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When the numerator and the denominator are both positive, the fraction is called proper if the numerator is less than the denominator, and improper otherwise. [11] The concept of an improper fraction is a late development, with the terminology deriving from the fact that fraction means piece , so a proper fraction must be less than 1. [ 10 ]
If the numerator y is of size m and the denominator x of size n, then the result can be laid out as either an m×n matrix or n×m matrix, i.e. the m elements of y laid out in rows and the n elements of x laid out in columns, or vice versa. This leads to the following possibilities: Numerator layout, i.e. lay out according to y and x T (i.e ...
If the odd denominator d of a rational is not a multiple of 3, then all the iterates have the same denominator and the sequence of numerators can be obtained by applying the "3n + d " generalization [26] of the Collatz function = {(), + ().
For example, the numerators of fractions with common denominators can simply be added, such that + = and that <, since each fraction has the common denominator 12. Without computing a common denominator, it is not obvious as to what 5 12 + 11 18 {\displaystyle {\frac {5}{12}}+{\frac {11}{18}}} equals, or whether 5 12 {\displaystyle {\frac {5 ...
If the divisor has a fractional part, one can restate the problem by moving the decimal to the right in both numbers until the divisor has no fraction, which can make the problem easier to solve (e.g., 10/2.5 = 100/25 = 4). Division can be calculated with an abacus. [14]
The number of Farey fractions with denominators equal to k in F n is given by φ(k) when k ≤ n and zero otherwise. Concerning the numerators one can define the function () that returns the number of Farey fractions with numerators equal to h in F n. This function has some interesting properties as [7]
The numerator and denominator are called the terms of the algebraic fraction. A complex fraction is a fraction whose numerator or denominator, or both, contains a fraction. A simple fraction contains no fraction either in its numerator or its denominator. A fraction is in lowest terms if the only factor common to the numerator and the ...
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. [1]