Infinity

Infinity is an abstract idea that, in essence, we cannot even picture. Infinity itself has no end, which is where its name comes from, but general infinity has no beginning either (although we can narrow it down – for example, a ray is also infinite, but it has a starting point). Infinity is usually written with the so-called “sideways figure eight” – ∞.

Calculating with infinities

Infinity is partly similar to zero, which is also peculiar in many ways and has special rules of its own. For example, you can't divide by zero – probably every small child knows that (by the way, do you know this joke? What is the best excuse when you haven't done your maths homework? “Miss, I accidentally started dividing by zero and my exercise book burst into flames…” ;o)).

Just as anything times zero is zero, it is also true that anything times infinity or anything plus infinity is infinity. If you subtract any number, however large, from infinity, you are always left with infinity. The same goes if you try to divide infinity. Infinity is simply so large that even if you subtracted millions from it every second, you would still always be left with infinity. Sometimes this is a little hard to grasp, for example take this little equation: ∞ + 1 = ∞. Is it true or not?

Logic tells us that the left-hand side of the equation is bigger, since if I add one to infinity, the whole expression must be bigger than plain infinity. It doesn't have to be :-). As I said, infinity plus anything is infinity. So infinity plus one is infinity, and infinity equals infinity. The equation is true.

There is another way to see it, through limits. Take the fraction 1/x and work out the limit from the right as x→0⁺, and let's assume our exercise book hasn't burst into flames. This limit would be equal to infinity. If you add one to this expression, you get the expression (1+x)/x, and the limit as x→0⁺ would again be infinity.

The fun starts when we start combining infinities more. For example, what does infinity minus infinity equal? Logic tells us that we are subtracting two equally large numbers from each other, so the result will be zero. I also said that if we subtract any number from infinity, the result is again infinity. On the other hand, if we subtract infinity from any number, we get minus infinity. So how do we get out of this? All the results are theoretically possible. Maths says that the expression ∞ − ∞ is not equal to anything in particular; it is an undefined or indeterminate expression. It is similar to trying to divide by zero – it simply makes no sense. If you come across an indeterminate expression like this, you have to rearrange it somehow (for some kinds of indeterminate expressions you can use, for example, L’Hôpital’s rule [pronounced “low-pee-tal’s rule”]).

Other undefined or indeterminate expressions:

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

By the way, the basic rules for signs do apply, for example −1 · ∞ equals minus infinity.

Want to go mad?

Then dive into set theory and work out which set is bigger. For example, is the set of even numbers bigger, or the set of odd numbers? Even numbers or positive integers? Positive integers or integers? Even numbers or integers? Positive integers or rational numbers? The answer is that they are all the same size. Strange as it may seem (there must surely be more integers than odd numbers), they are sets of the same size. The set of real numbers, however, is bigger than the sets mentioned. The devil himself couldn't make sense of it ;-)