Since learning about some of the basic properties of the real numbers, about how they form a line that is completely continuous, we’ve begun to talk about what those properties allow us to do with the real numbers. When we think about and talk about math, we tend to think and talk about manipulating numbers, not simply admiring their existence. The idea that “proving calculus” is important rests on the fact that our society requires (and our minds enjoy) this kind of numerical manipulation, and that we want the results of these endeavors to have some real meaning.
The first type of active manipulation of numbers that we talked about was the act of arranging real numbers into a sequence. A sequence of numbers is a subset of real numbers which are arranged in a particular order. Further, each element of the sequence can be constructed from some formula; that is, if you want to find the fifth term in the sequence, you can plug “5” into a given formula and acquire that term. For example, we can create a sequence of all numbers of the form {1/n} where each “n” that we plug in is a natural number (the kind of number that you can count on your hands). In the sequence, the first element is 1, the second element is 1/2, the third element is 1/3, and so on.
This sequence, like all sequences, contains an infinity number of terms. This is because there are an infinite number of natural numbers, and therefore an infinite number of “n”s to plug into the given formula. An infinitely long list of numbers all sharing a common formula is a pretty astounding, but it turns out that many sequences have an even more incredible property. This property is called convergence, and it means that after a certain term in the sequence, there are an infinite number of terms which are all practically equal to each other.
The sequence {1/n} is a sequence with this property. Eventually, as the natural numbers which we plug into the formula get very large, the term as a whole gets very small. And when n gets extremely large, when we start dividing 1 by numbers like 1,000,000,000, the elements of our sequence get very close to each other and to zero. Because of this, we say that the sequence {1/n} converges to zero. Interestingly, the terms are never actually equal to zero, and never actually equal to each other. 1/1,000,000,000 is a different number than 1/1,000,000,001, although they are very, very close together in value.
There are actually a lot of sequences that converge, and which converge to a variety of real numbers. This property of convergence is deeply tied to the structure of the real numbers- that line which we learned about at the very beginning of class. Since that line contains every possible number, it is possible for it to contain an infinite number of numbers which are as close together as we could want, without those numbers ever actually being equal. This allows for convergence: if the real line were missing values, there would be gaps in the line which inhibited the closeness of terms in a given sequence.
Sequences, while fascinating, might seem on the surface to not actually “do” much in terms of manipulating numbers. They hold a lot of power, however. They allow us to not only order information, but a potentially infinite amount of it. We can add terms of these sequences together to get new sequences, or to get sums of numbers that approximate complicated functions. These are all things we’re going to talk about in the next few days of class, as we keep looking at what the line of real numbers allows us to do.
~Anna
