Shadowing Practice: How real are the real numbers, really? - Learn English Speaking with Video
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Say, Achilles wants to run a race.
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First he'd have to run half the way, and then half the rest of the way, and half of that, and so on.
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But no, this video isn't about Xeno's paradox.
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I just wanted to point out, to get to the end, Achilles had to run through every fraction of the length at some point.
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So here's the question for the video.
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Does Achilles just run from fraction to fraction.
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If your first instinct is like mine, you might say of course not.
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He doesn't jump from one spot to another.
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He smoothly goes along all the points.
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But see, if he does just go along all the fractions, he doesn't have to jump.
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He starts at 0, but what's the closest fraction to 0?
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Well, pick any really small fraction, and there's always one that's closer.
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Just pick the one that's half of that.
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Then pick one that's closer by looking at the halfway point again and again and again to infinity.
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This means that there's never any distance between the fraction Achilles is on
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and the next one and so he never has any jumps.
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So it looks like we're done.
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Achilles can move smoothly along all the fractions.
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But see how we've introduced infinity here?
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The total distance is just a finite number, but for Achilles to move smoothly along it, he has to go to every fraction,
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and there are a lot of fractions.
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It's not just that there is an infinite number of fractions along this line, but that between any two fractions, no matter how tiny the distance is between them,
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there is an infinite number more fractions.
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That's an insane infinity.
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But don't worry, there is way way more.
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When we think of writing a fraction out in decimal, it's either some finite string of digits or sometimes an infinite one.
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But not all infinite strings of numbers are fractions.
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Of course, there are all the classic examples.
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We'll call these numbers irrational, the ones that can't be written as fractions but can be written as huge strings of numbers.
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We said that the fractions have no gaps, right?
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But consider this.
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Pick some irrational number, any.
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Let's say mine starts off like this.
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Then say Achilles runs with a constant speed, but at first he runs 0 .7 of the way, which is a fraction, so he can do that.
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Then he runs 0 .09 more, and again, that's a fraction, so it's fine.
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Then 0 .003, etc, etc. etc. Each bit of this race is a fraction and so he's allowed to run that much,
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but he'll stop when he reaches the end of this process.
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Now you might be saying he never will because there's an infinite number of steps, but don't forget he's running at a constant speed,
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that means the amount of time it takes him to run all these smaller steps is tiny.
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So he'll get there in a finite amount of time, in fact in slightly less time than it would take him to run 0 .8 of the way.
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So he gets there alright.
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But the question is, where is there?
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It's this irrational number.
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If he could only land on fractions, he would be fine throughout this whole process since he was landing on fractions.
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But he gets to the end and it's this point we didn't think existed.
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So does he just disappear?
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Since that doesn't seem very reasonable, we're forced to accept that all of these irrational numbers are there too,
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between the fractions, even though we showed that there is no gaps between the fractions.
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The fractions and the irrationals are collectively called the real numbers because they're the ones we need for the real world.
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The fact that we need real numbers might seem obvious now, but it was a long, long process in maths to invent the right numbers to support reality.
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This problem really puzzled mathematicians.
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Suppose we want to know how fast Achilles is running.
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If we want his average we just get the total distance and divide it by his time.
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But say Achilles tends to change his speed quite a lot during the race.
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We want to know exactly what his speed was at a particular point.
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Then what do we do?
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A better estimate is to start timing for a short amount of time after he's passed that point, and see how far he's gone then.
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Then find the average speed from that.
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But of course, that's not completely accurate either.
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His speed may have changed a bit even in that smaller time.
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You'd want to make it even more accurate by picking a smaller time still and doing the same thing.
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But of course, no matter how small a time you pick, this is never 100 % accurate.
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But we would kind of like to know his speed at this point in order to do physics.
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Well then, Leibniz and Newton came along with a great idea.
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They said why don't you look at what happens when that time gap is really small.
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Like basically zero, but not zero, but smaller than any number.
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Let's call it an infinitesimal.
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Then do the same calculation with your infinitesimal as your time.
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Done.
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Problem solved.
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Now if that sounds like magic to you, then don't worry, it is.
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But they cheerfully accepted dividing by zero because, well, it worked.
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But it eventually started to really bother mathematicians.
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They didn't like these weird infinitesimals and how they weren't zero sometimes but they were later.
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I mean, it's not very rigorous and mathematical sounding, is it?
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So they went and started upheaving big chunks of maths to start making it more rigorous,
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in the hope that they would eventually be able to explain calculus without the magic.
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Their hunt led them all the way back to the numbers themselves.
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They realised that what they needed was the real numbers.
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See, this property that the real numbers had, that if a sequence is getting closer and closer together, then it is going to converge to something,
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That's exactly what you need.
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Let's see why.
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Back to our speed calculation.
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If you do that process of averaging, each time halving the time that you use, and repeat to infinity, you get a long sequence of average times.
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Let's say something like this.
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Then you can see that the numbers are getting closer.
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But since we're dealing with the real numbers, we know that this is going towards some number number.
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And whatever number that is, define that to be the speed of Achilles at this point.
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So real numbers saved the day, but not without a price.
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We saw that even in a finite amount of space there is an infinite number of fractions
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and then an infinite number of irrationals slotted between them.
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That is much more infinity than you might have imagined fitting in there.
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But the plot thickens.
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See, mathematicians were very happy to discard all those nasty infinitesimals from maths.
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But in the 60s, a mathematician discovered that you can reformulate all of calculus completely rigorously using infinitesimals.
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Using that method, the dividing by zero but not really dividing by zero thing, is completely valid.
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But instead of getting completely rid of real numbers to do this, you actually add in another infinite set of numbers,
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called the hyperreels, that slot into the gaps between the reals.
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But that is a story for another time.
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The conclusion of this video is that there is an infinite number of numbers in the universe.
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But is the entire universe infinite, or does it have an edge?
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Answering that is going to take some astrophysics and some general relativity.
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That sounds really interesting.
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You should go check out Nick's video.
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Vocabulario y notas de pronunciación para esta lección
Este vídeo tiene 104 frases y 1297 palabras para practicar shadowing. La parte hablada dura 7:50. El hablante habla a un ritmo natural de unas 166 palabras por minuto, cercano a una conversación cotidiana. El 86 % de las palabras está entre las 3.000 más comunes del inglés; conviene repasar el resto antes de empezar.
Vocabulario clave de este vídeo
15 palabras menos comunes del vídeo, con su pronunciación y significado:
- fraction /ˈfɹæk.ʃən/ (sustantivo) — fracción. A part of a whole, especially a comparatively small part.
- infinite /ˈɪnfɪnɪt/ (adjetivo) — infinito. Indefinably large, countlessly great; immense.
- irrational /ɪˈɹæʃ.(ə.)nəl/ (adjetivo) — irracional. Not rational; unfounded, nonsensical or wrong-headed.
- infinity /ɪnˈfɪnɪti/ (sustantivo) — infinidad. Endlessness, unlimitedness, absence of a beginning, end or limits to size.
- divide /dɪˈvaɪd/ (verbo) — desunir, dividir. To split or separate (something) into two or more parts.
- finite /ˈfaɪ.naɪt/ (adjetivo) — limitado, finito. Having an end or limit; (of a quantity) constrained by bounds; (of a set) whose number of elements is a natural number.
- gap /ɡæp/ (sustantivo) — brecha, rehendija. An opening in anything, made by breaking or parting.
- string /stɹɪŋ/ (sustantivo) — cuerda, mecate ("cordel" or "cuerda"). A long, thin and flexible structure made from threads twisted together.
- mathematics /mæθ(.ə)ˈmæt.ɪks/ (sustantivo) — matemáticas, matemática. An abstract representational system studying numbers, shapes, structures, quantitative change and relationships between them.
- smoothly /ˈsmuːðli/ (adverbio) — suavemente, lisamente. in a smooth manner; smooth
- sequence /ˈsiː.kwəns/ (sustantivo) — secuencia. A set of things next to each other in a set order; a series.
- slot /slɑt/ (sustantivo) — ranura, hendidura. A narrow depression, perforation, or aperture; especially, one for the reception of a piece fitting or sliding into it.
- calculation /ˌkælkjuˈleɪʃn̩/ (sustantivo) — cálculo, calculación. The act or process of calculating.
- rigorous /ˈɹɪɡəɹəs/ (adjetivo) — riguroso, rigoroso. Showing, causing, or favoring rigour/rigor; scrupulously accurate or strict; thorough.
- calculus /ˈkæl.kjə.ləs/ (sustantivo) — cálculo. Any formal system in which symbolic expressions are manipulated according to fixed rules.
Phrasal verbs que vas a escuchar
- look at (verbo) — mirar, funar. To observe or watch.
- check out (verbo) — investigar, chequear. To record the departure or withdrawal of someone or something (such as guests, employees, books, etc.).
- deal with (verbo) — tratarse de, ocuparse. To handle verbally or in some form of artistic expression; to address or discuss as a subject.
- fit in (verbo) — caber en, encajar en. To be physically capable of going into a space.
- run through (verbo) — resumir. To summarise briefly.
- run with (verbo) — adaptarse, seguir / continuar con. To be streaming with a fluid.
Pronunciación a tener en cuenta
El hablante usa 22 contracciones y formas reducidas, como don't, doesn't, we're. Dilas en su forma corta, tal como las oyes.
- Los sonidos de “th”: mathematics /mæθ(.ə)ˈmæt.ɪks/, smoothly /ˈsmuːðli/, bother /ˈbɑðɚ/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/
- Los sonidos de “sh” y “zh”: fraction /ˈfɹæk.ʃən/, irrational /ɪˈɹæʃ.(ə.)nəl/, calculation /ˌkælkjuˈleɪʃn̩/, conclusion /kənˈkluːʒən/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/
- Palabras largas — cuida el acento: irrational /ɪˈɹæʃ.(ə.)nəl/, infinity /ɪnˈfɪnɪti/, mathematics /mæθ(.ə)ˈmæt.ɪks/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, infinitesimal /ˌɪnfɪnɪˈtɛsɪməl/
Cómo practicar con este vídeo
- Escucha el vídeo entero una vez sin hablar y anota las palabras que no conoces.
- Empieza a velocidad 0,75×, haz shadowing frase por frase y vuelve a la velocidad normal cuando te resulte fácil.
- Grábate y compara con el original, prestando atención a palabras como fraction, infinite, irrational.
¿Qué es la Técnica de Shadowing?
Shadowing es una técnica de aprendizaje de idiomas respaldada por la ciencia, desarrollada originalmente para la formación de intérpretes profesionales y popularizada por el políglota Dr. Alexander Arguelles. El método es simple pero poderoso: escuchas audio en inglés nativo y lo repites en voz alta de inmediato, como una sombra que sigue al hablante con solo 1-2 segundos de retraso. A diferencia de la escucha pasiva o los ejercicios de gramática, el shadowing obliga a tu cerebro y músculos de la boca a procesar y reproducir simultáneamente patrones de habla reales. Las investigaciones muestran que mejora significativamente la precisión de la pronunciación, la entonación, el ritmo, el habla conectada, la comprensión auditiva y la fluidez al hablar, convirtiéndola en una de las metodologías más efectivas para la preparación del IELTS Speaking y la comunicación en inglés en el mundo real.