I am a reporter by profession and I do not know much about mathematics except what I learned in schools. After a great success in teaching my son reading, I and my wife are trying to teach my son something about the numbers and mathematic notions. We did teach him the number line and he learnt it very quickly and in fact developed what he learnt. To him, the number line is not a straight line which goes in opposite directions to infinity. In his mind, it is a circle!

I and my son, Li Xiaotian, would look forward to knowing what you would think about all these. Li Guorong

It's all awfully fascinating. Of course you are right not to tell him he was wrong. My remark was not meant for him. I fully accept the idea of the positive/negative exchange across some points. This happens at 0 and also at infinity. What needs fixing now is thinking that infinity is a big 0. Is big 0 the same as small 0? Of course not. What's the point of introducing the adjectives? This does not serve any useful purpose. Thus this is something to work on.

Ask him what is 0. The idea is to lead him to accepting the notion that 0 has some arithmetic features, the main one being that adding it to any number does not modify the number. Is this true about the big 0 - infinity? Why not simply say that there is just one point - infinity - where the big positive and the big negative meet? What is gained by calling infinity a big 0?

Second point: this is true that however big a number is it is finite. This is exactly because we are talking about numbers. But infinity is something else. What makes a number a number is that a number exists as a memeber of a bigger collection - numbers. By itself, any number is just a name. But, as a member of a collection, it may be added to other members - other numbers. 0, for example, when added does not modify the number. Adding 1 increases the number. Adding negative numbers decreases the number. Can your son add infinity to a number? What would be
the result?

a+1 is on distance 1 from a
a+2 is on distance 2 from a
...

The bigger number you add to a, the farther the result is from a. So what could be the farthest point from a? Which is something he must get adding infinity to a.

Something else: I believe your son's terminology is inconsistent. There is 1 zero approched from two sides by positive and negative numbers. Why there are two infinities? More consistent would be to say that small positive and negative numbers meet at 0, while large positive and negative numbers meet at infinity. How does calling it a big zero help?

Please keep me informed on your conversations.

All the best,
Alexander Bogomolny

Yes, I discussed with my son today the questions you raised in your second message. The following is his answers: The 0 at the bottom of the circle, or the big 0, is a starting point while the one at the top of the circle, or the small 0, is an ending point. The big 0, which he also calls the positive 0, is different from the small 0, which he also calls the negative 0. Just like 1 has an exact opposite point known as -1, he argues 0 must also has its exact opposite point, which he says is the small 0 or negative 0, the point where the negative and the positive infinite meet each other. He says that 0 is a number which separate the negative and the positive numbers. He believes that 0+a=a, 0-a=-a, but -0+a=-0, -0-a=a. His talking of the negative 0 (what great an idea!) made me think of the Black Hole and his reasoning about the infinity made me think of the theory of the collapse of the Universe. He is just four years and eight months old. He can not give the reasons why he thinks so. I guess he is just using his power of imagination and ability of reasoning when he says all about the infinity. The inconsistency may be caused by the language problem: He speak Chinese and I act as a translator. I try my best to be as literal as possible but I know I can not put what he says into perfect English exactly as what he means. This is a problem we will continue to have in our future discussions.
Li Guorong
Beijing

This is probably a misprint. More consistently it'should be

-0-a=-0, right?

This is wonderful what you are doing with your son. It's of course not at all important that the boy will come up with the right theories. What's right any way?

>His talking of the negative 0 (what great an idea!)
>made me think of the Black Hole and his reasoning
>about the infinity made me think of the theory of
>the collapse of the Universe.

You make a nice pair of researchers.

Calling things names and thinking about them are two different activities. I believe you are trying to foster the latter.

To me an idea is good if it's fruitful. "Negative zero" has certain connatations: it's something of a negative and something of a zero.

You may pose a couple of questions:

1. What makes a number negative? Whatever the answer, does it apply to his negative zero?

2. What makes a number 0? Whatever the answer, does it apply to his negative zero?

I am not sure whether you should ask these questions. Judge for yourself - you've been doing a good job so far.

All the best,
Alexander Bogomolny

wow. i am really awestruck. your son is quite a philosopher for a 4 year old!

although you later wrote that you don't want to pressure your son into talking about a deep subject at his tender age unless he brings it up (very understandable), i noticed a striking semblance between your son's visualisation of numbers and infinity and the Reimann sphere of complex analyses. his conception is really quite nice.

while i don't know about the intricacies of a child's brain, and i certainly don't expect him to have an understanding of complex analyses, it is quite possible that a child's intuition is purer and greater than we think. i understand that you don't want to pressure your child, but i do hope that he will keep interest in the subject and open discussions and questions with you again. in any case, i don't doubt that he will be special in whatever he does (traditional Chinese painting included ). good job.

regards,
--gunes

"One must still have chaos in oneself to be able to give birth to a dancing star."
-- Friedrich Neit'sche

"One must still have chaos in oneself to be able to give birth to a dancing star."
-- Friedrich Neit'sche

Well, no one may know what goes on in the child's head. The other day, in a private correspondence, David Cantrell took me to task for failing to mention a mathematical construct that looks very close to what the boy might have had in mind. See

https://mathworld.wolfram.com/ExtendedRealNumberProjective.html

Dear Mr. Bogomolny,

Thank you for contacting me after such a long time. I am very happy to learn that you have posted our communications on your message boards. What we have discussed is something really interesting. My son is doing fine at kindergarten and he is learning traditional Chinese painting. He has never picked up the subject since your last email. So I have also stopped asking him anything about it because I do not want to press him too hard. He is still a child. Too much pressure does no good to him. I think you will agree with me.
I have read the attached file from you and I appreciate the response to the letters although I can not really get it. It is a subject too abstract for me.

Just keep in touch. Li Guorong

Well, coincidentally enough, Aristotle had a similar way of thinking. Plato believed in a perfect infinity, but Aristotle didn't. He had this way of thinking about infinity (I'll phrase it as though it were his words):

"Let's say I'm walking down the street and construction workers are building houses in front of me. These construction workers finish a house, then they slide down the road a few feet and build another house. These guys are so fast that I never actually see them finish; they build as I walk! So, at any point, although there are a finite number of houses, I never actually see the houses end, so wouldn't it appear infinite to me?"

This might make an interesting story, if not for your son, then for you at least. Hope you enjoy.