play a difficult piano concerto with the Julliard Symphony. I guess it lasted about twenty minutes. How did he memorize it let alone get his fingers to operate correctly? Mozart wrote a symphony when he was seven; Daniel Boone killed a bear when he was only three.
In my teaching career I met some really bright kids, maybe not in the Norbert Weiner class but they had no problem with abstraction. But they are a small minority. The teaching problem is how to get through to kids who have difficulty with the transition