If you divided infinity by 2. Wouldn’t it still be infinity? Is there any mathematic pricipals that are attributed to infinity?
Because of Zeno, mathematics avoided “actual” infinities, and stuck to “potential” infinities. Cantor was one of the pioneers in the study of actual infinities. As long as you are careful defining your operations, actual infinities can be introduced into a number system without introducing contradictions.
There are several systems of arithmetic that allow infinities. You need to specify which one you mean. There are the cardinal and the ordinal systems of transfinite numbers (Cantorian set theory). These only deal with integers-like numbers. In both of these, there are many infinities, and an infinity divided by a finite non-zero number is unchanged. There are also systems that have a continuum of values (such as non-standard analysis), in some of these an infinity divided by two would be less than the original infinity, but still be infinite.
You can’t use infinity like a number. It can’t be divided by 2 as if it were just a big value.
However, you could take some infinite set of numbers and divide each of them by 2. For instance, say you start with the set of positive even whole numbers [2,4,6,8,…]. If you were to divide each of them by 2, you would end up with the set of positive whole numbers [1,2,3,4,…]. It’s still an infinite set, since for every large even number there is a corresponding half-value.
It gets a lot weirder when you get into sets that can’t be inumerated. For instance, the set of rational numbers is infinite, but it’s ‘more infinite’ than the set of whole numbers. This is because you can’t list all the rational numbers in any range, whereas you can do that with the whole numbers. For any list you try to make, you can show there is an infinite number of possible rational numbers left out.
Now, hopefully, we’ll get some real math majors in here to tear into the subject.
Yes, half infinity is still infinity. For that matter, there is an infinite number of points on a number line between 0 and 1 (and between 0 and 0.5).
I’m no mathematician, but it seems like there are different definitions/attributes. There is the infinity that goes on forever. And there are also levels of infinity (like your first question).
Actually, the rationals are the “same size” as the whole numbers. It’s the real numbers that give you trouble when you try to make a “list” of them.
DrMatrix posted while I was writing my comment. Just wanted to make sure no one thinks I don’t count him among the ‘real math majors’.
Yes, you can! In calculus infinity is a formal symbol (the figure 8 that fell over), not a number and you cannot manipulate like a number, but there are several systems that use infinities as numbers that can be divided by two.
Saltire, Actually, I am a real math minor. Feel free to call me a real math geek, though. 