Approximation/Introduction/Section

A basic thought of mathematics is the idea of approximation, which occurs in many contexts and which is important both for mathematics as an auxiliary science for the empirical sciences and for the construction of mathematics as a pure science, in particular in analysis.

The first example for this is measuring, say the length of a line segment or the duration of time. Depending on the context and the aim, there are quite different ideas of what an exact measurement is, and the desired accuracy has an impact on the measuring device to take.

The result of a measurement is given in respect to a unit of measurement by a decimal fraction, that is as a number with finitely many digits after the point. The number of digits after the point indicates the claimed exactness of the measurement. To describe results of a measurement one neither needs irrational numbers nor rational numbers with a periodic decimal expansion.

Let's have a look at meteorology. From measurements at several different measurement stations, one tries to set up the weather forecast for the following days with mathematical models and computer simulation. In order to make better forecasts, one needs more measurement stations.

Let's have a look at approximations as they appear in mathematics. A certain line segment can (at least ideally) be divided into ${}n$ parts of the same length and one may be interested in the length of the parts, or in the length of the diagonal in the unit square. These lengths could also be measured, however, mathematics offers better descriptions of these lengths by providing rational numbers and irrational numbers (like ${}{\sqrt {2}}$). The determination of a good approximation is then pursued within mathematics. Let us consider the fraction ${}q={\frac {3}{7}}$. An approximation of this number with an exactness of nine digits is given by

${}0,428571428<{\frac {3}{7}}<0,428571429\,.$

The decimal fractions on the left and on the right are both approximations (estimates) of the true fraction ${}{\frac {3}{7}}$ with an error which is smaller than ${}{\frac {1}{10^{9}}}$. This is a typical accuracy of a calculator, but depending on the aim one sometimes wants a better accuracy (a smaller error). The computation in this example rests on the division algorithm, and one can go on to achieve any wanted error bound (here, it is an additional aspect that because of the periodicity we can just read off the digits and repeat them and do not have to compute further). The approximation of a given number by a decimal fraction is also called rounding.

In the empirical and in the mathematical situation, we have the following principle of approximation.

Principle of approximation: There does not exist a universal accuracy for an approximation. A good approximation method is not a single approximation, but rather a method to produce for any given wanted accuracy (error, level of exactness, deviation) an approximation within the given accuracy. To increase the accuracy (make the error smaller) one has to increase the effort.

With this principle at the back of one's mind, many difficult concepts like convergent sequence and continuity become comprehensible.

Approximations appear also in the sense that empirical functions for which a certain sampling is known, shall be described by a mathematically easy function. An example for this is the interpolation theorem. Later we will also encounter the Taylor formula, which approximates a given function in a small neighborhood of one point by a polynomial. Also, here the mentioned principle of approximation occurs, that in order to get a better approximation one has to increase the degree of the polynomials. In integration theory, the graph of a function is bounded by upper and lower staircase functions, in order to approximate the area below the graph. With finer staircase functions (shorter steps) we get better approximations.

How good an approximation is becomes sometimes clear if we want to compute with the approximations. For example, given certain estimates for the side lengths of a rectangle, what estimate does hold for the area of the rectangle? If we want to allow a certain error for the area of a rectangle, what error can we allow for the side lengths?