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{"id":59364,"date":"2024-09-20T02:46:38","date_gmt":"2024-09-20T11:16:38","guid":{"rendered":"https:\/\/radiocubalibre.live\/los-desconcertantes-numeros-anomalos-ciencia\/"},"modified":"2024-09-20T02:46:38","modified_gmt":"2024-09-20T11:16:38","slug":"los-desconcertantes-numeros-anomalos-ciencia","status":"publish","type":"post","link":"https:\/\/radiocubalibre.live\/los-desconcertantes-numeros-anomalos-ciencia\/","title":{"rendered":"Los desconcertantes n\u00fameros an\u00f3malos | Ciencia"},"content":{"rendered":"

\n<\/p>\n

\n

\"N\u00fameros — Simon Newcomb (1835-1909).<\/span>Alamy Stock Photo<\/span><\/figcaption><\/figure>\n
Nuestro comentarista habitual Rafael Granero da las siguientes respuestas a las preguntas de la semana pasada<\/a>:<\/p>\n
Los n\u00fameros cuyos logaritmos decimales est\u00e1n comprendidos entre 0 y 2 son aquellos que se encuentran entre 1 y 100. Esto se debe a que:<\/p>\n
logd(1) = 0<\/p>\n
logd(100) = 2<\/p>\n
Por lo tanto, cualquier n\u00famero x que cumpla 1 < x < 100 tendr\u00e1 un logaritmo decimal entre 0 y 2.<\/p>\n
Logaritmo decimal de 0,01<\/p>\n
El logaritmo decimal de 0,01 es -2.<\/p>\n
Esto se puede entender de la siguiente manera: logd(0,01) = logd(1\/100) = logd(1) \u2013 logd(100) = -2<\/p>\n
Logaritmos decimales de 9, 30 y 1\/3<\/p>\n
Sabiendo que logd(3) = 0,477, podemos calcular los dem\u00e1s:<\/p>\n
Logaritmo decimal de 9: logd(9) = logd(3^2) = 2 * logd(3) = 2 * 0,477 = 0,954<\/p>\n
Logaritmo decimal de 30: logd(30) = logd(3 * 10) = logd(3) + logd(10) = 0,477 + 1 = 1,477<\/p>\n
Logaritmo decimal de 1\/3: logd(1\/3) = logd(1) \u2013 logd(3) = -0,477<\/p>\n
Por su parte, Manuel Amor\u00f3s encuentra de esta ingeniosa manera el valor de x cuando x elevado a la potencia x3 es igual a 3:<\/p>\n
x^(x^3) = 3<\/p>\n
(x^(x^3))^3 = 3^3<\/p>\n
(x^3)^(x^3) = 3^3<\/p>\n
x^3 = y<\/p>\n
y^y = 3^3<\/p>\n
y = 3, ergo x = ra\u00edz c\u00fabica de 3<\/p>\n
Y Bretos Burs\u00f3 propone una interesante interpretaci\u00f3n del n\u00famero e, base de los logaritmos neperianos que, aunque solo es apta para personas con ciertos conocimientos matem\u00e1ticos, no he resistido la tentaci\u00f3n de incluirla:<\/p>\n
Supongamos que vemos c\u00f3mo se forma al azar una fila de personas arbitrariamente larga, y que somos capaces de distinguir siempre, dadas dos personas, cu\u00e1l es la m\u00e1s alta (por peque\u00f1\u00edsima que sea la diferencia). Contamos la cantidad de personas que llegan hasta que la \u00faltima en hacerlo es m\u00e1s alta que la pen\u00faltima (dicha cantidad ser\u00e1 siempre mayor o igual a 2). Entonces:<\/p>\n
– el valor esperado o promedio de esta cantidad variable es el n\u00famero e.<\/p>\n
– la probabilidad de que esa \u00faltima persona sea adem\u00e1s m\u00e1s alta que todas las anteriores es e-2.<\/p>\n
(Cada una de las dos afirmaciones anteriores equivale a que la suma de la serie de 1\/n! para n = 0, 1, 2, 3\u2026 es el n\u00famero e).<\/p>\n
\n
M\u00e1s informaci\u00f3n<\/header>\n<\/aside>\n
La ley de los n\u00fameros an\u00f3malos<\/h3>\n
Como vimos, las tablas de logaritmos, al permitir la conversi\u00f3n de las multiplicaciones en sumas y de las divisiones en restas, facilitaban considerablemente los c\u00e1lculos cuando no hab\u00eda ordenadores; pero hace mucho que cayeron en desuso, junto con las maravillosas reglas de c\u00e1lculo que asomaban en el bolsillo superior de todo ingeniero que se preciara.<\/p>\n
En el siglo XIX, las tablas de logaritmos estaban entre los manuales m\u00e1s consultados de cualquier biblioteca t\u00e9cnica o cient\u00edfica, y ese uso continuo permiti\u00f3 al gran astr\u00f3nomo y matem\u00e1tico Simon Newcomb darse cuenta de que las primeras p\u00e1ginas de todas las tablas que examin\u00f3 mostraban m\u00e1s se\u00f1ales de uso que las siguientes, y que el nivel de uso decrec\u00eda regularmente a medida que se pasaban las p\u00e1ginas. Eso significaba que hab\u00eda m\u00e1s n\u00fameros consultados que empezaban por 1 que por cualquier otra cifra, seguidos en cantidad por los que empezaban por 2, a continuaci\u00f3n ven\u00edan los que empezaban por 3\u2026<\/p>\n
A partir de sus observaciones, Newcomb enunci\u00f3 una ley sobre la frecuencia de los n\u00fameros en relaci\u00f3n con las mantisas (partes decimales) de sus logaritmos, que le permiti\u00f3 estimar que la probabilidad de que un n\u00famero tomado del mundo real empiece por 1 es de aproximadamente el 30 %, la de que empiece por 2 es del 18 %, la de que empiece por 3 es del 12 %\u2026 y as\u00ed, siempre decreciendo, hasta llegar al 9, cuya probabilidad de encabezar un n\u00famero no llega al 5 %.<\/p>\n
Las contraintuitivas conclusiones de Newcomb cayeron en el olvido hasta que, en 1938, el ingeniero estadounidense Frank Benford, tras comprobar m\u00e1s de 20.000 n\u00fameros de 20 muestras diferentes (tales como n\u00fameros de habitantes de una lista de ciudades, cotizaciones de bolsa, constantes f\u00edsicas, pesos moleculares, tasas de mortalidad, n\u00fameros de direcciones postales\u2026), enunci\u00f3 la que denomin\u00f3 \u201cley de los n\u00fameros an\u00f3malos\u201d, hoy conocida como ley de Benford (aunque algunos preferimos llamarla ley de Benford-Newcomb), seg\u00fan la cual la primera cifra n en una muestra de n\u00fameros tomados del mundo real aparece con una probabilidad dada por la f\u00f3rmula: logd(n+1) \u2013 logd(n). (Por limitaciones tipogr\u00e1ficas, el logaritmo decimal se indica como logd).<\/p>\n
\u00bfNo deber\u00edan las primeras cifras distribuirse de forma equiprobable entre los nueve d\u00edgitos (obviamente, se excluye el cero)? \u00bfPor qu\u00e9 es m\u00e1s probable que el n\u00famero de habitantes de una ciudad empiece por 1 que por 9? \u00bfSe te ocurre alguna explicaci\u00f3n para este resultado tan \u2014aparentemente\u2014 arbitrario?<\/p>\n<\/div>\n

\n
Source link <\/a><\/p>\n","protected":false},"excerpt":{"rendered":"
Simon Newcomb (1835-1909).Alamy Stock Photo Nuestro comentarista habitual Rafael Granero da las siguientes respuestas a las preguntas de la semana pasada: Los n\u00fameros cuyos logaritmos decimales est\u00e1n comprendidos entre 0… <\/p>\n","protected":false},"author":1,"featured_media":59365,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[64,15],"tags":[],"class_list":["post-59364","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-ciencia-y-salud","category-ultimas-noticias"],"aioseo_notices":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"https:\/\/radiocubalibre.live\/wp-content\/uploads\/2024\/09\/DJNPQGY6KFHELMJIKKAFRQSP44.jpg","uagb_featured_image_src":{"full":["https:\/\/radiocubalibre.live\/wp-content\/uploads\/2024\/09\/DJNPQGY6KFHELMJIKKAFRQSP44.jpg",1200,651,false],"thumbnail":["https:\/\/radiocubalibre.live\/wp-content\/uploads\/2024\/09\/DJNPQGY6KFHELMJIKKAFRQSP44-150x150.jpg",150,150,true],"medium":["https:\/\/radiocubalibre.live\/wp-content\/uploads\/2024\/09\/DJNPQGY6KFHELMJIKKAFRQSP44-300x163.jpg",300,163,true],"medium_large":["https:\/\/radiocubalibre.live\/wp-content\/uploads\/2024\/09\/DJNPQGY6KFHELMJIKKAFRQSP44-768x417.jpg",768,417,true],"large":["https:\/\/radiocubalibre.live\/wp-content\/uploads\/2024\/09\/DJNPQGY6KFHELMJIKKAFRQSP44-1024x556.jpg",1024,556,true],"1536x1536":["https:\/\/radiocubalibre.live\/wp-content\/uploads\/2024\/09\/DJNPQGY6KFHELMJIKKAFRQSP44.jpg",1200,651,false],"2048x2048":["https:\/\/radiocubalibre.live\/wp-content\/uploads\/2024\/09\/DJNPQGY6KFHELMJIKKAFRQSP44.jpg",1200,651,false]},"uagb_author_info":{"display_name":"Radio Cuba Libre","author_link":"https:\/\/radiocubalibre.live\/author\/caballerodeparis\/"},"uagb_comment_info":0,"uagb_excerpt":"Simon Newcomb (1835-1909).Alamy Stock Photo Nuestro comentarista habitual Rafael Granero da las siguientes respuestas a las preguntas de la semana pasada: Los n\u00fameros cuyos logaritmos decimales est\u00e1n comprendidos entre 0…","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/radiocubalibre.live\/wp-json\/wp\/v2\/posts\/59364","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/radiocubalibre.live\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/radiocubalibre.live\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/radiocubalibre.live\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/radiocubalibre.live\/wp-json\/wp\/v2\/comments?post=59364"}],"version-history":[{"count":0,"href":"https:\/\/radiocubalibre.live\/wp-json\/wp\/v2\/posts\/59364\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/radiocubalibre.live\/wp-json\/wp\/v2\/media\/59365"}],"wp:attachment":[{"href":"https:\/\/radiocubalibre.live\/wp-json\/wp\/v2\/media?parent=59364"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/radiocubalibre.live\/wp-json\/wp\/v2\/categories?post=59364"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/radiocubalibre.live\/wp-json\/wp\/v2\/tags?post=59364"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}