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division, in mathematics
a form of reproduction of organisms and cells that are part of multicellular organisms.
In bacteria, division takes place by the formation of a transverse septum, which is preceded by replication of the nucleoid DNA strand. In unicellular algae and animals possessing a typical cell nucleus, division is also asexual reproduction. Division can take place both in active and in resting (encysted) states. Besides division into two, in protozoans the cytoplasm frequently separates into numerous uninuclear cells immediately after several successive divisions of the nucleus. This process is called schizogony. Division occurs in unicellular organisms (with rare exceptions, for example, in infusorians) as mitosis. It is preceded by the replication of DNA and doubling of the chromosomes. In multicellular organisms, cell division is the basis of individual development, or ontogeny, and sexual reproduction. Multicellular plants and animals are characterized by various secondary forms of reproduction, which are accomplished by the maternal organism dividing into parts of the same or different sizes (budding). Reproduction by division or budding is invariably accompanied by the regeneration of missing parts of the body. Among multicellular animals, reproduction by division occurs in certain ciliated worms.
IU. I. POLIANSKII
the inverse operation to multiplication; it is the process of finding one of two factors when their product and the other factor are given. Thus, to divide a by b means to find x such that bx = a or xb = a. The result of dividing by x is called the quotient of a and b. The given product a is called the dividend and the given factor b, the divisor. Division is denoted by a colon (a:b) or a horizontal (sometimes slanted) line a/b or a/b).
Division is not always possible in the system of numbers consisting of the integers (6 is divisible by 2 and 3 but not by 5), but in those cases where it is, the result is always uniquely determined. In the system of all rational numbers (that is, the integers and fractions) division is not only unique but is always realizable with one exception—division by zero. On the basis of the definition of division given above, it is apparent that it is not possible to divide a number different from zero by zero. The result of dividing zero by zero, according to the definition, can be any number since c . 0 = 0 in all cases. It is usually preferable in algebra (in order not to violate the uniqueness of division) to consider division by zero to be impossible for all cases.
Division with remainder differs from the exact division that has been discussed up to now. This is in essence a very special operation, which differs from the division in the sense defined above. If a and b are non-negative integers, then the operation of division with remainder of the number a by the number b consists in determining the non-negative integers x and y that satisfy the requirements (1) a = xb + y and (2) y < b. Here, a is called the dividend, b the divisor, x the quotient, and y the remainder. This operation is always realizable and unique. If y = 0, it is then said that a is divisible by b without a remainder. The operation of division with remainder is defined similarly for polynomials of the form
P(x) = a0xn + a1xn-1 + .... + an
It consists in finding for the two polynomials P(x) and Q(x) two other polynomials S(x) and R(x) that satisfy the requirements (1) P(x) = S(x)Q(x) + R(x) and (2) the degree of R(x) is less than the degree of Q(x). This operation is also always realizable and unique. lf R(x) =0, then P(x) is divisible by Q(x) without a remainder.
REFERENCESDepman, I. la. Istoriia arifmetiki, 2nd ed. Moscow, 1965.
Kurosh, A. G. Kurs vysshchei algebry, 9th ed. Moscow, 1968.
a large tactical unit in the ground forces, the air force, and the navy of various states. Among others there are infantry (rifle, motorized rifle, motorized, and motorized infantry) divisions, mechanized divisions, cavalry divisions, artillery divisions, antiaircraft divisions, tank (armored) divisions, air divisions, airborne divisions, air mobile divisions, and air defense divisions. The organization of troops into divisions appeared in Russia and France in the early 18th century and was firmly adopted by the armies of most states in the course of the 19th century.
Before World War I (1914-18) an infantry division usually had four infantry regiments, one or two cavalry squadrons, from 36 to 72 division artillery guns, and a total strength of 15,000-16,000 men. During the war the infantry division became a large combined arms unit with infantry, cavalry, artillery, engineer, and signal communication units. In the 1930’s tank (armored) divisions and air divisions were created in the armed forces of a number of states (the USSR, the USA, Great Britain, and Germany). In some armies (for example, the French Army) tanks were even included in infantry divisions. In most armies an infantry (rifle) division had three infantry (rifle) regiments.
In the USSR the tables of organization of the division were repeatedly changed during the Great Patriotic War (1941-45), its organization was refined because of new combat materiel and weaponry, its maneuverability and fire power increased, and its management was improved. According to the tables of organization of 1943-44 the total strength of a rifle division was to be 9,435 men and of a guards rifle division, 10,670 men. But the actual strength was as a rule below the figure given in the tables of organization. In the postwar period, when the motorization of the ground troops was completed, rifle divisions in the USSR began to be called motorized rifle divisions, and cavalry divisions were abolished. The present-day division in the armed forces of various states consists organizationally of regiments, brigades, or brigade groups. It includes units (subunits) of various combat arms and special troops, as well as various services. The strength and composition of divisions vary. For example, the mechanized division in the USA has over 18,000 men, about 190 tanks, about 2,800 motor vehicles, 850 armored personnel carriers, 57 helicopters, and 234 units of artillery, mortar, and missile weaponry. In the armed forces of many states an air division is composed of several regiments of one or various arms of aviation.
M. G. ZHDANOV
(1) In plant taxonomy, the highest conventionally used taxonomic category in the plant kingdom.
(2) In animal taxonomy, a taxonomic category sometimes used in constructing a system of higher taxons—phyla. It is not included among the taxons accepted by the International Code of Zoological Nomenclature.
(3) In zoogeography, the term “division” designates the Nearctic (New World) and Palearctic (Old World) parts of the Holarctic zoogeographic region.
(4) In anatomy and morphology, the term “division” is used to designate regions of the body or of its parts (anterior, or cranial, division; posterior, or caudal, division; trunk division; cervical division).