Friday, June 15, 2007

It's Time For...

...another frightening look inside my head.

As an engineer I occasionally have to deal with numbers.

Yesterday, I somehow spent my thinking time while I ate my lunch pondering prime numbers.

Just go with me here, OK?

I'll just ask y'all; can one type of infinity be less infinite than another type of infinity?

This is the conundrum that has me asking this.

A) There are an infinite number of integers.
B) Some integers are what we call even numbers, all perfectly divisible by the number 2 (no remainder).
C) Some integers are what we call prime numbers and can only be divided evenly by themselves and the number 1.

With me so far?

Every other integer is an even number; 2, 4, 6, 8, ..., 2200, 2202, 2204, ...

But prime numbers happen less often; 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, ...

So, again my question, rephrased: If there are an infinite number of even integers, and an infinite number of prime integers, and if prime integers "happen" less often than even integers, are there fewer prime integers than there are even integers, despite the fact that they both continue on to infinity?

In short, are prime integers "less infinite" in number than "even integers" are?

I was just wondering.

Then I finished my lunch and had to go back to work.

I needed a work break to get away from my lunch break.

Qtpies7 said...

My son thinks that one infinity can be less infinite than another.
My brain just hurts.
He says you have to go a lot farther into infinity to get the same amount of prime integers.
My brain still hurts.

We got our power back on, less than 48 hours, it wasn't too bad. 9 days twice in a month would be awful! The first day we still had the sort nice leftover from AC but yesterday wasn't so kind. We plugged our tv into our van outlet and watched movies, it was family bonding.

Babystepper said...

This is exactly the kind of thing my husband sits and think about. You engineers have issues.

I'll have him read you post for a little extra input. I'm with qtpies. My brain hurts.

Daughter two said...

You lost me at numbers.

Travis said...

Now you made me get my dictionary.

Infinity is defined as unlimited extent of time, space, or quantity. There's a paragraph of text to define this word, but let's go with this aspect.

By this definition, neither can be more infinite than the other because infinity simply is unlimited. Prime numbers may not catch up to integers in terms of the quantity of them, yet they will continue extending in an unlimited fashion such that one cannot measure which is longer. We can't say how many of each one there will ultimately be, therefore the only measure of them is infinite.

There are more integers than prime numbers, but there is no end to either set of numbers.

Or, more simply, a ton of feathers weighs the same as a ton of concrete.

I don't think it's the substance being measured that is the center of the question. Rather it is the measurement itself. Infinite is infinite and a ton is a ton, no matter what you measure.

That's my 2 cents, adjusted for inflation. I'm going now to take some aspirin.

JAM said...

Thanks all. Yeah, I know, infinity is infinity and to wonder if something is less infinite than another kind of infinity is really a meaningless question. But it had me going there for a while. And I was only exaggerating a little bit when I said I had to go back to work to get a break from lunch. I'm like a dog with a bone on things like that.

Plus, since I'm always walking a tightrope on whether or not I have a blog topic, I used it.

Jenny Ryan said...

These are exactly the kinds of questions my husband will randomly break out into in the midst of, say, a relaxing day on the beach.

I just smile and look pretty and nod a lot.