Zero is a strange beast. It is the only number that is neither positive nor negative, it is arguably neither real nor imaginary (or maybe both), and it’s the only number that in modern mathematics no number can be divided by.
The way that 0 sometimes behaves like any other number and sometimes doesn’t reminds me of words like “nowhere”, “never” and “nothing”. “Nothing” is a word that doesn’t refer to any thing, and yet it can often be coherently used in places where you would normally expect a word that does refer to some thing; for instance, a question like “What are you doing?” seems to demand an action as an answer, yet it can also be answered by the word “nothing,” which doesn’t refer to any action at all.
Regarding division by 0, it is often said (though never by actual mathematicians) that 1/0=infinity. This is based on the observation that in the function 1/x=y, as x approaches 0, y approaches infinity (assuming x approaches 0 from the positive side. If it approaches 0 from the negative side, y approaches -infinity). For a number of reasons, however, 1/0=infinity is rejected by serious mathematicians, as it simply doesn’t work in the system of mathematics that’s been built up over millennia.
Whether or not it’s valid, however, it obviously seems intuitively true to a lot of people, as does the inverse equation, 1/infinity=0. I also note that infinity is another concept that can sometimes be used like a number and sometimes can’t, and it’s very interesting to see these two concepts which parallel each other in this way bound together by such a simple equation.
Now, the problem with this equation surfaces when you realize that if this equation were valid then 0*infinity=1 would also be valid, as would 0*infinity=2, 3, or any other number, and this is one of the main reasons why these aren’t considered valid equations by mathematicians.
0 and infinity could be seen as being equivalent to nothing and everything, respectively, while all other numbers might be thought of as being different somethings. Something I find interesting about this is that the idea all existing somethings being a product of nothing and everything is reminiscent of some mystical texts I’ve read, such as in Helena Blavatsky’s The Secret Doctrine, which states that the absolute, unmanifest reality from which all concrete, manifest things spring can be described as either absolute being and yet nonbeing, or absolute consciousness and yet unconsciousness.
Under the Spell of Words 