Taming InfinityYouTube↗  IA↗(long shot)
Science & Technology  1.8M views 8.6K likes 497 dislikes   19 Sep 2011
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There's been some discussion about the video where I claimed that 1+2+4+8+16 and so on
Ha habido cierta discusión sobre el vídeo en el que afirmé que 1+2+4+8+16+...
is equal to negative one - apparently I'm on my way to math jail.
es igual a menos uno. Parece que me van a llevar a la cárcel matemática. Pero en realidad, la suma de los términos
But actually, adding up divergent series is an old topic in math, and more importantly,
de una serie divergente es algo ya antiguo en las matemáticas y, lo que es más importante, ¡es algo irrelevante para la física!
it's actually relevant to physics! In physics, there are a lot of hard problems where we can't find the whole answer - but
En física, hay muchos problemas difíciles para los que no somos capaces de encontrar una solución completa.
we can find the answer to a simplified version of the problem, and then correct it step by
Pero sí sabemos resolver una versión simplificada del problema, y después corregirla
step so we get closer and closer to the real answer.
paso a paso para ir acercándonos a la verdadera solución.
Check it out: we can approximate this hyperbola by starting with a line, adding a curve, a
Compruébalo: Podemos aproximar esta hipérbola partiendo de una línea recta y añadiendo luego una curva,
wiggle, and onwards… except that our step-by-step process only works for a little part of the
una ondulación, y así sucesivamente. Con la salvedad de que este proceso paso a paso solo sirve para una pequeña porción de la curva.
curve! Outside this "area of convergence", we're totally wrong!
Fuera de esta "zona de convergencia", nuestra solución es completamente inválida.
Ok, back to physics - Some problems, like figuring out how electrons scatter in quantum
Vale. Volvamos a la física. Algunos problemas, como los cálculos relacionados con la dispersión de electrones en la teoría
field theory, are really hard to solve! But we know there must be an answer, because we do experiments and see that something happens. So we take the step-by-step approach - except that most of the time, the corrections get bigger and bigger, adding up to infinity!
cuántica de campos, son muy difíciles de resolver. Pero sabemos que debe existir una solución, porque realizamos experimentos y vemos que suceden cosas. Así que probamos a aplicar un enfoque paso a paso, pero en la mayoría de los casos las correcciones son cada vez más grandes, ¡y su suma es infinita!
What went wrong?
¿Dónde está el error? Estamos fuera de la "zona de convergencia". Pero, por suerte, no todo está perdido.
We're outside the 'area of convergence'. But luckily, all is not lost! Because, just as the series 1+2+4+8+16 etc. contains enough information to tell us that
Porque, igual que la serie 1+2+4+8+16+... contiene información suficiente como para que podamos saber que
it really means negative one, physicists use clever techniques to tease the real answer
en realidad es equivalente a menos uno, los físicos utilizan técnicas ingeniosas para extraer una solución real
out of infinity.
a partir del infinito.
For example, they've calculated quantities, like the strength of the electromagnetic force,
Por ejemplo, han calculado magnitudes como la intensidad de la fuerza electromagnética
to incredible detail - and the calculations agree with experiment!
con un grado de precisión increíble. ¡Y sus cálculos concuerdan con los experimentos! Este es uno de los resultados
This is one of the most accurate and precise results in all of science – and that's the
más precisos de toda la ciencia, que pone de manifiesto lo mucho que se puede conseguir si
power of taming infinity.
somos capaces de domesticar los infinitos.
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