Bizzare number series
Tuesday, March 01, 2005
1 + 1 = 2
2 + 2 = 3
3 + 3 = 4
4 + 4 = 5
5 + 5 = 6
6 + 6 = 7
7 + 7 = 8
8 + 8 = 9
9 + 9 = 10
10 + 10 = 11
This is probably the most bizarre number series I've ever encountered.
Number series come in all shapes and sizes. One of the most notable ones is the Fibonacci numbers:
0, 1, 1, 2, 3, 5, 8, 13,...
To get the next number, you just add the last two numbers. I.e. 13 + 8 = 21. And you keep going on ad infinitum.
In more general terms, you can use a discrete function on the form of a recurrence relation:
Fn = Fn - 2 + Fn - 1, n = 3, 4, 5, ..., and F0 = 0, F1 = F2 = 1
- discrete, is a fancy term Mathematicians invented to say that you only use Integers as the input of a function
- recurrence relation, means that the function calls itself to generate the next term
From a normal person's perspective, number series appear to be pass time activities for bored Mathematicians. Perhaps we wouldn't be so wrong thinking that, most sequences were probably found by playing around with Integers.
However, numbers are related to everything and the Fibonacci sequence is observed in a great number of very important natural phenomena. For a quick introduction, if you don't feel like reading a real book, visit
Fibonacci Numbers and Nature.
Now, back to my bizarre number series: at first sight it looks like non-sense. After all, the only true statement is 1 + 1 = 2.
So, where did the rest of the terms come from?
I've been thinking about it for a couple of days now and I have a theory. But first, the story behind the numbers.
The StoryMy four year old son, Gabriel, and I were reading
"Desmond's Birthday Party." In the story, Desmond (a little dog) goes to town to buy materials to bake himself a birthday cake. His day is broken down into one hour periods. At 12 noon, the story goes, the clock tower begins to toll its bell for 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 DONGS. During this part of the story, Gabriel enumerates the DONGS and off we go to the 1 O'clock page "which is lunch time," we both repeat.
Two days ago, before moving on, I asked Gabriel what "1 + 1" equaled to.
He replied "1 + 1 equals 2."
Great, I thought, he's learning how to add - This is a big step for him - So far he is only able to count from 1 up to around 31 without making any sequential mistakes.
After he correctly guessed (or added) "1 + 1," I asked him what 2 + 2 equaled to. He stopped for a few seconds and said "3." And then, without me asking, he kept on saying out loud: "3 + 3 = 4; 4 + 4 = 5; 5 + 5 = 6; 6 + 6 = 7; 7 + 7 = 8; 8 + 8 = 9; 9 + 9 = 10; 10 + 10 = 11."
I promptly asked him why he thought that "2 + 2 = 3."
He wasn't so interested in analyzing his results. He just wanted me to continue reading, to which request, I obliged and continue on. The story ends on a happy note: everyone eats the "delicious cake" Desmond baked at 3 PM (I've always wondered where his mom is in all this). And at 5 PM, his friends give him a wrist watch for his birthday present and, apparently, they all go home.
Gabriel loves the story. We read it every night before going to bed - Well, I do most of the reading - His job is more of a supervisory nature. I.e. Telling me that I'm in the wrong page when I'm skipping pages to make the reading session go a bit faster :)
The TheoryMy theory on the number sequence is a follows: I think Gabriel figured out that "1 + 1" is actually 2, the only true mathematical statement in the series. Then he "reasoned" that because "1 + 1" gives 2, then "2 + 2" must give 3, and that "3 + 3" must give 4, et al.
His little mind followed, albeit wrongly, a discovered pattern from the result on one correct observation: adding the same number twice gives you the next number in the Integers following the number you just added, which is only true for the number "1."
In other words, the number "1" is the only number that added twice gives as a result the next number in the sequence of Integers after itself. I.e. the next number after 1, is 2. So, 1 + 1 = 2.
Try it: every other number, except 1, when added twice doesn't give you the next number in the sequence of Integers (See below, for the proof). BTW, I don't think Gabriel knows this particular fact.
So, in Gabriel's little mind, the pattern he is following makes sense (He still says that 2 + 2 = 3). And so, I propose the "Gabriel Number Sequence," officially discovered by my four year old son. Which leads to the "Gabriel Theorem."
The Grabriel Theorem and The ProofTheorem: Let n be an Integer > 0. Then, the only possible solution for the equation n + n = n + 1, is n = 1.
I.e. the only number n, where its immediate successor n + 1 in a sequence of Integers, that when added twice gives as a result the next number in the sequence of Integers (or n + 1), is n = 1.
Proof 1We can prove this fact with the aid of 7th grade algebra:
Let n be an Integer > 0. I.e. 1, 2, 3, ...
In order for n + n to have be the next Integer in the sequence, the result must be n + 1. I.e.
n + n = n + 1
2n = n + 1
2n - n = 1
n = 1
Substituting back, we get:
1 + 1 = 1 + 1
1 + 1 = 2
Therefore, n cannot be anything, but 1 and so, we've proved the "Grabriel Theorem."
Proof 2A more sophisticated proof (I'm trying to be funny here), by reductio ad absurdum:
Assume that there is an Integer n > 0, that satisfies our requirement. Namely, that n + n = n + 1.
Assume that such Integer exist, and so we can try the next Integer in the sequence. Namely (n + 1). Thus:
(n + 1) + (n + 1) should give (n + 1) + 1, or the next Integer after (n + 1).
Hence:
(n + 1) + (n + 1) = (n + 1) + 1
2(n + 1) = (n + 1) + 1
2(n + 1) - (n + 1) = 1
n + 1 = 1
n = 1 - 1
n = 0
Which, is contradictory to our definition: n > 0.
Extra geeky stuff: what would Piaget sayThinking about Gabriel's discovery, I started recalling Jean Piaget's theory of early childhood development. To put it in context, what follows is a minimalist and bastardized version of Piaget's research:
In the sensorimotor stage (0-2 years), intelligence takes the form of motor actions.
Intelligence in the preoperation period (3-7 years) is intuitive in nature.
The cognitive structure during the concrete operational stage (8-11 years) is logical but depends upon concrete referents.
In the final stage of formal operations (12-15 years), thinking involves abstractions.
The stage of interest for this entry is the period between 3-7 years of age. Piaget proposes that children's intelligence is developing via intuition. Interesting stuff.
In this particular case I'll let you make your own conclusion, however, Gabriel's discovery kind of makes sense when you really think about it: he's making an assumption based on one single correct observation. Instinctively, he's sure that "3 + 3 = 4," otherwise, he wouldn't continue on with his number sequence, all the way up to 11.
To put things in perspective and to give a more "real" example of intuition and reasoning: imagine telling someone, who doesn't know what a prime number is, that 3, 5, and 7 are prime numbers and then asking what is the next number in the sequence.
A most sensible answer would be 9.
If I didn't know what a prime number is, I'd would rationalize that 3, 5, and 7 are odd numbers, and thus 9 is the next prime. Not only that, I would confidently say that all odd numbers are prime - Of course, we all know that this is horribly wrong, however, the observation is made based on a widely known fact: 3, 5, and 7 are odd numbers.
I can't remember when was the first time I learnt about prime numbers, but, I'm almost certain I made such horrible mistake.
For those of us, who don't know or remember: a prime number is an Integer that can only be divided by 1 and itself, and must be greater than 1. For example, 4 is not prime: 4 = 2 * 2.
BTW, if this silly fact has been proved before, I have never seen it and hence the claim of the "Grabriel Theorem." If this trivial piece of Mathematics has already being published, shame on the publisher, and obviously my apoligies to the other author. The result of this "theorem" was the result of a few minutes of leasure time.
BTW2, I haven't broken the news to Grabiel that 2 + 2 is not 3. I haven't had the heart. I'm also hoping he'll eventually figure out the truth all by himself.
Comments:
dude, you could have written the whole thing in 2 paragraphs. ghee man, lots of bla bla for nothing ;p. BTW, what about extending this to the imaginary numbers. Heck, maybe even to the prime numbers. Maybe your son will solve rieman. good luck man.