Infinity

Infinity is an abstract concept that is basically impossible to imagine. Infinity itself has no end, hence the name, but general infinity has no beginning (we can, however, make it more specific, for example, a semi-infinite line is also infinite but has a beginning). An infinity is usually written with a "lying eight" - ∞.

We count infinities

An infinity is partly similar to zero, it is also special in many ways and special rules apply. For example, you can't divide by zero, probably every little kid knows that (by the way, you know this joke? What's the best excuse for not having math homework? "Teacher, I accidentally started dividing by zero and burned my notebook...";o)).

Just like anything times zero is zero, anything times or anything plus infinity is infinity. If you subtract any high number from infinity, you are always left with infinity. Likewise, if you try to divide infinity. In short, infinity is so big that even if you subtract millions from it every second, you're still always left with infinity. Sometimes it's a little hard to understand, for example, let's have this equation: ∞ + 1 = ∞. Is it valid or not?

Logic tells us that the left-hand side of the equation is greater, just that if I add one to infinity, the whole expression must be greater than just infinity. It doesn't have to :-). Like I said, infinity plus whatever is infinity. So infinity plus one is infinity and infinity equals infinity. The equation holds.

Another way can be found, through limits. Now, let's have a fraction 1/x and calculate the limit for x→0 and assume that we haven't burned the notebook. Such a limit would be equal to infinity. If you add one to this expression, you have the expression (1+x)/x and the limit for x→0 would again be infinity.

The fun begins if we start combining infinities more. For example, what does the expression infinity minus infinity equal? Logic tells us that we are subtracting two equal numbers from each other, so the result will be zero. I also said that if we subtract any number from infinity, the result will be infinity again. Conversely, if we subtract infinity from any number, the result is minus infinity. So how do we get out of this now? All results are theoretically possible. Mathematics says that the expression ∞ -∞ does not equal anything concrete, it is an undefined/undefined expression. It's a similar example to trying to divide by zero, it just doesn't make sense. If you come across such an indeterminate expression, you have to modify the expression in some way (for some kinds of indeterminate expressions, you can use, for example, L'Hospital's rule [read "lopital's rule"]).

Other undefined/vague expressions:

• ∞ − ∞
• ∞/∞
• 0/0
• 0 · ∞
• 0⁰
• ∞⁰
• $1^∞$

Incidentally, though, basic operations on signs apply; for example, −1 · ∞ equals minus infinity.

Do you want to go crazy?

Then get into set theory and calculate which set is bigger. For example, is the set of even numbers bigger or the set of odd numbers bigger? Even numbers or natural numbers? Natural or integer? Even or whole? Natural or rational? The answer is that they are all the same size. Strange as it may seem (there must be more integers than odd numbers), they are equally large sets. The set of real numbers is larger than the aforementioned sets. The devil to know ;-)