A dichotomy translated from Greek means "sequential bisection" or "bifurcation." Dichotomous division is quite successfully used in mathematics and logic for the classification of elements, and in philosophy and linguistics - for the formation of subsections of the same term, mutually exclusive of each other.
The dichotomy method must be distinguished from ordinary division. For example, the word “man” can be divided into the concepts of “men” and “women”, and can be divided into “men” and “not men”. So, in the first case, the two concepts do not contradict each other, so there is no dichotomy here. In the second case, “man” and “not man” are two definitions that contradict each other and do not intersect, and this is the definition of a dichotomy.
The dichotomy method is attractive because of its simplicity, since there are always only two classes that are exhausted by the volume of the divisible concept. In other words, proportionality is always present in dichotomous division. The next main property is the exclusion of each other by the members of the division due to the fact that each dividend set can fall into only one of the classes “b” or “not b”, and division is carried out only on one basis, connected with the presence or absence of a certain characteristic.
For all its merits, the dichotomy method also has a drawback consisting in the uncertainty of that part of it that has a “not” particle. For example, if all scientists are divided into mathematicians and not mathematicians, then there is a certain ambiguity regarding the second group. In addition to this drawback, there is another one, which consists in the difficult establishment of a concept that contradicts the first value, according to the degree of distance from the first pair.
As already mentioned above, a dichotomy is often used as an auxiliary technique in the classification of any concepts. The dichotomy method is actively used to find the values of functions determined by a certain criterion (for example, comparison to the maximum or minimum).
Quite often, the dichotomy method is used unconsciously, the algorithm of which can be described literally step by step. For example, in the game “Guess the Number” one of the players makes a number in the range from 1 to 100, and the other makes attempts to guess it based on the “less” or “more” of the first. If you think logically, the first number is always called 50, and in the case of a smaller one - 25, the larger - 75. Therefore, at each stage the uncertainty of the hidden number is halved, and even an unlucky person will guess this unknown in about 7 attempts.
When using the dichotomy method in solving various equations, finding the right solution is possible only when it is reliably known to find a single root on a given interval. This does not mean at all that the use of this method is possible to find the roots of only linear equations. When solving equations of a higher order using the half division method, it is first necessary to separate the roots into segments. Moreover, the process of their separation is carried out by finding the first and second derivatives of the function and equating the resulting equations to zero (f '(x) = 0, f' '(x) = 0). The next step is to determine the values of f (x) at the boundary and critical points. The result of all the calculations is the interval | a, b |, on which the sign of the function value changes and where f (a) * f (b) <0.
When considering a graphical method for solving an equation using a dichotomy, the solution algorithm is quite simple. For example, there exists a segment | a, b | within which there is one root x.
The first step is to calculate the algebraic mean x = (a + b) / 2. Next, the value of the function at a given point is calculated. If f (x) <0, then [a, x], otherwise - [x, b]. Thus, the narrowing of the interval is carried out, as a result of which a certain sequence x is formed. The calculation stops when the difference ba reaches a smaller error.