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In the mathematics of one's elementary K-12 education there are a revolting number of fundamental assumptions that one is taught to make, by virtue of the fact that the student is not taught to think about math but rather only to use it as a tool. Even if one did stop to question the very fundamental ideas of numbers, there would be no one capable of answering any really good questions or proving any interesting theorems. In my very limited experience, these fundamental questions turn out to be extremely interesting and lead to very beautiful explanations of what numbers really are. I was so excited when I read this one that I have to relate it to the reader.

First we must ask the question, what is a real number? No one really stops to explain what they actually are. Sure, teachers will say that they are the set of all rational and irrational numbers, where rational numbers are those numbers that can be written as fractions and irrational numbers are anything else. But that doesn't really tell much about what a real number actually is? How can I even know they exist? It seems to be an awfully heinous and unrealistic complication to make numbers that can't be written as fractions and go on indefinitely. Well, maybe this method won't really "prove" the existance of real numbers so much as define their existance, but it's pretty amazingly cool. So here it is, sans set notation because that isn't really fully available in the character map.

Theoretically, anyone should be able to understand this, there's no opaque shroud of obfuscatory notation in the way. It's pure logic. But just so everyone can understand that this is cool, I should say a few things. Sets are bunches of things--usually numbers--basically arranged in a list. If a set 1 is a subset of set 2, then all of the elements of set 1 are also elements set 2. The union of two sets is another set containing all the elements from both sets. The intersection of two sets is another set containing the elements in common between both sets. An empty set is a set containing nothing.

Define a "cut" (a Dedekind cut, to be more specific) of sets A and B, where A and B are subsets of the set of rational numbers so that:
1.) The union of A and B = the set of rational numbers, A is not an empty set, B is not an empty set, and the intersection of A and B is an empty set.
2.) If a is an element of A, and b is an element of B, then a < b.
and
3.) A contains no largest element.

A real number is thus defined as a cut in the set of rational numbers.

Think about it. It's cool. It basically defines the set of real numbers as the set of rational numbers plus all the infinitely numerous and infinitely small gaps in between the rational numbers. It's like having an infinite string that has been cut in a bizillion places and defining the real numbers as the string and all the space from the cuts in between. Maybe that doesn't give much more insight into real numbers, but you have to admit that it's just a little bit exciting.