fractionFraction2002-04-06 12:06:102004-12-12 22:42:20Definitionsolidusproper fractionnumeratordenominatorimproper fractionlowest terms% this is the default PlanetMath preamble. as your knowledge
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% define commands hereA \emph{fraction} is a rational number expressed in the form $\frac{n}{d}$ or $n/d$, where $n$ is designated the \emph{numerator} and $d$ the \emph{denominator}. The slash between them is known as a \emph{solidus} when the fraction is expressed as $n/d$.
The fraction $n/d$ has value $n \div d$. For instance, $3/2 = 3 \div 2 = 1.5$.
If $n$ and $d$ are positive, and $n/d < 1$, then $n/d$ is known as a \emph{proper fraction}. Otherwise, it is an \emph{improper fraction}. If $n$ and $d$ are relatively prime, then $n/d$ is said to be in \emph{lowest terms}. Each rational number can be expressed uniquely as a fraction in lowest terms. To get a fraction in lowest terms, simply divide the numerator and the denominator by their greatest common divisor:
$$\frac{60}{84} = \frac{60 \div 12}{84 \div 12} = \frac{5}{7}.$$
The rules for manipulating fractions are
\begin{eqnarray*}
\frac{a}{b} & \qquad = \qquad & \frac{ka}{kb}\\
\frac{a}{b} + \frac{c}{d} & \qquad = & \frac{ad + bc}{bd}\\
\frac{a}{b} - \frac{c}{d} & \qquad = & \frac{ad - bc}{bd}\\
\frac{a}{b} \times \frac{c}{d} & \qquad = & \frac{ac}{bd}\\
\frac{a}{b} \div \frac{c}{d} & \qquad = & \frac{ad}{bc}.
\end{eqnarray*}