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# Fractions

(Redirected from Ratio)

By fraction we mean a rational number, that is the ratio (or division) of one integer (the numerator) by another integer (the denominator), generally represented vertically as $\scriptstyle \frac{n}{d} \,$ or horizontally as $\scriptstyle n/d \,$.

If the numerator and the denominator are coprime, then the fraction is said to be in reduced form, i.e. written in lowest terms. For example, $\scriptstyle \frac{12}{16} \,=\, \frac{3}{4} \,$. If $\scriptstyle n \,>\, d \,>\, 0 \,$, the fraction is greater than 1, in which case it might be rewritten as $\scriptstyle \lfloor \frac{n}{d} \rfloor + \frac{(\frac{n}{d} - \lfloor \frac{n}{d} \rfloor)d}{d} \,$. For example, $\scriptstyle \frac{4}{3}\,=\,1 + \frac{1}{3} \,$, though generally the addition sign becomes a tacit operator.

One important sequence of fractions is the sequence of even-index Bernoulli numbers

$\Bigg\{1, \frac{1}{6}, \frac{-1}{30}, \frac{1}{42}, \frac{-1}{30}, \frac{5}{66}, \frac{-691}{2730}, \frac{7}{6}, \cdots\Bigg\} \,$

In the OEIS, sequences of fractions[1] are generally rendered as two sequences of integers, one for the numerator, the other for the denominator. Both sequences then have Keyword frac and cross-references to each other. In the Bernoulli example, A000367 gives the numerators and A002445 gives the denominators.

However, for sequences of unit fractions, we could say that such and such sequence has the denominators and A000012 has the numerators for all of them.

## Reduced form

Given a fraction $\scriptstyle \frac{N}{D} \,$, one obtains the reduced form (fraction in lowest terms) $\scriptstyle \frac{n}{d} \,=\, \frac{N}{D} \,$ by respectively dividing the numerator $\scriptstyle N \,$ and the denominator $\scriptstyle D \,$ by the gcd of $\scriptstyle N \,$ and $\scriptstyle D \,$

$n = \frac{N}{\gcd(N, D)},\quad d = \frac{D}{\gcd(N, D)}. \,$

## Proper, improper and mixed fractions

Fractions are either

• Proper fractions: fractions $\scriptstyle \frac{N}{D} \,$ s.t. $\scriptstyle N \,<\, D \,$;
• Improper fractions: fractions $\scriptstyle \frac{N}{D} \,$ s.t. $\scriptstyle N \,\ge\, D \,$;
• Mixed fractions: fractions $\scriptstyle I + \frac{N}{D} \,$ s.t. $\scriptstyle I \,\in\, \Z \,$ and $\scriptstyle N \,<\, D \,$.