Saturday, December 29, 2012

A Monograph on the Special Number 36 and the Fundamental Theorem of Arithmetic

A monograph is specialist work of writing on a single subject or an aspect of a subject, usually by a single author. Well, I’m not single … in fact … today, I’ve been married for 36 years. That alone makes 36 special.

Now most non-mathematicians focus on “decades.” For example, a tenth anniversary, a fiftieth anniversary, a centennial. And that is fine. Some like the “half-decade,” such as a twenty-fifth anniversary. Those are meaningful milestones. Why, just last year, we had our 35th wedding anniversary. That’s the Coral or Jade Anniversary (US is confused between the two, in the UK it is just Jade). I didn’t get her Jade or Coral, but I did get her Tourmarine earrings (with lots of diamonds … always get those ladies lots of diamonds.)

But, as an integer, 35 is rather boring. It belongs in the class of numbers that have only two prime factors. Five time seven. That’s it.

The Fundamental Theorem of Arithmetic, also called the "unique factorization theorem" or the "unique-prime-factorization theorem," states that every integer greater than 1 is either prime itself or is the product of prime numbers, and that, although the order of the primes in the second case is arbitrary, the primes themselves are not.

Stated a little less technically, any integer can be factored into prime factors in one and only one way. (Order of factors doesn’t matter.) By the way, integers are numbers that can be written without a fractional or decimal component. For example

…, -2, -1, 0, 1, 2, 3, …

and prime numbers are a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number is an integer greater than zero, or

1, 2, 3, …

I’m not going to define divisors. This is turning into an infinite series.

So, let’s examine (but definitely not prove … you don’t prove by example) the Fundamental Theorem of Arithmetic.

For example, 36 can be factored as 6 times 6. And 6 is 2 x 3. So 36 = 2 x 2 x 3 x 3. (Remember, order doesn’t matter in multiplication.)

But 36 also is 2 x 18. And 18 = 2 x 9. And 9 = 3 x 3. So that yields 2 x 2 x 3 x 3. The same thing.

Wait, isn’t 36 = 3 x 12? Why yes it is. And 12 = 3 x 4, and 4 = 2 x 2, therefore 2 x 2 x 3 x 3.

So, whether you start with 2 x 18 or 3 x 12 or 4 x 9 or 6 x 6, you always end up at the basic factorization of 2 x 2 x 3 x 3.

Now 1 is not considered a prime factor. It would just “get in the way” of the math because one times any number is the original number, and any number can be divided by one … actually over and over and over again. 1 x 1 x 1 x 1 x N = N. No change; so that sorta takes the fun out of one.

Still, you can consider 36 = 1 x 1 x 2 x 2 x 3 x 3. In the words of the church lady, “Isn’t that special.” By the way, that's 12 x 22 x 32. Really special!

Prime integers aren’t just mathematician’s wet dreams. Why the whole system of computer encryption is based on the difficulty of factoring large numbers made up of only two primes. Without that encryption, there would be no secret messages nor secure computer communications. So primes … and the very fact they’ve been studied since ancient (Greek) times … is at the fundamental foundation of our modern digital society.

Yes, 36 is a very special number. And my wife Linda is a very, very, very special lady. I love her this much = 10 raised to the 100 power, raised to the 100 power, raised to the 100 power. That’s a hypergoogleplex. No, on second thought, I love her a lot more than that.

(Let's see if html can handle a hypergoogleplex: 10100100100. Wow … I think I love html as much as I love math.)

Why I love her "This Much" … holding out arms like a fisherman explaining a large fish … assume length of left arm is minus infinity and length of right arm is positive infinity. Now we're cooking.

Of course, infinity isn't a number, it's more like a "limit." Oh great, now I'm going to have to explain "limits." This will take a while. Better sit down. … (to be continued?!?)

Happy Anniversary love of my life = LRC.