Pratique du Shadowing: How real are the real numbers, really? - Apprendre l'anglais à l'oral avec la vidéo
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104 phrases
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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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Vocabulaire et conseils d’expression pour cette leçon
Cette vidéo contient 104 phrases et 1297 mots à répéter en shadowing. La partie parlée dure 7:50. Le locuteur parle à un rythme naturel d’environ 166 mots par minute, proche d’une conversation courante. 86 % des mots font partie des 3 000 mots les plus courants en anglais ; le reste mérite d’être vu avant de commencer.
Vocabulaire clé de cette vidéo
15 mots moins courants de la vidéo, avec leur prononciation et leur sens :
- fraction /ˈfɹæk.ʃən/ (nom) — fraction. A part of a whole, especially a comparatively small part.
- infinite /ˈɪnfɪnɪt/ (adjectif) — infini. Indefinably large, countlessly great; immense.
- irrational /ɪˈɹæʃ.(ə.)nəl/ (adjectif) — irrationnel. Not rational; unfounded, nonsensical or wrong-headed.
- infinity /ɪnˈfɪnɪti/ (nom) — infinité. Endlessness, unlimitedness, absence of a beginning, end or limits to size.
- divide /dɪˈvaɪd/ (verbe) — diviser, fendre. To split or separate (something) into two or more parts.
- finite /ˈfaɪ.naɪt/ (adjectif) — fini. Having an end or limit; (of a quantity) constrained by bounds; (of a set) whose number of elements is a natural number.
- gap /ɡæp/ (nom) — brèche, écart. An opening in anything, made by breaking or parting.
- string /stɹɪŋ/ (nom) — corde. A long, thin and flexible structure made from threads twisted together.
- mathematics /mæθ(.ə)ˈmæt.ɪks/ (nom) — mathématiques. An abstract representational system studying numbers, shapes, structures, quantitative change and relationships between them.
- smoothly /ˈsmuːðli/ (adverbe) — souplement, doucement. in a smooth manner; smooth
- sequence /ˈsiː.kwəns/ (nom) — suite, séquence. A set of things next to each other in a set order; a series.
- slot /slɑt/ (nom) — emplacement, fente. A narrow depression, perforation, or aperture; especially, one for the reception of a piece fitting or sliding into it.
- calculation /ˌkælkjuˈleɪʃn̩/ (nom) — calcul. The act or process of calculating.
- rigorous /ˈɹɪɡəɹəs/ (adjectif) — rigoureux. Showing, causing, or favoring rigour/rigor; scrupulously accurate or strict; thorough.
- calculus /ˈkæl.kjə.ləs/ (nom) — calcul. Any formal system in which symbolic expressions are manipulated according to fixed rules.
Les verbes à particule que vous entendrez
- look at (verbe) — regarder. To observe or watch.
- check out (verbe) — mater, regarder. To record the departure or withdrawal of someone or something (such as guests, employees, books, etc.).
- deal with (verbe) — traiter. To handle verbally or in some form of artistic expression; to address or discuss as a subject.
Prononciation à surveiller
Le locuteur utilise 22 contractions et formes réduites, comme don't, doesn't, we're. Prononcez-les sous leur forme courte, telles que vous les entendez.
- Les sons « th »: mathematics /mæθ(.ə)ˈmæt.ɪks/, smoothly /ˈsmuːðli/, bother /ˈbɑðɚ/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/
- Les sons « sh » et « zh »: fraction /ˈfɹæk.ʃən/, irrational /ɪˈɹæʃ.(ə.)nəl/, calculation /ˌkælkjuˈleɪʃn̩/, conclusion /kənˈkluːʒən/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/
- Mots longs — placez bien l’accent: irrational /ɪˈɹæʃ.(ə.)nəl/, infinity /ɪnˈfɪnɪti/, mathematics /mæθ(.ə)ˈmæt.ɪks/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, infinitesimal /ˌɪnfɪnɪˈtɛsɪməl/
Comment s’entraîner avec cette vidéo
- Écoutez la vidéo en entier une fois sans parler et notez les mots que vous ne connaissez pas.
- Commencez à la vitesse 0,75×, répétez phrase par phrase, puis revenez à la vitesse normale quand cela devient facile.
- Enregistrez-vous et comparez avec l’original, en faisant attention à des mots comme fraction, infinite, irrational.
Qu'est-ce que la technique du Shadowing ?
Le Shadowing est une technique d'apprentissage des langues fondée sur la science, développée à l'origine pour la formation des interprètes professionnels. Le principe est simple mais puissant : vous écoutez de l'anglais natif et le répétez immédiatement à voix haute — comme une ombre suivant le locuteur avec un décalage de 1 à 2 secondes. Les recherches montrent une amélioration significative de la précision de la prononciation, de l'intonation, du rythme, des liaisons, de la compréhension orale et de la fluidité.