Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science. Alexey Stakhov
steps n of the algorithm and the number of “indicatory elements” k, involved in the measurement, i.e.,
Further, the function F(n, k) will also be called the efficiency function of the measurement algorithm.
In the modern measurement technology (in particular, in the technique of analog-to-digital conversion), the following measurement algorithms are widely used:
1.8.1. Counting algorithm
This algorithm uses only one IE (k = 1) and is implemented in the n steps; at the same time, the segment AB is divided into n + 1 equal parts in the n steps, i.e., in this case, the (n, 1)-exactness of the (n, 1, S)-algorithm is determined by the following efficiency function:
It is important to emphasize that this measurement algorithm takes its origin in measurement praxis and has deep roots in ancient mathematics. It is this algorithm that underlies the Eudoxus exhaustion method and the Euclidean definition of the natural number:
which defines not only the natural numbers but also all the problems of the elementary theory of numbers, the foundations of which are set out in Euclid’s Elements.
1.8.2. “Binary” algorithm
This algorithm also uses the only one IE (k = 1) and is implemented in the n steps; at the same time, the initial segment AB is divided into the 2n equal parts, i.e., in this case, the (n, 1)-exactness of the algorithm is determined by the following efficiency function:
Note that this algorithm “generates” the binary representation of integers (the “binary system”), which underlies the modern information technology:
where
Thus, the above classic measurement algorithms (counting algorithm and “binary” algorithm), on the one hand, generate elementary theory of numbers (counting algorithm) and on the other hand, are the basis of modern computer science (“binary” algorithm).
1.8.3. Readout algorithm
This algorithm is realized in one step (n = 1) and uses k IE; at the same time, by using k IE, the segment AB is divided into k + 1 equal parts, i.e., in this case, the (1, k)-exactness of the algorithm is determined by the following efficiency function:
As an example, let’s consider more in detail the operation of the classic measurement algorithms: the counting algorithm (Fig. 1.5), the “binary” algorithm (Fig. 1.6) and the readout algorithm (Fig. 1.7).
Fig. 1.5. The 3-step “counting algorithm”.
Fig. 1.6. The “binary” measurement algorithm.
Fig. 1.7. Readout algorithm.
The counting algorithm, shown in Fig. 1.5, is realized in three steps and uses only one IE (k = 1). It divides the segment [0, 4] into four equal parts.
The first step consists in the enclosing IE to point 1. In this case, depending on the indication of the IE, two situations may arise: [0, 1] and [1, 4].
The second step:
(a) If the “indication” of IE at point 1 is equal to 0 (IE shows to the left), this means that the “measurable” point X is on the segment [0, 1]. In this situation, the measurement process ends, because the coordinate of point X (X [0, 1]) has been defined with “exactness” equal to the measurement unit.
(b) If the “indication” of IE at point 1 is equal to 1 (IE shows to the right), this means that the “measurable” point X is located on the interval [1, 4]. In this situation, the measurement process continues and IE in the next step is enclosing to point 2; as a result, we get two new situations: [1, 2] and [2, 4].
The third step:
(a) If the “indication” of IE in point 2 is equal to 0 (IE shows to the left), this means that the “measurable” point X is on the segment [1, 2]. In this situation, the measurement process ends, because the coordinate of point X (X [1, 2]) has been determined with the “exactness” equal to the measurement unit.
(b) If the “indication” of IE in point 2 is equal to 1 (IE shows to the right), this means that the “measurable” point X is on the segment [2, 4]. In this situation, the measurement process continues and IE in the next step enclosing to point 3. As a result, we get two new situations: [2, 3] and [3, 4], depending on the “indication” of IE at the last step of the algorithm.
Figure 1.6 shows the operation of the 3-step “binary” algorithm on the segment [0, 8]. The essence of the algorithm is clear from Fig. 1.6 and reduces to enclosing the IE to the middle of the uncertainty interval relative to point X obtained in the previous step on the basis of the “indication” of IE.
Finally, Fig. 1.7 presents an example of the readout algorithm widely used in measurement practice. This algorithm underlies the traditional measuring ruler.
The readout algorithm, shown in Fig. 1.7, is realized in one step (n = 1) and uses three IE, which are enclosed simultaneously to the points of the segment [0, 4], as shown in Fig. 1.7.
1.8.4. Restrictions S
In