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For example, measuring the length of a table using a measuring tape involves comparing the table to the markings on the tape. This is conceptually equivalent to dividing the length of the table by a unit of length, the distance between markings.
For example, density (mass divided by volume, in units of kg/m 3) is said to be a "quotient", whereas mass fraction (mass divided by mass, in kg/kg or in percent) is a "ratio". [8] Specific quantities are intensive quantities resulting from the quotient of a physical quantity by mass, volume, or other measures of the system "size". [3]
In the example, 20 is the dividend, 5 is the divisor, and 4 is the quotient. Unlike the other basic operations, when dividing natural numbers there is sometimes a remainder that will not go evenly into the dividend; for example, 10 / 3 leaves a remainder of 1, as 10 is not a multiple of 3.
In Spain, Italy, France, Portugal, Lithuania, Romania, Turkey, Greece, Belgium, Belarus, Ukraine, and Russia, the divisor is to the right of the dividend, and separated by a vertical bar. The division also occurs in the column, but the quotient (result) is written below the divider, and separated by the horizontal line.
Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.
Its existence is based on the following theorem: Given two univariate polynomials a(x) and b(x) (where b(x) is a non-zero polynomial) defined over a field (in particular, the reals or complex numbers), there exist two polynomials q(x) (the quotient) and r(x) (the remainder) which satisfy: [7]
In this example, we see that 30 divided by 4 is 7 with a remainder of 2. The number written above the bar (237) is the quotient, and the last small digit (2) is the remainder. ) ¯ The answer in this example is 237 with a remainder of 2.
In the above theorem, each of the four integers has a name of its own: a is called the dividend, b is called the divisor, q is called the quotient and r is called the remainder. The computation of the quotient and the remainder from the dividend and the divisor is called division, or in case of ambiguity, Euclidean division.