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 = 51/3, 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.
division, in taxonomy: see classification.

(German panzer, “armoured”) Self-contained military unit of the German army, built around the capabilities of armoured vehicles. In World War II, it consisted of a tank brigade with four battalions; a motorized infantry brigade with four rifle battalions; an artillery regiment; and reconnaissance, antitank, and military-engineering battalions and service units. Germany had six panzer divisions in 1939 and 20 by 1941. It remains the principal offensive element of the German army.

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Specialization in the production process. Complex jobs can usually be less expensively completed by a large number of people each performing a small number of specialized tasks than by one person attempting to complete the entire job. The idea that specialization reduces costs, and thereby the price the consumer pays, is embedded in the principle of comparative advantage. Division of labour is the basic principle underlying the assembly line in mass production systems. See Émile Durkheim.

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