Dividing by zero

Originally posted by FabianFnas
Rationals are a countable infinite set.
Zero is an even number.

whoops, you beat me to it. Didn't see this post.

Originally posted by moonbus
(Though some states have defined it as even for legal purposes because some states ration gasoline to even-numbered license plates or odd-numbered license plates on certain days.)

And you can get a number plate that is just zero?

Originally posted by moonbus
PS Don't think of infinity as a quantity--not even a very large one. If you do, you'll either ask silly questions ('is infinity odd or even' ) or you'll just go bonkers. Think of infinity as a property (e.g., limitlessness).

In the "extended" real number system you could define +∞ as "the number greater than every real number" and -∞ as "the number less than every real number."

Most of what's being talked about in this thread has to do with definitions. Whether or not you want to call ∞ a number is largely a matter of semantics. In calculus and elementary differential equations courses I think ∞ is best not thought of as a number, while in measure theory it may be more naturally treated as a number.

Also (and this has sort of been mentioned), most mathematicians would not say the set of integers and the set of even integers have the "same number of elements," but rather would say they have the same cardinality. All that means is there exists a one-to-one correspondence between the elements of the two sets. Look up the "Hilbert Hotel" to see how this can be done. For example, the sets {1,2,3,...} and {2,3,4,...} have the same cardinality since the function f(n)=n+1 serves as a one-to-one correspondence between the elements of {1,2,3,...} and the elements of {2,3,4,...}.

Originally posted by Soothfast
In the "extended" real number system you could define +∞ as "the number greater than every real number" and -∞ as "the number less than every real number."

Most of what's being talked about in this thread has to do with definitions. Whether you want to define ∞ as a number is largely a matter of semantics.

No, you cannot define oo as being a number, as any other number, without introducing problems.

Is oo + 1 larger than oo? Is oo * 2 larger than oo? Is oo*oo larger than oo? It has been proven that they are all equal. That you cannot do with any other real number.

Originally posted by FabianFnas
No, you cannot define oo as being a number, as any other number, without introducing problems.

Is oo + 1 larger than oo? Is oo * 2 larger than oo? Is oo*oo larger than oo? It has been proven that they are all equal. That you cannot do with any other real number.

You need to break out of your semantic shackles, young Padawan. 😉

Look up the hyperreal number system for some mind-blowing stuff. I'm not a big fan of the system, but it's a fine example of "nonstandard analysis".

Originally posted by FabianFnas
No, you cannot define oo as being a number, as any other number, without introducing problems.

Is oo + 1 larger than oo? Is oo * 2 larger than oo? Is oo*oo larger than oo? It has been proven that they are all equal. That you cannot do with any other real number.

Also I should point out that, in set theory and other areas of mathematics, the various "sizes" of infinity are usually referred to as "transfinite numbers".

Originally posted by Soothfast
Also I should point out that, in set theory and other areas of mathematics, the various "sizes" of infinity are usually referred to as "transfinite numbers".

Then you know that there are no cardinalities between that of integers and that of reals.

Originally posted by humy

Without Cantor, I wouldn't believe it either.

I would assert that Cantor was at least partially wrong! The ratio of one infinite set and another infinite set of the same order is always undefined and completely meaningless.

Two sets of countable infinitive numbers [b]are always in a one-to-one relationship.

I ...[text shortened]... te]
No. And I can give a mathematical proof of that on request if anyone is really interested.[/b]

Proof of the conjecture! We demand proof! Please, please, please!

The post that was quoted here has been removed

Ah, this is a competition? Now I understand!
Who makes the rules...? You?

Originally posted by FabianFnas
Rationals are a countable infinite set.
Zero is an even number.

I stand corrected. Rational numbers can be mapped to integers, and are therefore countable. I was momentarily confusing real numbers (which are non-denumerable) with rational numbers (which are). In any fraction x/y, x can be mapped to an integer and y can also be mapped to an integer; so rational numbers are denumerable in the sense required. The set of rational numbers does however have the interesting property that there is no next-number.

RE zero being even: http://math.stackexchange.com/questions/15556/is-zero-odd-or-even

Originally posted by moonbus
I stand corrected. Rational numbers can be mapped to integers, and are therefore countable. I was momentarily confusing real numbers (which are non-denumerable) with rational numbers (which are). In any fraction x/y, x can be mapped to an integer and y can also be mapped to an integer; so rational numbers are denumerable in the sense required. The set of rat ...[text shortened]... number.

RE zero being even: http://math.stackexchange.com/questions/15556/is-zero-odd-or-even

This notion "countable" is somewhat misleading. Because it tells you that you can count them as integers: One two three.

A better word would be "listable". Because when you list something they don't necessarily be in order. Rationals are easy to list, one after another, if you don't mind what item is larger and what is lesser.

One interesting property about rationals is:
If you chose two different rationals, there are always infinitely many rationals between them. How close you chose the two rationals doesn't matter, there are always another rational in between, even infinitely many of them.

Same goes for reals. Chose two real numbers and you will always find one rational in between, even infinitely many of them. It doesn't matter how close you chose them to be.

Originally posted by FabianFnas
This notion "countable" is somewhat misleading. Because it tells you that you can count them as integers: One two three.

A better word would be "listable". Because when you list something they don't necessarily be in order. Rationals are easy to list, one after another, if you don't mind what item is larger and what is lesser.

One interesting proper ...[text shortened]... onal in between, even infinitely many of them. It doesn't matter how close you chose them to be.

Yes, I agree; "countabiity" or "denumerability" apply only in a peculiar way to infinities. But "listable" also doesn't quite catch the essential property of numbers being in series. We'd have to specify an "ordered list" to ensure that sequenciality is perserved.

The fact that there is no next-rational-number is what led me to mistakenly think that rationals could not be put into one-to-one correspondence with integers. I did a quick search and found the diagonal method, which is obvious once you've seen it.

Infinities are confusing. It is easy to go astray with them if one is not careful. I think it is most fruitful to think of infintiites not as quantities at all, but rather as sets with properties: such a limitlessness, ordered listableness, etc.

Originally posted by twhitehead
And you can get a number plate that is just zero?

http://search.yahoo.com/search?ei=utf-8&fr=aaplw&p=license+plate+ending+in+zero+image

If the link does not work, search on "license plate ending in zero image"

scroll down and click on "Oregon 1937"

EDIT:" there is another one from Maryland

Originally posted by moonbus
Yes, I agree; "countabiity" or "denumerability" apply only in a peculiar way to infinities. But "listable" also doesn't quite catch the essential property of numbers being in series. We'd have to specify an "ordered list" to ensure that sequenciality is perserved.

The fact that there is no next-rational-number is what led me to mistakenly think that ratio ...[text shortened]... all, but rather as sets with properties: such a limitlessness, ordered listableness, etc.

Well, an ordered list isn't important here. The importance with the listable (countable) proparty of an infinite set is to establish a one-to-one relationship with elements in another infinite set, namely the integer set.

Cantor himself had much trouble to explain the infinity and its pecularities to the other matematicians of is time. It was too counterintuitive for people to accept. The criticism toward him led to a depression and later a place in a mental hospital. He died without the recognision of being right, that he got later. Now it is a perfectly valid branch of mathematics.