[garsh]

Today's topic is infinity.

First question - Are there more integers than natural numbers?

Potentially useful definitions:

• Integer: Any positive or negative number that does not include a fraction or decimal, including zero.
• Natural numbers: the set of positive integers (1, 2, 3, etc.).
• Infinite set: a set which has a proper subset that is as large as the set itself.
• Proper subset: a subset of a set that lacks at least one element that is in the set.

If you are already familiar with the topic, please do not post answers. Give others a shot at guessing.

[sullise]

Well, since natural numbers only comprise the set of POSITIVE integers, then no, there can't be more Natural Numbers then integers. On the other hand, if you consider the negative and postive of a integer to be 1, then there are the same number of natural numbers as integers.

[garsh]

Well, since natural numbers only comprise the set of POSITIVE integers, then no, there can't be more Natural Numbers then integers.

Agreed.

On the other hand, if you consider the negative and postive of a integer to be 1, then there are the same number of natural numbers as integers.

What about the integer value 0?

[sullise]

Good point....overlooked that 0 as being an integer..that's the trick to the question..lol.

[alphynewman]

Both the ranges are infinite (infinite integers and infinite natural numbers)

infinite = infinite.

So mathematically, there cannot be more integers than natural numbers.

Am i right?

[penguin110]

I'll be back, let me get my calculator.

[tolik]

Both the ranges are infinite (infinite integers and infinite natural numbers)

infinite = infinite.

So mathematically, there cannot be more integers than natural numbers.

Am i right?

I'm pretty sure sullise is on the right track here...

[WillyNilly]

Both the ranges are infinite (infinite integers and infinite natural numbers)

infinite = infinite.

So mathematically, there cannot be more integers than natural numbers.

Am i right?

But wouldn't there will always be twice as many integers as natural numbers?

(I don't see how the negative and positive versions of a number are considered to be one)

If N=set of natural numbers and I=set of integers based on N, the number of constituents in I will always be twice as many natural numbers plus one (the zero thing).

So as number of N approaches infinity, the number of I approaches two times infinity plus one (or basically always more integers than natural numbers).

*whew*....my brain hurts now. I'm going to back to sleep

This post has been edited by WillyNilly: 6-6-05, 9:00am

[wmspringer]

Both the ranges are infinite (infinite integers and infinite natural numbers)

infinite = infinite.

So mathematically, there cannot be more integers than natural numbers.

Am i right?

You're not, sorry :-)

It's true that both sets are infinite, but one infinite set is bigger than the other infinite set. :-)

[garsh]

It's true that both sets are infinite, but one infinite set is bigger than the other infinite set.

Actually, both of these infinite sets are exactly the same size!

The way to prove it is to find a way to map every natural number to every integer. If it can be done, then both sets are the same size. So here's one way to do the mapping:

Naturals, Integers
1, 0
2, 1
3, -1
4, 2
5, -2
6, 3
7, -3
8, 4
9, -4
10, 5
11, -5
...

As you can see, we can keep matching all the natural numbers to all the integers in this fashion, so these sets are exactly the same size!

But wouldn't there will always be twice as many integers as natural numbers?

Mathematically speaking, two times infinity is still infinity. It's similar to saying that two times zero is still zero. It's still the same size - infinite.

[garsh]

Now, once you wrap your head around that answer, here are the ones where the answers really blow my mind:

2. Are there more rational numbers than natural numbers?
3. Are there more real numbers between 0 and 1 than there are natural numbers?

Potentially useful definitions:

• Natural numbers: the set of positive integers (1, 2, 3, etc.).
• Rational numbers: numbers that can be expressed as quotients of integers (1/2, 3/4, -8/7, etc.)
• Real numbers: number which can be represented as a non-terminating decimal (pi, e, 1.11111..., etc.)
• Infinite set: a set which has a proper subset that is as large as the set itself.
• Proper subset: a subset of a set that lacks at least one element that is in the set.

[tolik]

oops.

EDIT: Comment deleted, I was posting it at the same time garsh posted his next question.

This post has been edited by tolik: 6-6-05, 11:06am

[wmspringer]

Actually, both of these infinite sets are exactly the same size!

The way to prove it is to find a way to map every natural number to every integer.  If it can be done, then both sets are the same size.  So here's one way to do the mapping:

But that just shows that there exists *one* mapping of the natural numbers onto the integers. I could set it up similarly to show a mapping from the reals between zero and one to all integers (where the first set is actually infinitely bigger)

[wmspringer]

3. Are there more real numbers between 0 and 1 than there are natural numbers?

Ooops, used this as an example above before I read this post and realized you'd asked it :-)

Answering them both off the top of my head:

2) Yes, the natural numbers are a subset of the rationals

3) Yes, infinitely more; we actually had to prove this in a class I took back in 2001

This post has been edited by wmspringer: 6-6-05, 11:24am

[alphynewman]

You're not, sorry :-)

It's true that both sets are infinite, but one infinite set is bigger than the other infinite set. :-)

I was right... na na na na naaaa na...
they need better icons for bragging

[garsh]

But that just shows that there exists *one* mapping of the natural numbers onto the integers.

Sorry, I should have been more precise - that was a one-to-one mapping between the natural numbers & the integers. Once you've found a one-to-one mapping, you've shown that both sets are the same size.

I could set it up similarly to show a mapping from the reals between zero and one to all integers

But, can you show us a one-to-one mapping?
And how about the rational numbers?

[wmspringer]

This is a fun topic; I haven't tried this stuff in years :-)

[plankton]

most of us think of infinity as one big number, some infinities are bigger ... But the first infinity is bigger. T

http://www.cs.dartmouth.edu/~rockmore/WSJ.pdf

[dejavu]

Today's topic is infinity.

yes. I agree. that is my topic every day. my total existence. infinite demands from infinite kids and infinite animals and infinite bills and infinite friends with too many expecations of me; infinite madness.
infinite no end to this infinite sapping of my energy and resources. I'm gonna die.


(don't worry about me, it's just hurricane season)

[wurlybird9]

well, don't worry, deja. according to chaos theory, one more deal posted, and those hurricanes and all that other infinite bothersome stuff might head somewhere else.

[garsh]

Now, once you wrap your head around that answer, here are the ones where the answers really blow my mind:

2. Are there more rational numbers than natural numbers?
3. Are there more real numbers between 0 and 1 than there are natural numbers?

These are much harder.
The answers turn out to be:

2. There are just as many rational numbers as there are natural numbers.
3. There are more real numbers between 0 and 1 than there are natural numbers!

Yes, there are actually different sizes of infinity!

Read more about the proofs here.

[kar522]

This is a fun topic; I haven't tried this stuff in years :-)

Maybe you need to get out infinitely more....

[garsh]

Maybe you need to get out infinitely more....

Bah! Go stroll the feed & seed some more and leave us alone!

[momalisa76]

Bah!  Go stroll the feed & seed some more and leave us alone!

I'm heading up to Paul Taylors soon...anybody need anything?

[kar522]

I'm heading up to Paul Taylors soon...anybody need anything? 

About 20 lbs of bulk niger thistle, that is, if you're buying.... And a couple of hog panels, while you're at it....I need to build a strong trellis....

City folk just don't understand the wonderfulness of feed & seed stores...

[momalisa76]

i've never been to paul taylors, but i've heard wonderful things...it was described to me as the WM of horse and western supplies...if it's not there, you better get ready to order it online

[Keggster22]

[dejavu]

Yes, there are actually different sizes of infinity!

Read more about the proofs here.

OMG this cracks me up. I needed something new to add to my life!

great! now I have to go read.

[dejavu]

hey, can you explain to me why I have a friend who is a rocket scientist, yet cannot figure out how to change a light bulb? it's so amazing when I go to her house and find that she put a lamp in the bathroom rather than changed the light bulb in the ceiling fixture. the girl is mensa. I don't get it.

(well, the question is not why I have a friend like this, but how can she be like that?)



This post has been edited by dejavu: 6-7-05, 7:04pm

[wurlybird9]

that is rather puzzling. While some types of geniuses may not be mechanically inclined, you would expect a rocket scientist to.

How do you know that she doesn't know how to change the bulb, though? Maybe she just couldn't reach and figured it was easier to just use a lamp, than have to grab a stepladder every time.