RAZONAMIENTO MATEMATICO TEORIA Y PROBLEMAS 1a EDICIÓN
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WALTER PARDAVÉ LIVIA
RAZONAMIENTO MATEMATICO TEORIA Y PROBLEMAS 1a EDICIÓN
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PRIMERA EDICION Enero de 1999 DIAGRAMACIÓN E IMPRESIÓN Sistemas & Computadores Ltda. Centro Empresarial Chicamocha Of. 303 Sur Telf: (97) 6343558 - Fax (97) 6455869 Bucaramanga - Colombia
ISBN: 958-8037-19-0
Prohibida la reproducción parcial o total de esta obra, por cualquier medio, sin autorización escrita del autor Impreso en Colombia
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A mi esposa Chelita Chelita y a mis hijas Maria Camila Camila y Dina Marcela Marcela
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TABLA DE CONTENIDO PREAMBULO ............................. ............................................ .............................. ............................... ............................... .............................. ............... 11 COMO RESOLVER RESOLVER UN PROBLEMA? ....................................... ........................................................... .......................... ...... 13 CAPITULO I: OPERACIONES OPERACIONES FUNDAMENTALES FUNDAMENTALES ........................................ .......................................... .. 21 CAPITULO II: OPERADORES MATEMA MATEMATICOS TICOS ........................................ ................................................. ......... 39 CAPITULO III: SERIES NUMERICAS ....................................... ........................................................... .......................... ...... 61 CAPITULO IV: IV: HABILIDAD OPERATIV OPERATIVA A ....................................... ........................................................ ................. 107 CAPITULO V: RAZONAMIENT RAZONAMIENTO O GEOMETRICO GEOMETRICO ......................................... ........................................... .. 127 CAPITULO VI: CRIPTOGRAMA CRIPTOGRAMA ........................................ ............................................................ ................................ ............ 149 CAPITULO VII: RAZONAMIENTO RAZONAMIENTO LOGICO ....................................... ................................................... ............ 171
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PREAMBULO El razonamiento matemático, es un tipo particular de exigencia a nivel pre-universitario pre-universitar io y superior, que se ha generalizado en lo últimos años, por ello este libro habrá de permitirles ejercitar sus capacidades en las áreas matemáticas, además de brindarles importantes mecánicas resolutivas para que superen con exitos los exámenes o evaluaciones que han de someterse. Pretendo llegar al estudiantado y público en general con un trabajo novedoso en cuanto a contenidos y metodología de presentación de temáticas como: operadores matemáticos, criptograma, trazo de figuras, habilidad operativa, figuras análogas, etc. Quiero agradecer a las personas que han hecho propicia esta edición, a quienes me favorezcan con su preferencia y dedicar la obra a mis alumnos pasados, presentes y futuros; de quienes aprendo mucho cada día.
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COMO RESOLVER UN PROBLEMA? ‘Todos ‘Todos tenemos problemas”. Es una frase, muy conocida y aceptada por todos. En efecto la vida nos enfrenta a situaciones que debemos resolver continuamente, algunas veces nosotros solos, otras con ayuda de los demás. Pero no solo existen problemas a nivel personal, las sociedades en las cuales nos desenvolvemos y en las de todo el mundo afrontan dificultades o problemas, los cuales pueden ser económicos, culturales, religiosos, etc., en fin, de una gran diversidad. El planeta tierra se enfrenta a múltiples problemas, llámese por ejemplo la existencia de vida en otros planetas, hasta cuando durará la energía solar etc. No hay ciencia que no se enfrente a problemas en sus respectivos campos. Es decir, existe una inmensa variedad varieda d y cantidad de problemas que afectan al ser humano, los hay desde muy simples hasta realmente complejos, tanto que al hombre le toma muchos años resolverlos. Pero así como van surgiendo los problemas, van surgiendo las soluciones, al desarrollo que actualmente el hombre ha resuelto con esa admirable tenacidad e inteligencia que le caracterizan. Toda esta variedad y cantidad de problemas que existen han hecho también que muchos hombres se pregunten: ¿Existe un método universal para su solución ?. Se ha buscado la respuesta en las Matemáticas. Sabemos que casi todas las cuestiones matemáticas son susceptibles de solución, pero para resolver un
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una ecuación en la que él reemplace determinados datos y al despejar la ecuación aparezca como respuesta que enfermedad padecemos y que hay que hacer para curarnos. El médico tiene que resolver el problema de curarla apoyado en su experiencia personal y en el estudio de otros casos análogos. Así sucede con varias otras ciencias que no pueden resolver sus problemas matemáticamente. Para suerte nuestra, por ahora, vamos a enfrentarnos a problemas susceptibles de ser resueltos matemáticamente, ellos al igual que cualquier otro tipo de problema, tienen datos e incógnitas. Podemos decir que los datos son las informaciones de las cuales disponemos para poder resolver el problema, informaciones que tienen que ver con la incógnita, es decir lo que desconocemos. Hablemos ahora de los problemas que habremos de resolver durante nuestra preparación. Ellos pertenecen a las matemáticas elementales, pueden incluso resolverse por distintos métodos, y como ahora estamos tratando de aumentar nuestra capacidad de razonamiento, razonamient o, vamos a dar una serie de pautas que habremos
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1. QUERER RESOLVER RESOLVER EL PROBLEMA Es el primer paso y consiste en una actitud fundamentalmente de carácter anímico. Debe Ud. estar predispuesto psicológicamente a resolver las dificultades, por más simples o complicados que el problema le parezca, Ud. debe estar dispuesto a enfrentarse a él y a vencerlo. Cada problema es un reto a su capacidad; es un reto que Ud. acepta y que hará para que para superarlo, haga uso de todas las armas disponibles que le permitan resolverlo. Su actitud en estos casos es determinante: ¡El hombre no logra nada que no se proponga! Recordemos sino aquella célebre frase del Dr. Barnard “Si piensas que estas vencido lo estas”, “Si piensas que puedes, podrás”. Querer resolver el problema es haberlo resuelto ya en un 50%.
2. ENTEND ENTENDER ER EL PROBL PROBLEMA EMA
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* ¿Cuál es la incógnita?
Apuntela.
* ¿Cuáles son los datos?
Apuntelos
* ¿Hay alguna condición?
Señalela
* ¿Con los datos podré satisfacer satisfacer la condición condición
¡ Pienselo! Pienselo!
* ¿Son los datos datos suficie suficientes? ntes? ¿Son pocos? pocos? ¿O son son demasiado demasiados? s? * ¿El problem problemaa pertenece pertenece al Algebra? Algebra? a la la Geometría Geometría?? A que área área de las las matemáticas pertenece?. * ¿De que se trata este problema problema?? ¿Que cantidad cantidades es interviene intervienen n en él? ¿Intervi ¿Intervienen enen personas, edades, porcentajes, áreas?. * ¿Recuerda ¿Recuerda a leerlo leerlo a algún algún otro problema? problema? ¿En ¿En qué se parecen? parecen?.. * ¿Puede ¿Puede cambiarle cambiarle de de datos a este proble problema, ma, sin sin que su estructura estructura varíe?. varíe?. Es decir lo que usted debe hacer es investigar, y ésta, es una actitud mental
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empezará a esbozar un plan para cruzarlo, ¿lo hará a nado?, ¿Se construirá una embarcación?, ¿Intentará buscar un lugar adecuado para cruzarlo?. Note que estas son formas de resolver la dificultad que le afecta y que usted está viendo cuál le conviene más. ¿No es así?. De la misma manera proceda ante un problema matemático, después del paso (2) ya está usted convertido en un investigador, inves tigador, para lo cual deber estar su capacidad de esfuerzo original trabajando al máximo. Tal Tal vez este usted desorientado en el camino que le lleve a la solución, pregúntese: * ¿Que relación existe entre mis datos y mis incógnitas? * ¿Puedo representar matemáticamente la relación existente entre datos e incógnitas? ¿Si? ¿Cómo?, ¿No?, ¿Porqué?. * ¿Puedo escribir los datos en función de las incógnitas? * ¿Y la condición? ¿La puede representar?
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* Puede usted usted imagina imaginarr algún proble problema ma más accesib accesible, le, pero pero relacionad relacionado o con el que tiene enfrente? ¿O un problema más general? ¿O un problema más especial?. * Esta utilizando todos los datos? datos? ¿La ¿La condición, condición, la ha entendido íntegramente? íntegramente? ¿Está usándola bien? ¿No se olvidó de algún dato o propiedad?. Si se trata de un problema geométrico, es siempre si empre recomendable dibujar las figuras que intervienen y señalar en ellas los datos e incógnitas que le den. Es decir inicialmente debe de recurrir a la anologia, revisar sus conocimientos, apelando a su sagacidad y habilidad mental que algunos psicólogos llaman “Iluminación” “Iluminaci ón” o “Bright idea” (Idea Brillante), que consiste en dar, dar, por intermedio del razonamiento con la idea que ha de constituir la clave órientadora de su futura acción en procura de la solución.
4. REALIZAR EL PLAN QUE LE LLEVARA LLEVARA A LA SOLUCIÓN
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Finalmente después de una interesante aventura intelectural habra logrado dar con la solución y no lo oculte, ahora le invade una gran satisfacción, una gran alegría, que premio el esfuerzo realizado y le insta a seguir resolviendo la mayor cantidad de problemas, pues sabe usted que en la práctica está la clave del éxito.
5. REPASA REPASANDO NDO EL PROB PROBLE LEMA MA
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CAPITULO I OPERACIONES FUNDAMENTALES Para resolver los problemas pertenecientes a este capitulo, no necesitamos más que conocer los principios fundamentales que rigen a la suma, resta, multiplicación y división; además de tener rapidez para efectuar los cálculos numéricos necesarios.
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* La cifra cifra de las unidade unidadess del resultad resultado o de la suma suma total, total, está dada dada por la suma suma de las cifras de los sumandos. * Las cifras cifras que le le siguen siguen son el resulta resultado do de sumar sumar las respec respectivas tivas column columnas, as, aumentándoles en cada caso lo que se lleva de la suma anterior. anterior. * Realiz Realizada ada la suma suma de cada cada column columna, a, Si Si es mayor que 9 unidades; la cifra de la derecha se coloca debajo de la columna sumada (“se pone”, “se queda”, y la (a) otra (s) se le agrega (n) a la columna siguiente (“se lleva”)
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* Ahora coloquemos los sumandos en forma vertical, tenemos: abc + bca cab ??? ??? * De donde donde notamos notamos que que la cifra cifra de de las unidad unidades es será igual al valor valor de la suma: suma:
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Como puede usted observar cada producto parcial es resultado de multiplicar al multiplicando por cada cifra del multiplicador, y los respectivos resultados
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5.
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Solución
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exacta; una vez que lo logremos ya sabemos que el número buscado (N) será el cociente de la división realizada.
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