Thursday, August 25, 2011

Cauchy sequences and intervals

The counting numbers are fundamental to all of computation. Using collections of counting numbers we can construct more complex objects, for example, the rational numbers can be constructed with three counting numbers: a numerator, a denominator, and a sign.

The rational numbers allow us to effectively confront most challenges; however, there are some numbers, known as the irrational numbers, in which can never be expressed as rational numbers. Just as you can never express infinity because there is always a greater number, there is always a greater rational approximation for any irrational number. Such values that can never be expressed because they always have another step that can always bring them closer to a target point are the convergence value of a cauchy sequence.

For example, one of the first irrational numbers discovered was the square root of two, which can be expressed in terms of the following cauchy sequence:

$$\sqrt{2} = \sum _{n = 0}^{\infty} {(-1)^{n + 1} \frac{(2n-3)!!}{(2n)!!}}$$

Most other popular irrational numbers like e, pi, or the golden ratio can be expressed in terms of cauchy sequences:

$$e = \sum _{n=0}^{\infty} {\frac{1}{n!}}$$
$$\pi = 4 \sum _{n=0}^{\infty} {\frac{(-1)^n}{2n+1}}$$
$$\varphi = [1; \overline{1}]$$

Although we can never acquire any of these values, we can approximate them in terms of intervals. For example, we know pi is in the interval between 3 and 4, and as we use our cauchy sequence we progressively narrow that range. Once we have constructed intervals, we are going to want to perform some operations on them, leading to interval arithmetic. These same methods approximation can be used for physical measurements.