tag:blogger.com,1999:blog-17849938123650165552024-03-05T14:25:00.057-08:00banco didactico matematicosolanohttp://www.blogger.com/profile/02741968945810651934noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-1784993812365016555.post-90780477537371967082010-05-06T12:30:00.000-07:002010-05-06T13:45:20.391-07:00HISTORIA DE LOS NÚMEROS RACIONALES.apuntes
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<br /><o:p></o:p></span></h1> <p class="MsoNormal"><span style=""><o:p> </o:p></span></p> <p class="MsoNormal"><b style=""><span style="line-height: 115%;font-size:14pt;" >¿Por qué usaban los egipcios solo fracciones unitarias en sus cálculos? </span></b></p> <p class="MsoNormal"><span style="line-height: 115%;font-size:14pt;" ><span style=""> </span>Salvo la excepcionalidad constituida por el 2/3 y la más tardía del 3/4, los escribas egipcios sólo utilizaro</span><span style="line-height: 115%;font-size:14pt;" >n en sus cálculos fracciones de la forma 1/n</span><span style="line-height: 115%;font-size:14pt;" >.<span style=""> </span>Se han dado diferentes teorías para justificar el uso de este tipo de fracciones en Egipto. <span style="">Para exp</span></span><span style="line-height: 115%;font-size:14pt;" ><span style="">licar por qué hay que remitirse al origen funcional de las fracciones, es decir, los contextos y situaciones en </span></span><span style="line-height: 115%;font-size:14pt;" ><span style="">que se inscribe su uso.</span><o:p></o:p></span></p> <p class="MsoNormal"><span style="line-height: 115%;font-size:14pt;" ><span style=""> </span>En matemática moderna se emplea el uso de </span><span style="line-height: 115%;font-size:14pt;" >fracciones unitarias en determinadas situaciones, y el argumento más convincente para el empleo por parte de los egipcios es la facili</span><span style="line-height: 115%;font-size:14pt;" >dad de dividir un todo en n partes.<o:p></o:p></span></p> <p class="MsoNormal"><span style="line-height: 115%;font-size:14pt;" ><span style=""> </span>Básicamente, la fracción surge en un contexto de </span><span style="line-height: 115%;font-size:14pt;" >medida y en otro de reparto. Supóngase un ejercicio sencillo como </span><b><span style="line-height: 115%;font-size:16pt;" >dividir dos panes entre ocho personas</span></b><span style="line-height: 115%;font-size:14pt;" >. </span><span style="line-height: 115%;font-size:14pt;" >Para hacerlo, basta dividir cada uno en cuatro partes (1/4). Más sencillo de </span><span style="line-height: 115%;font-size:14pt;" >efectuar en la práctica sería el dividir cada pan en dos partes iguales y cada una de estas partes </span><span style="line-height: 115%;font-size:14pt;" >en otras dos (1/2 de 1/2 es igual a 1/4).<o:p></o:p></span></p> <p class="MsoNormal"><span style="line-height: 115%;font-size:14pt;" ><span style=""> </span>La acción de reparto es particularmente sencilla por este procedimiento de </span><span style="line-height: 115%;font-size:14pt;" >divisiones sucesivas por la mitad, lo que es el motivo de que las fracciones de Horus (1/2, 1/4, 1/8, 1/16, 1/3</span><span style="line-height: 115%;font-size:14pt;" >2, 1/64) hayan sido de uso tan frecuente.</span><span style="line-height: 115%;font-size:14pt;" ><o:p></o:p></span></p> <p class="MsoNormal"><span style="line-height: 115%;font-size:14pt;" ><span style=""> </span>La cuestión se complica si el número de personas entre las que hay que repartir l</span><span style="line-height: 115%;font-size:14pt;" >os dos panes es distinto de una potencia de dos. </span><b><span style="line-height: 115%;font-size:16pt;" >Dividir dos panes entre seis </span></b><b><span style="line-height: 115%;font-size:16pt;" >personas</span></b><span style="line-height: 115%;font-size:14pt;" >, por ejemplo, supondría partir cada pan en dos partes y cada una de ellas en tres partes iguales (1/3 de 1/2 es igual a 1/6). <o:p></o:p></span></p> <p class="MsoNormal"><span style="line-height: 115%;font-size:14pt;" ><span style=""> </span>Pero ¿qué sucede cuando el </span><span style="line-height: 115%;font-size:14pt;" >número de personas es impar?. Por ejemplo</span><span style="line-height: 115%;font-size:16pt;" >, </span><b><span style="line-height: 115%;font-size:16pt;" >dividir dos panes entre cinco personas</span></b><span style="line-height: 115%;font-size:14pt;" > </span><!--[if gte vml 1]><v:shapetype id="_x0000_t75" coordsize="21600,21600" spt="75" preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f"> <v:stroke joinstyle="miter"> <v:formulas> <v:f eqn="if lineDrawn pixelLineWidth 0"> <v:f eqn="sum @0 1 0"> <v:f eqn="sum 0 0 @1"> <v:f eqn="prod @2 1 2"> <v:f eqn="prod @3 21600 pixelWidth"> <v:f eqn="prod @3 21600 pixelHeight"> <v:f eqn="sum @0 0 1"> <v:f eqn="prod @6 1 2"> <v:f eqn="prod @7 21600 pixelWidth"> <v:f eqn="sum @8 21600 0"> <v:f eqn="prod @7 21600 pixelHeight"> <v:f eqn="sum @10 21600 0"> </v:formulas> <v:path extrusionok="f" gradientshapeok="t" connecttype="rect"> <o:lock ext="edit" aspectratio="t"> </v:shapetype><v:shape id="Imagen_x0020_4" spid="_x0000_s1026" type="#_x0000_t75" alt="http://personal.us.es/cmaza/egipto/recto2.JPG" style="'position:absolute;" allowoverlap="f"> <v:imagedata src="file:///C:\Windows\Temp\msohtmlclip1\01\clip_image001.jpg" title="recto2"> <w:wrap type="square" anchory="line"> </v:shape><![endif]--><!--[if !vml]--><!--[endif]--><span style="line-height: 115%;font-size:14pt;" >
<br />En este caso, se puede dividir </span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjh8r2jwkiBHV4pfzsDSNPa87avOzRwSyi9uOgjyJIm0aOcUgiqYQ5EBEFBleIeORDEx8j_Bot4iCL1eQqOauM7d_XBclLZh17rY-FLNeaLUTQFADRtaXJy_Dn1I5L4T4uDgOgG4RscGiI/s1600/fracciones+egipcias1.png"><img style="margin: 0pt 0pt 10px 10px; float: right; cursor: pointer; width: 152px; height: 200px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjh8r2jwkiBHV4pfzsDSNPa87avOzRwSyi9uOgjyJIm0aOcUgiqYQ5EBEFBleIeORDEx8j_Bot4iCL1eQqOauM7d_XBclLZh17rY-FLNeaLUTQFADRtaXJy_Dn1I5L4T4uDgOgG4RscGiI/s200/fracciones+egipcias1.png" alt="" id="BLOGGER_PHOTO_ID_5468249660053077218" border="0" /></a><span style="line-height: 115%;font-size:14pt;" >cada pan en tres partes iguales de manera que, en un primer reparto, se da 1/3 de pan a cada persona. Con ello sobraría una de las tres partes correspondiente a un pan que, a su vez, habría que dividir en cinco partes iguales para repartir por igual. Cada uno de los trozos resultante supondría 1/5 de 1/3 de pan, es decir, 1/15 de pan.
<br /><span style=""> </span>En resumen, cada persona no se llevaría 2/5 de pan sino 1/3 + 1/15 , lo que lleva a establecer para el escriba egipcio igualdad: 2/5 = 1/3 + 1/15<o:p></o:p></span></p> <p class="MsoNormal"><b><span style="line-height: 115%;font-size:16pt;" ><span style=""> </span>Dividir tres panes entre cinco personas</span></b><span style="line-height: 115%;font-size:14pt;" >. Nosotros aceptamos que a cada persona le corresponde exactamente 3/5 del total, pero si aplicamos el método de fracciones unitarias a 3/5 obtenemos 3/5 = 1/3 + 1/5 + 1/15 por lo que para empezar podemos dividir un pan en tres partes iguales, los otros 2 en 5 partes y luego cada una de las 5 partes de uno de estos últimos la dividimos a su vez en 3 partes.</span><b><span style="line-height: 115%;font-size:16pt;" ><o:p></o:p></span></b></p> <p class="MsoNormal"><span style="line-height: 115%;font-size:14pt;" ><span style=""> </span>Este concepto es más sencillo para un niño que fácilmente aprende a dividir un todo en n partes iguales y tomar una de ellas que un intento de aplicar 3/5 directamente o bien como suma de 2/5 + 1/5. <o:p></o:p></span></p> <p class="MsoNormal"><span style="line-height: 115%;font-size:14pt;" ><span style=""> </span>Dentro del contexto de reparto, por consiguiente, la fracción no es un número susceptible de ser generalizado, sino <i style="">la expresión de una acción de reparto</i>. Y en el reparto tal como ha sido expuesto sólo son admisibles las fracciones unitarias. Es por ello que, debido al origen de la fracción y a la limitación contextual del mismo, el egipcio nunca pudo superar la noción de la fracción en relación a la acción que la fundamenta.<o:p></o:p></span></p> <p class="MsoNormal"><span style="line-height: 115%;font-size:14pt;" ><span style=""> </span>Las fracciones con numerador distinto de 1 se reducían a sumas de fracciones conocidas, con numerador 1, pero siempre los sumandos tenían que ser diferentes. Así Ahmes en el </span><a href="http://www.egiptologia.org/ciencia/matematicas/papiro_rhind.htm"><span style="color:#000000;">papiro Rhind </span></a><span style="line-height: 115%;font-size:14pt;" >escribe 2/5 como 1/3 + 1/15 y nunca se podría emplear 1/5 + 1/5. La propia expresión 2/5 no tenía sentido en el pensamiento egipcio. Cualquier cantidad se expresaba como una parte entera mas una suma de fracciones unitarias, y a lo sumo 2/3<o:p></o:p></span></p> <p class="MsoNormal"><span style="line-height: 115%;font-size:14pt;" ><span style=""> </span>Cuando un egipcio se encontraba con una fracción 5/8 no pensaba ¿cómo puedo transformar 5/8 en una suma de fracciones unitarias?, sino que se limitaba a dividir 5 entre 8 <o:p></o:p></span></p> <p class="MsoNormal"><b><u><span style="line-height: 115%;font-size:14pt;" >¿Por qué emplearon el 2/3 como excepción</span></u></b><a name="¿Por_qué_emplearon_el_2/3_como_excepción"><b style=""><u><span style="line-height: 115%;font-size:14pt;" >?</span></u></b></a><b style=""><u><span style="line-height: 115%;font-size:14pt;" ><o:p></o:p></span></u></b></p> <p class="MsoNormal"><span style="line-height: 115%;font-size:14pt;" ><span style=""> </span>Una de las acciones matemáticas frecuentes consistía en el cálculo de la capacidad de graneros y depósitos en general.Para ello muchas veces debían de hacer equivalencias entre unidades de medidas, era allí donde utilizaban 2/3;asi para los egipcios 2/3 no es una fracción expresión de un reparto, como en el caso de las fracciones unitarias, sino que tiene una naturaleza de tipo operativo: Es el operador por el que hay que multiplicar los “codos cúbicos” para obtener su expresión en “khar.<o:p></o:p></span></p> <p class="MsoNormal"><span style="line-height: 115%;font-size:14pt;" >FUENTES:<o:p></o:p></span></p> <p class="MsoNormal"><span style="line-height: 115%;font-size:14pt;" > </span><a href="http://personal.us.es/cmaza/egipto/fracciones.htm"><span style="color:#000000;">http://personal.us.es/cmaza/egipto/fracciones.htm</span></a><span style="line-height: 115%;font-size:14pt;" ><o:p></o:p></span></p> <span style="line-height: 115%;font-family:";font-size:11pt;" ><a href="http://www.egiptologia.org/ciencia/"><span style="color:#000000;">http://www.egiptologia.org/ciencia/</span></a></span>solanohttp://www.blogger.com/profile/02741968945810651934noreply@blogger.com0