2012-10-06

The connection 0 and other


A boy (T.) is working on 7. The question is what two numbers make 7. The material using here is shown in Figure 1. This is called ``Zahlenhaus''. One house has two rooms and each room has some people. How many people are in the house? is the question and the students think about addition.
Figure 1. Zahlenhaus: 7 = 4 + 3
Figure 2. Zahlenhaus: Questions
Beside the house figure, there is a equation (Figure 2), usually missing one part. For example,

  • 7 = 3 + ?
  • 7 = 4 + ?.

The leaner fills in ?. T. has no problem to fill in those questions.

However, he doesn't understand the following.

  • 7 = 0 + ?

By the way, this is a easy case. Since he understands what he doesn't understand. I, myself, have a problem to find when I lost in a mathematics book. If I know where I lost, it is rather easy to return to the track. But, in many cases, I didn't know where I lost. Then I first need to find out where to return in the book.

OK, let's back to the Zahlenhaus. In this case, there is no one in the left room. I asked him, ``How many people in the right room?'' He did not answer a while, then, said, ``2 + 5'', then, ``3 + 4''. I have a problem. T. answered,

  • 7 = 0 + 2 + 5.

This is mathematically correct. I didn't said we can only use one number for one room since one house can be represented two numbers, why one room should be only one number assigned? At the first place, I asked seven is what and what? This is a practice. You can answer seven is seven. It is not a math, this is rather a game.

Because of this implicit rule, which is one room only has one number, 7 = 0 + 2 + 5 is not correct. Zahlenhaus has no three rooms. If this Zahlenhaus has three rooms, this is correct. But, isn't it more confusing that the calculation depends on the number of the room?

I recall about a book by Charles Dodgson (Lewis Carroll), the title is ``Through the Looking-Glass, and What Alice Found There.'' The following is a conversation when Alice met the King.
``... Just look along the road, and tell me if you can see either of  them.''
``I see nobody on the road,'' said Alice.
``I only wish I had such eyes,'' the king remarked in a fretful tone. ``To be able to see Nobody!'' [Quote 1]
There is nobody in one of the rooms in the Zaehlenhause. How do you show nobody is there. This is not a simple concept, human being took a lot of time to find zero. ``There is nobody'' means ``There are zero people.'' It is not ``There aren't zero people.'' If ``There aren't zero people,'' there should be somebody since it is negation of zero people's existence. We are talking about zero people exist. If this was his problem, it would be a tough problem for me. But, I found it is OK for him that if zero people exist in the room, he can write 0. I am not sure he thinks zero people's existence means 0 or he thinks 0 is another name of ``nobody''.

Then, is the problem that one room can be represented by one number? ``Normally'' we use as less number as possible since this makes easier to compare the results. It is more difficult to compare 1 + 1 + 1 + 1 + 1 + 1 + 1 with 1 + 1 + 1 + 1 + 1 + 1 than to compare 7 and 6. Therefore, lazy mathematician ``normally'' writes 0 + 7 as 7. ``Normally'' style is chosen since it is shorter or simpler. But ``normality'' and ``correctness'' are sometimes not the same, and ``correctness'' is more important in mathematics.

I thought, ``Alice again.'' I tried to tell my friends that how mathematical the Alice books are.
``Can you do Addition?'' the White Queen asked. ``What's the one and one and one and one and one and one and one and one and one and one?''
``I don't know,'' said Alice. ``I lost count.''
``She ca'n't do Addition,'' the Red Queen interrupted. ``Can you do Subtraction?' Take nine from eight.'
``Nine from eight I ca'n't, you know.'' Alice replied very rapidly: ``but ---''
``She ca'n't do Subtraction,'' said the White Queen. ``Can you do Division? Divide a loaf by a knife -- what's the answer to that?'' [Quote 2]
If a room of Zahlenhaus has more numbers, it becomes the White Queen's question. Still the White Queen is correct. Any natural numbers are sum of ones. How can I explain this to him?

I asked T., if you want to buy an ice cream that costs two Euro. You have one Euro, how much do you ask to your mother? ``One Euro'', he answered. Then, if you don't have any, zero Euro you have? ``Two Euro'', he answered. If you need 7 Euro, but, you have zero Euro, how much do you ask?  He thought a while and answered, ``I don't know''. You don't know? ``One plus six?'', he answered. His answers are following,
  • 2 = 1 + 1
  • 2 = 0 + 2
  • 7 = 1 + 6
  • 7 = 0 + ? I don't know.
I don't know that what kind of model he has in his brain. Why does he think he doesn't know at the last case?

To make sure, I repeat this.
  • You want one Euro. You have zero. You want? ``one'', OK.
  • You want two Euro. You have zero. You want? ``two'', OK.
  • You want three Euro. You have zero. You want? ``three'', OK.
  • You want seven Euro. You have zero. You want? ``seven?'' Yes.
After this, T. answered the questions of 0 + x correctly. But I didn't understand what was his problem. I still wonder, he really understood this. Maybe it is difficult that the numbers are all the same as numbers. It is a level of abstraction. one and two are surely different, but, as a number, for example, made by ones, you can add them... all numbers shears these properties. By the way, If you work on mathematics, you will forget the numbers. Nowadays, numbers are for computers.

Quote 1

This quote is in Chapter 7, The Lion and the Unicorn, Through the Looking Glass and What Alice Found There. by Lewis Carroll.

The text is taken from ``The Annotated Alice, the definitive edition,'' edited by Martin Gardner, Penguin books, page 234, ISBN-13: 978-0140289299.

Quote 2

This quote is in Chapter 9, Queen Alice, Through the Looking Glass and What Alice Found There. by Lewis Carroll.

The text is taken from ``The Annotated Alice the definitive edition,'' pp. 265-266, Penguin books, ISBN-13: 978-0140289299.

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