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<h1>Calculadora de fracciones continuas</h1>
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<label for="num">a</label><input type="text" id="num" value="" class="input" aria-label="numerador"/>
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<input type="button" id="calc" value="Calcular fracción continua" />
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Cualquier número real <var>x</var> se puede representar de manera única mediante la fracción continua:
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	<span class="offscr">"x igual a a sub 0 más 1 sobre a sub 1 más 1 sobre a sub 2 más 1 sobre a sub 3 más etcétera</span>
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<use x='184.934' xlink:href='#dot2' y='115.746'/>
<use x='189.531' xlink:href='#dot2' y='118.746'/>
<use x='194.129' xlink:href='#dot2' y='121.746'/>
</g>
</svg>
</p>
<p>donde <var>a</var><sub>1</sub>, <var>a</var><sub>2</sub>, <var>a</var><sub>3</sub>, ... son números enteros mayores que cero. Una representación más compacta es:</p>
<p class="mysvg">
<span class="offscr">x es igual a a sub 0 más doble barra a sub 1, a sub 2, a sub 3, etcétera doble barra</span>
<svg height='13.5093pt' version='1.1' viewBox='0 0 153.013 13.5093' width='153.013pt' xmlns='http://www.w3.org/2000/svg' xmlns:xlink='http://www.w3.org/1999/xlink'>
<defs>
<path d='M2.34 0.048C2.34 -0.648 2.112 -1.164 1.62 -1.164C1.236 -1.164 1.044 -0.852 1.044 -0.588S1.224 0 1.632 0C1.788 0 1.92 -0.048 2.028 -0.156C2.052 -0.18 2.064 -0.18 2.076 -0.18C2.1 -0.18 2.1 -0.012 2.1 0.048C2.1 0.444 2.028 1.224 1.332 2.004C1.2 2.148 1.2 2.172 1.2 2.196C1.2 2.256 1.26 2.316 1.32 2.316C1.416 2.316 2.34 1.428 2.34 0.048Z' id='comma'/>
<path d='M5.148 -8.556C5.148 -8.568 5.22 -8.748 5.22 -8.772C5.22 -8.916 5.1 -9 5.004 -9C4.944 -9 4.836 -9 4.74 -8.736L0.72 2.556C0.72 2.568 0.648 2.748 0.648 2.772C0.648 2.916 0.768 3 0.864 3C0.936 3 1.044 2.988 1.128 2.736L5.148 -8.556Z' id='over'/>
<path d='M2.304 -3C2.304 -3.348 2.016 -3.636 1.668 -3.636S1.032 -3.348 1.032 -3S1.32 -2.364 1.668 -2.364S2.304 -2.652 2.304 -3Z' id='dot'/>
</defs>
<g id='page3' transform='matrix(1.12578 0 0 1.12578 -63.986 -64.41)'>
<use x='56.6248' xlink:href='#let-x' y='66'/>
<use x='66.6351' xlink:href='#equal' y='66'/>
<use x='79.1072' xlink:href='#ltr-a' y='66'/>
<use x='85.2752' xlink:href='#sub-0' y='67.8'/>
<use x='92.6919' xlink:href='#plus' y='66'/>
<use x='104.497' xlink:href='#over' y='66'/>
<use x='110.372' xlink:href='#over' y='66'/>
<use x='116.247' xlink:href='#ltr-a' y='66'/>
<use x='122.415' xlink:href='#sub-1' y='67.8'/>
<use x='127.165' xlink:href='#comma' y='66'/>
<use x='132.429' xlink:href='#ltr-a' y='66'/>
<use x='138.597' xlink:href='#sub-2' y='67.8'/>
<use x='143.347' xlink:href='#comma' y='66'/>
<use x='148.611' xlink:href='#ltr-a' y='66'/>
<use x='154.779' xlink:href='#sub-3' y='67.8'/>
<use x='159.529' xlink:href='#comma' y='66'/>
<use x='164.793' xlink:href='#dot' y='66'/>
<use x='170.126' xlink:href='#dot' y='66'/>
<use x='175.459' xlink:href='#dot' y='66'/>
<use x='180.793' xlink:href='#over' y='66'/>
<use x='186.668' xlink:href='#over' y='66'/>
</g>
</svg>
</p>
<p>
Si el número a representar es racional, hay una cantidad finita de términos en la fracción continua. Si el número es una irracionalidad cuadrática de la forma <span class="offscr">fracción donde el numerador es a más la raíz cuadrada de b y el denominador es c</span>
<svg height='16.0902pt' version='1.1' viewBox='0 0 62.7131 16.0902' width='62.7131pt' xmlns='http://www.w3.org/2000/svg' xmlns:xlink='http://www.w3.org/1999/xlink'>
<defs>
<path d='M3.9 2.916C3.9 2.88 3.9 2.856 3.696 2.652C2.496 1.44 1.824 -0.54 1.824 -2.988C1.824 -5.316 2.388 -7.32 3.78 -8.736C3.9 -8.844 3.9 -8.868 3.9 -8.904C3.9 -8.976 3.84 -9 3.792 -9C3.636 -9 2.652 -8.136 2.064 -6.96C1.452 -5.748 1.176 -4.464 1.176 -2.988C1.176 -1.92 1.344 -0.492 1.968 0.792C2.676 2.232 3.66 3.012 3.792 3.012C3.84 3.012 3.9 2.988 3.9 2.916Z' id='lpar'/>
<path d='M3.384 -2.988C3.384 -3.9 3.264 -5.388 2.592 -6.78C1.884 -8.22 0.9 -9 0.768 -9C0.72 -9 0.66 -8.976 0.66 -8.904C0.66 -8.868 0.66 -8.844 0.864 -8.64C2.064 -7.428 2.736 -5.448 2.736 -3C2.736 -0.672 2.172 1.332 0.78 2.748C0.66 2.856 0.66 2.88 0.66 2.916C0.66 2.988 0.72 3.012 0.768 3.012C0.924 3.012 1.908 2.148 2.496 0.972C3.108 -0.252 3.384 -1.548 3.384 -2.988Z' id='rpar'/>
</defs>
<g id='page4' transform='matrix(1.12578 0 0 1.12578 -63.986 -61.8195)'>
<use x='56.6248' xlink:href='#lpar' y='66'/>
<use x='61.1942' xlink:href='#ltr-a' y='66'/>
<use x='70.0288' xlink:href='#plus' y='66'/>
<use x='81.8342' xlink:href='#sqrt' y='55.1875'/>
<rect height='0.47998' width='4.99576' x='91.8343' y='54.7075'/>
<use x='91.8343' xlink:href='#ltr-b' y='66'/>
<use x='96.83' xlink:href='#rpar' y='66'/>
<use x='101.399' xlink:href='#over' y='66'/>
<use x='107.274' xlink:href='#ltr-c' y='66'/>
</g>
</svg>,
entonces la fracción continua es periódica. Esta calculadora puede encontrar el desarrollo en fracciones continuas de números racionales e irracionalidades cuadráticas.
</p>
<p>Puede escribir números o expresiones numéricas en las cajas de entrada.</p>
<p>La calculadora acepta números de hasta 10000 dígitos.</p>
<p>
Si necesita que la raíz cuadrada reste al número de la izquierda, simplemente cambie el signo de <var>a</var> y <var>c</var>.
</p>
<p>
Si <var>b</var> es negativo, el resultado no es un número real, así que no se podrá representar como fracción continua.
</p>
<p>La calculadora puede hallar todos los convergentes para números racionales. En el caso de irracionalidades cuadráticas, la calculadora se detiene después de hallar el convergente número 100000 si el período es más largo.</p>
<h2>Expresiones</h2>
<p>Se pueden entrar expresiones que usen los siguientes operadores y paréntesis:</p>
<ul>
<li> + para suma
<li> - para resta
<li> * para multiplicación
<li> / para división entera
<li> % para el resto de la división entera
<li> ^ o ** para exponenciación (el exponente debe ser mayor o igual que cero).
<li> <strong>&lt;</strong>, <strong>==</strong>, <strong>&gt;</strong>; <strong>&lt;=</strong>, <strong>&gt;=</strong>, != para comparaciones. Los operadores devuelven cero si es falso y -1 si es verdadero.
<li> <strong>AND</strong>, <strong>OR</strong>, <strong>NOT</strong> para lógica binaria.
<li> <strong>n!</strong>: factorial (<var>n</var> debe ser mayor o igual que cero).
<li> <strong>p#</strong>: primorial (producto de todos los primos menores o iguales a <var>p</var>).
<li> <strong>B(n)</strong>: Número probablemente primo anterior a <var>n</var></li>
<li> <strong>F(n)</strong>: Número de Fibonacci F<sub>n</sub>
<li> <strong>L(n)</strong>: Número de Lucas L<sub>n</sub> = F<sub><var>n</var>-1</sub> + F<sub><var>n</var>+1</sub>
<li> <strong>N(n)</strong>: Número probablemente primo posterior a <var>n</var></li>
<li> <strong>P(n)</strong>: particiones irrestrictas (cantidad de descomposiciones de <var>n</var> en sumas de números enteros sin tener en cuenta el orden).
<li> <strong>Gcd(m,n)</strong>: Máximo común divisor de estos dos números enteros.
<li> <strong>Modinv(m,n)</strong>: inverso de <var>m</var> modulo <var>n</var>, sólo válido cuando gcd(m,n)=1.
<li> <strong>Modpow(m,n,r)</strong>: halla <var>m</var><sup><var>n</var></sup> módulo <var>r</var>.
<li> <strong>IsPrime(n)</strong>: returna cero si <var>n</var> no es un primo probable y -1 si lo es.
<li> <strong>NumDigits(n,r)</strong>: cantidad de dígitos de <var>n</var> en base <var>r</var>.
<li> <strong>SumDigits(n,r)</strong>: suma de dígitos de <var>n</var> en base <var>r</var>.
<li> <strong>RevDigits(n,r)</strong>: halla el valor que se obtiene escribiendo para atrás los dígitos de <var>n</var> en base <var>r</var>.
</ul>
<p>Puedes usar el prefijo <em>0x</em> para números hexadecimales, por ejemplo 0x38 es igual a 56.</p>
<h2>Código fuente</h2>
<p>
Se puede bajar el código fuente de este programa y el del viejo applet de fracciones continuas desde <a href="https://github.com/alpertron/calculators">GitHub</a>. El código fuente está escrito en lenguaje C, por lo que es necesario <a href="https://kripken.github.io/emscripten-site/docs/getting_started/downloads.html">Emscripten</a> para generar Javascript.
</p>
<p>Escrito por Dario Alpern. Actualizado el 21 de enero de 2018.</p>
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Si encuentra algún error o tiene algún comentario, por favor llene el <A HREF="FORMULAR.HTM?Calculadora+de+fracciones+continuas">formulario</A>.
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