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Division is also not, in general, associative, meaning that when dividing multiple times, the order of division can change the result. [7] For example, (24 / 6) / 2 = 2 , but 24 / (6 / 2) = 8 (where the use of parentheses indicates that the operations inside parentheses are performed before the operations outside parentheses).
In a fraction, the number of equal parts being described is the numerator (from Latin: numerātor, "counter" or "numberer"), and the type or variety of the parts is the denominator (from Latin: dēnōminātor, "thing that names or designates"). [2] [3] As an example, the fraction 8 / 5 amounts to eight parts, each of which is of the ...
The lowest common denominator of a set of fractions is the lowest number that is a multiple of all the denominators: their lowest common multiple.The product of the denominators is always a common denominator, as in:
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.
Fractions are written as two integers, the numerator and the denominator, with a dividing bar between them. The fraction m / n represents m parts of a whole divided into n equal parts. Two different fractions may correspond to the same rational number; for example 1 / 2 and 2 / 4 are equal, that is:
A common technique used for division is called long division. Other methods include short division and chunking. [72] Integer arithmetic is not closed under division. This means that when dividing one integer by another integer, the result is not always an integer. For instance, 7 divided by 2 is not a whole number but 3.5. [73]
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.
An equivalent definition is sometimes useful: if a and b are integers, then the fraction a / b is irreducible if and only if there is no other equal fraction c / d such that | c | < | a | or | d | < | b |, where | a | means the absolute value of a. [4] (Two fractions a / b and c / d are equal or equivalent if and ...