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Related to Division: long division


fundamental operation in arithmetic; the inverse of multiplication. Division may be indicated by the symbol ÷, as in 15 ÷ 3, or simply by a fraction, 15/3. The number that is being divided, e.g. 15, is called the dividend and the number dividing into it, e.g. 3, the divisor. The result of division is called the quotient. If the dividend is an exact (integral) multiple of the divisor, then the division will be exact, the quotient being the factor by which the divisor must be multiplied to yield the dividend (in the above example the quotient 5 multiplied by the divisor 3 equals the dividend 15). If the dividend is not an exact multiple of the divisor there will be a remainder expressed as a fraction with the divisor as the denominator; e.g., 16-3 = 5 1-3, where 1-3 is the remainder. A division in which the divisor b is larger than the dividend a is simply indicated by the fraction a/b, with no actual operation being carried out. In terms of multiplication either of the symbols 1/b or b−1 is called the multiplicative inverse of b with the property that the product of a number and its inverse equals 1, or b · b−1 =1. The division of a by b is equivalent to the multiplication of a by the multiplicative inverse of b, i.e., a ÷ b = a · (1/b) = a · b−1; for example, when a = 25 and b = 5, then 1/b = 1/5 and 25 ÷ 5 = 25 · (1/5) = 5.


in taxonomy: see classificationclassification,
in biology, the systematic categorization of organisms into a coherent scheme. The original purpose of biological classification, or systematics, was to organize the vast number of known plants and animals into categories that could be named, remembered, and
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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.




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.


Depman, 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.




(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).


(computer science)
One of four required parts of a COBOL program, labeled identification, environment, data, and procedure, each with a set of rules governing the contents.
The inverse operation of multiplication; the number a divided by the number b is the number c such that b multiplied by c is equal to a.


One of the sixteen basic organizational subdivisions used in the AIA uniform system for construction specifications, data filing, and cost accounting. See illustration for contract documents.


1. a part of a government, business, country, etc., that has been made into a unit for administrative, political, or other reasons
2. a formal vote in Parliament or a similar legislative body
3. a mathematical operation, the inverse of multiplication, in which the quotient of two numbers or quantities is calculated. Usually written: a ? b, a/b
4. Biology (in traditional classification systems) a major category of the plant kingdom that contains one or more related classes
5. Horticulture any type of propagation in plants in which a new plant grows from a separated part of the original
6. Logic the fallacy of inferring that the properties of the whole are also true of the parts, as Britain is in debt, so John Smith is in debt
7. (esp in 17th-century English music) the art of breaking up a melody into quick phrases, esp over a ground bass
References in classic literature ?
Michael had inherited his father's temper, unredeemed by his father's better qualities: his second letter reiterated the charges contained in the first, and declared that he would only accept the offered division as an act of atonement and restitution on Andrew's part.
Raised in unrivalled prosperity, we inherit an economy that is still the world's strongest, but is weakened by business failures, stagnant wages, increasing inequality, and deep divisions among OUR OWN people.
It was remarked in the preceding paper, that weakness and divisions at home would invite dangers from abroad; and that nothing would tend more to secure us from them than union, strength, and good government within ourselves.
The guild-masters were pushed on one side by the manufacturing middle class; division of labour between the different corporate guilds vanished in the face of division of labour in each single workshop.
If you march fifty LI in order to outmaneuver the enemy, you will lose the leader of your first division, and only half your force will reach the goal.
The general, whom the boys knew as the com- mander of their division, looked at the other officer and spoke coolly, as if he were criticising his clothes.
I mean those species which are distinguished each from each by one and the same method of division.
A rustling sound of uprising greeted our entrance; without looking to the right or left, I walked straight up the lane between two sets of benches and desks, and took possession of the empty chair and isolated desk raised on an estrade, of one step high, so as to command one division; the other division being under the surveillance of a maitresse similarly elevated.
This proposition he assented to; so I took the silk kerchief from my neck, and cutting it with my knife into half a dozen equal pieces, proceeded to make an exact division.
The causes of atheism are: divisions in religion, if they be many; for any one main division, addeth zeal to both sides; but many divisions introduce atheism.
But the same person cannot at once be poor and rich: for which reason the most obvious division of the city is into two parts, the poor and rich; moreover, since for the generality the one are few, the other many, they seem of all the parts of a city most contrary to each other; so that as the one or the other prevail they form different states; and these are the democracy and the oligarchy.
There we shall encounter difficulties, and there," thought he, "I shall be sent with a brigade or division, and there, standard in hand, I shall go forward and break whatever is in front of me.