3 Paradoxes That Gave Us Calculus

[@Nofxthepirate]

3 years into my Math degree, and I still find elegant explanations of calculus to be highly fascinating. Calculus is the reason I pursued math in the first place.

[@arctic_haze]

In the fourth year you will learn that for mathematicians, the actual solution is not interesting, only the fact that a class of equation has a solution 😁

[@singh2702]

@@arctic_haze Agreed , example Finding the integral makes total sense when you reverse the derivative and increase the exponent by 1 , because that gives the total area. However the denominator(which divides the total area to give you the area under the graph) you must arrive at is a rule that you just have to accept no explanation as to why areas work like this but it's just a fact. I know it works like that in order to get back to the derivative but how it arrives at the area is a rule you have to accept.

[@singh2702]

Interesting or not

[@ShrekMeBe]

Especially when the "explanation" smiles and jingles at you like you're the one!

[@GaZonk100]

I explained to my daughter when she was in late high school that calculus was all about slicing...lots of 'slicing' into lots of 'slices'

[@ronsa6783]

As an engineer I never have to be right; just close enough.

[@HoD999x]

pi is 3

[@awsm1680]

π=e=3

[@mikeg3660]

Within “limits”

[@VivekYadav-ds8oz]

lim π->e (e/π) = 1

[@brandonlee6536]

Haha i love engineers

[@lehoff]

I use a lot of these maths principles weekly and you explained it better than my lecturers ever did! Even though I know this stuff it's fun to hear it explained so well and in a fun and engaging way!

[@allenalsop6032]

Would you then hunt deers or photograph geeses?

[@WendyPingree]

@@allenalsop6032 Do you think the singular of mathematics is mathematic? Or physics, physic?

[@ookjannesplanting1296]

"Where are you going with this, Zeno?" Not to the end, that's for sure

[@wilsonli5642]

"And that's why it is impossible for me to get up to take out the garbage!"

[@ThePeaceableKingdom]

@Roger Loquitur Zeno is often thought to be a student and proponent of Parmenides, rather than Empedocles. (that "ardent soul who leapt into Etna and was roasted whole") Parmenides is often thought to have argued that change is impossible; Zeno seems to be giving examples in his paradoxes of why change is impossible. But it's hard to know exactly what Parmenides argued for two reasons: 1) Half his book is missing. The Way of Truth survives, wherein Aurora, goddess of the dawn, gives him divine knowledge of the true nature of things, but the other half, The Way of Seeming, is lost. And 2) his Greek is obscure (I am told - I don't read Greek) and difficult to understand even for those who are fluent in the ancient language. He pronounces that "Is is; and is not is not" - often translated as "Being is, and Non-Being is not." But that's not crystal clear. One author has suggested it's a description of Euclidian space: there are no voids in Euclidian space that the compass and straight edge can't traverse, no "is not" in the geometric plane or volume. Likewise it cannot be altered. You can move your desk across the room, but you cannot move the place your desk used to be. You can't alter it at all. But there are other interpretations, and the most common is that he argued that change is impossible because change involves something that is not being but rather "becoming," somewhat friendlier to the objections in Zeno's paradoxes.

[@rosairedubrule60]

Is Zee no a definitive from his mom

[@eyeballs50]

your toenail is longer than the gap sooner or later, so you really do get there

[@singh2702]

@@eyeballs50 Your toenail or your shoe or whatever is foremost will always be on the beginning of a new gap, just like when runners do the 100m dash when they in position to start their fingers are positioned before the white line.

[@rexwhitehead8346]

"Calculus" was the word used for any systematic method of calculating. Newton speaks of "an arithmetical calculus" being used by Dr Halley to do a better calculation of the orbit of Halley's comet than he himself could do by geometrical means using very large sheets of paper. The calculus you're speaking of was really called the "Infinitesimal Calculus". The little Roman stones (calculi) were used as counters, not to divide big things into lots of little things.

[@allenalsop6032]

"Infinitesimal Calculus of a Single Variable " to be more precise. Many different types of Calculus exist ie., Calculus of Variations, Vector Calculus, Complex Calculus etc.

[@Jabrils]

Im so happy the animations are back. 🤗 but Zeno is looking a little mad haha, I wouldn't trust a man looking like that either.

[@FM_GOBi]

Good to see you here Jabrils . :)

[@domainofscience]

Yes I agree. I love your drawing style!

[@lucianoosinaga2980]

that actually plato tho

[@pingnick]

🚀🚀🚀🚀🚀🤯🗽🌈☮️💟🎬

[@scienceium5233]

@@domainofscience i am a big fan

[@shawniscoolerthanyou]

I remember thinking about Zeno's paradox in undergrad and it made me think that the fact that we can move around at all is because the infinite sum 1/2^n converges.

[@allenalsop6032]

Well, that and friction.

[@albertrenshaw4252]

1:50 - Xenos paradox is much better represented by saying before you can first travel that half, you must travel half of that.This variant has you never even starting to move, as it shrinks backwards to zero infinitely.

[@carlsanders7824]

That is a brilliant point. Using Zeno's own logic, one could never take a step.

[@MrDiaxus]

@@carlsanders7824 Perhaps that's why he called movement an illusion?

[@christianosminroden7878]

Well, since both representations apply to arbitrary lengths including everything approaching zero, they are basically the same - at least at the limit… 😉

[@salamanca1954]

Which, is, I think, a tautology.

[@thomasreedy4751]

How can you move half the distance that is smaller than the distance taken up by your mass? You can’t. The paradox doesn’t make sense in a quantum world where mass takes up a discrete amount of space. So no paradox - just a poorly defined mathematical approximation of the physical world.

[@dshack4689]

Wonderful explanations. I also liked how at 4:20 an emergency siren is heard approaching (but never quite arriving) at the filming location; it had its limit.

[@irrelevant_noob]

also at 6:25 ^^

[@joebloggsgogglebox]

Zeno's paradox is easily resolved when you remember that movement involves time as well as distance. The statement "you never reach the end" has an implicit assumption of the existence of a time dimension.

[@whoarethebrainpigs]

well it seems like a simple X,Y graph showing distance and velocity. tending to Zero ,,but not getting there,

[@joebloggsgogglebox]

@@whoarethebrainpigs because you are implicitly slowing time down to a halt

[@FerunaLutelou]

Okay, let's say it takes you 2 hours to travel from point A to point B, an hour to travel to reach a halfway point between them, and so on as with a "travel distance" example. This paradox works both with space and time.

[@joebloggsgogglebox]

@@FerunaLutelou Imagine recording a video of a person walking from point A to point B, and then playing back the first 1/2, then the next 1/4, then the next 1/8, etc. We'll never reach the end because of the endless pauses between playbacks, but we could just watch it all in one go with no pauses. This doesn't seem paradoxical to me at all, it's just saying endless pauses never end, a tautology.

[@FerunaLutelou]

@@joebloggsgogglebox there are no real "pauses" in this problem. Really bad example.

[@integza]

The 17 years old version of me would find this video very useful but the now version of me appreciates it even more Shout out to the animations! Love the style

[@poglad1557]

I wish I could have watched this before I learnt calculus. It would have been super useful!

[@mienaikoe]

The now version of integza knows a bunch of cool things to use calculus for!

[@scienceium5233]

yeah the 5 months ago version of me would also find this more useful

[@davidegaruti2582]

Love your work integza

[@inorite4553]

The Engineer in me appreciates it even less after 2 years of math.

[@pbp6741]

It’s wonderful how excited she gets when she imagines we get the right answer to her questions. Limits!

[@crhu319]

It's like we are babies and we actually slid onto the pot before we ...said anything stupid. LOL

[@Jaryism]

Yes, I love being treated like a child..

[@AdrianColley]

"On the Threshold of Infinity" would be a good title for a mathematician's memoirs.

[@TheElectra5000]

Or a short story by H.P. Lovecraft.

[@garymartin9777]

There is already a movie titled "The Man Who Knew Infinity" about Srinivasan Ramanujan.

[@whoarethebrainpigs]

George Cantor...

[@jrock4737]

Like, “something deeply hidden” by -Sean Carroll

[@Phobero]

Or for a prog rock album

[@williamkz]

I love her enthusiasm for learning and knowledge. It's pure joy listening to her.

[@NoirpoolSea]

Disclaimer. It's been over 45 years since I had my calculus classes.. BUT this is the best and most concise introduction to the subject that I've ever seen. It brings together the great "philosophical" (at the time) blockbuster of Zeno's paradox with modern mathematical tools AND keeps showing how the idea of a limit is a tool used in all of these. GENIUS!! It's like turning math into LEGO bricks! I know someone who definitely doesn't know higher maths and will be trying this out on them.

[@allenalsop6032]

Great attitude but calculus is still in the realm of elementary math. And unless you say things like "5 deers and 8 geeses and 4 mices....math is already both singular and plural. "Maths", was just a mistake someone made that caught on with those who didn't know any better and then spread to others of the same ilk.

[@kanjakan]

@@allenalsop6032 Starting a reply with a condescending remark? Check. Going on to nitpick a trivial "mistake" (it's not) that makes no difference to anything whatsoever? Check. What a sad person you must be.

[@maggyfrog]

for every step, you are moving through an infinite number of half steps, which is a subset of the bigger infinity that is the total distance you will walk, therefore, you are definitely moving and not just frozen at any point.

[@alexjohnson2527]

I really wish I saw this before taking cal 1 and 2. You explained the real-life implications behind the mathematics so much better than my professors. Makes it a lot easier to grasp what exactly we are calculating in real-life. Well done!

[@anujarora0]

What a wonderful day! You and Kurzgesagt uploaded on the same day.

[@Rocinster]

I'll toast to that bro

[@gauravpallod4768]

YES!

[@samkes1061]

Ted Ed and Action lab also

[@vmitsi]

Years of university maths and I never fully grasped what I was calculating. Thank you for giving real life examples and making this super simple to understand!

[@allenalsop6032]

And unless you say things like "5 deers and 8 geeses and 4 mices....math is already both singular and plural. "Maths", was just a mistake someone made that caught on with those who didn't know any better and then spread to others of the same ilk. It is basically baby talk similar to "minus it", "plus it" and "times it". not to mention the grammar faux pas. (or is it faux pases there? lol)

[@dagwoodsystems]

I love that you're so passionate and determined to teach me something that you totally ignore the sirens in the background. Awesome work. New subscriber!

[@Anuchan]

In a future video you could have police breaking into your house to arrest you, but you keep talking faster before they take you out of the room. Love your videos!

[@riersonjohnson1547]

Impressed at how you managed to get footage of a nerf dart mid-flight that wasn't blurry

[@upandatom]

thanks it was so hard lol

[@IvanKalamazoo55]

she borrowed the camera from the SlowMo guys)

[@nigeldepledge3790]

If you look closely, you can see a tiny bit of motion blur....

[@bokkenka]

@@nigeldepledge3790 I thought you were going to say 'you can see a tiny bit of thread'.

[@nigeldepledge3790]

@@bokkenka LOL

[@piratatazmania]

As an engineer, I have to say that I love the way you talk. Usually when people explain this things, they do it with serious and deep voices, which subconsciously makes people think that this are serious, deep and difficult topics. You on the other hand, explain this things in such a cheerful and happy manner, that it makes me feel that this are cheerful topics, that we should be happy learning about this things. Please don't change!!!!!!!!

[@tuckergary1516]

75 yr old engineer has that same curse. "gotta make it better."

[@nnaammuuss]

"these".

[@veerleswartebroekx6816]

Hey but calculus (and math and physics and cosmology and chemistry) ARE all cheerful and wonderfully interesting topics! My mom introduced me to these ideas when I was just a 3 year old toddler and I kept asking about concepts like infinity as it was so hard to wrap my head around it. I still feel the same wonder now 45 years later.

[@tigrayrimey6418]

oh, brother, I cant tell you how hard I laugh reading ur comments though it make sense

[@davidh6300]

Yes!

[@KhAnubis]

8:48 When someone says that they don't like physics

[@mreese8764]

She didn't even blink. Terminator style.

[@gabrielamba2151]

Kaget

[@xtramoist9999]

Anyone else duck? No? Just me? 😒

[@gnnvoauekes6320]

For you it might be Berlin

[@bethmarsicano7574]

I kid you not. Years ago I performed in a play by Aristophanes. It's a Greek anti-War play "Lysistrada" and the men are involved in the Peloponnesian war. The women decide that withholding sex will bring them home and end the war. Since the men could not actually set fire on the stage and we (the females) had seized the Acropolis... How you ask? Nerf guns. Just like you see here. The men had Nerf guns too. It wasn't 411 B.C.E. but it was the 1990's and not taken out of period. I was a scientist before and after a few years returned to my roots. Nerf guns are educational; math and literature.

[@peanutnozone]

As a life-long lover of calculus who sometimes had trouble grasping the central themes, you were able to help me visualize why integrals, derivatives and the limit actually work together to actively solve the problems I could arrive at the answer to, but not understand in an intuitive sense why it worked or the underlying logic. Thank you thank you thank you, you have re-ignited my love and admiration for mathematics!

[@robertglas5854]

Finally, after decades, someone explains this in a clear, simple and great way! Thank you!

[@DavidCarter-ib3vw]

Same here. I scored a B in 4 courses of Calculus and then a B in differential equations. I could solve the problems but never understood why calculus works. I lay golf so I used calculus to figure the angle the ball must take from the tee for maximum distance. It is 45 degrees which seems intuitive. However, you font know unto you know. There’s is a highly successful pro golfer that says all his success is based on physics and math ( don’t know his name because I never watch pro anything.

[@jasonf8910]

Great job! Finally a good 5th grade level explanation of calculus I can show my nephews!

[@davidchidester5463]

This would have been so helpful while trying to wrap my head around calculus in school. Limits are one of those things that make no sense up until the moment it clicks. Then you look back and can't understand how you couldn't understand it before.

[@johnmcgee6297]

I was thinking how useful watching a few hours of instruction in this format from different personalities (especially hers) would have helped me in school.

[@PeterManger]

Awesome “why” video. Whenever teaching, I always think students aren’t given enough “but why am I learning this”. And sirens made me look out my window! Twice!

[@nickromo8195]

I don't know why no one else is talking about the sirens..I wonder if they were on purpose or just happen to be going by while she was filming

[@BrightBlueJim]

And I just then realized why sirens are called that. It's because they distract you from your goals!

[@sherylbegby]

I like your explanation. I also like thinking about it like this: Find how far you have come, in stades, after each step: Step 1: 1/2 stade Step 2: 1/2 + 1/4 = 3/4 stades; Step 3: 3/4 + 1/8 = 7/8 stades; Step 4: 7/8 + 1/16 = 15/16 stades. So we get the next number in the series by adding half the remaining distance to 1 each time The pattern becomes 15/16, 31/32, 63/64, 127/128 etc. This is just (2^n - 1)/2^n, which approaches 1 as n --> ∞ I also like the explanation I heard in a German school that because our eyes tell us the arrow/runner does IN FACT reach the finishing line/ target, the sums of the ever-decreasing distances must indeed add up to 100% of the total distance (1 stade). Lovely video! Thank you

[@boredpwner]

My calculus professor boasted a 50% fail rate for his Calculus 101 class at my college. I was a Biology major and I barely passed, but I would have salvaged my GPA if I had seen this video back then. Math is fun when it is taught by someone who loves not only the content, but also the TEACHING aspect. You are a phenomenal teacher, and I’m glad I can still learn complex topics outside of school 😊

[@Auoric]

I'm an engineering freshman and this really helped me understand the concept of calculus. Thank you!

[@jsmith294]

If you are an engineering freshman without an understanding of calculus... time to start thinking about a career in social work.

[@hectornonayurbusiness2631]

@@jsmith294 you only really need calculus in college not so much in industry. Quit being a jerk.

[@c.LOSER.to.G0.d]

@@jsmith294 y so rude?

[@twenty-fifth420]

Up and Atom: "I couldn't find a bow and arrow so I am using a nerf gun instead." Me: "Modern problems require modern solutions."

[@jimdecamp7204]

I think they were going a little too PC. It is a lot easier and safer to tape a Nerf gun, too.

[@irrelevant_noob]

"modern" problems? o.O

[@charlesdye8367]

KZbin's geeky goddess. Thanks for getting my confusion as close as possible to zero yet never, ever actually getting me there. Expanding my limits!

[@aljoschalong625]

When I heard Zeno's paradox for the first time as a child, my first thought was: That's silly. He's looking at increasingly smaler times. So all he's saying is: When time stands still there is no movement.

[@KenJackson_US]

I was thinking he could easily solve it by setting the mathematical finish line beyond the actual finish line.

[@murilovsilva]

I have to say, the passion with which you talk about mathematics is truly remarkable. You obviously don't simply know what you are talking about, you love it, and that is the mark of the true professional. It's a rare trait in people who teach for a living, and considerably rarer in people who teach math. You've definitely earned my like and my subscription. Kudos! Also, after watching so many science and tech videos on KZbin that advertise Curiosity Stream and Nebula, you were the first presenter who made me interested in actually checking them out.

[@superlukey3]

Great introduction to Calculus! Definitely worth showing to prospective math students.

[@anujarora0]

An engineer, a physicist, and a mathematician are at a bar and see a beautiful woman across the room. They're all too nervous to talk to her so the physicist devises a plan to work up the necessary courage. Walk half the distance from them to her, then half the remaining distance, and again, and again, and again. The mathematician says it won't work because they will never actually get to her. The engineer says, "Well, it's close enough for practical purposes."

[@AB-ee5tb]

😂😂😂😂😂

[@xpqr12345]

A physicist, a biologist and a mathematician were in a bar, looking at the empty house across the street. After a short while, a man and a woman enter the house, coming out an hour later with a baby. The physicist comments: "The house can't have been empty from the beginning!" The biologist replies: "They must have reproduced!" The mathematician states the obvious: "If exactly one person enters the house, it will be empty again!"

[@le_science4all]

Great video!! Infinity is cool 😁 Also, I meant to say that my favorite drawing is the one on the left 😋#Entropy

[@HoD999x]

ah, mr. laplace/bayes :D

[@mikebauer6917]

Paradox via philosophical oversimplification. Zeno’s mistake is measuring continuous distance, when in reality a runner takes discrete strides so at some point the runner gets with a single stride of the finish and does so. No mystery or paradox. Basically, Zeno assumed some sort of wacky world where the runner’s stride shrinks so that they can never span the remaining distance.

[@cuebj]

9:40. Actually, the film is moving through the projector gate, the paper moves using the spring in the paper and the work done by your finger. Energy is imparted into both those examples of multiple still images

[@samuelowens000]

This is the first time the "sponsored by nebula" thing has actually gotten me to want to get it. Great video as always!

[@ben07643]

same!

[@NL2500]

Same, but they only accept credit cards. Even though Google play on their app, Paypal is not accepted. It is all very limited

[@RadeticDaniel]

Knowing Atom here and Tom Scott are there teased me, but hell, say Hannah Fry and I know there's gold beyond KZbin depth =)

[@lukelutio1246]

Your enthusiasm and ways of presenting challenges is heart warming and makes me reconsider my math phobia :-D

[@rv706]

Wait, no. "Calculus" comes from the Latin word for "small stone" ( _calculus_ , _-i_ ) because small stones were the most elementary object used as an abacus in ancient times. Or at least so we were taught in (Italian) high school! This is supported, for example, by the fact that in many romance languages a word directly derived from "calculus" is used to mean "computation" (Italian _calcolo_ , French _calcule_ , Spanish _cálculo_ , Romanian _calcul_ ). The "breaking a big problem into smaller things" hypothesis for the etymology seems particularly weird to me, also given that the term "calculus" originates from the ancient Romans, not from the first users/inventors of infinitesimal analysis, centuries later. Or do you have some evidence for your guess? - Anyway, you have a very lovely and enjoyable channel!

[@craigwall9536]

She's just shallow and quoting shallow references. Probably NPR.

[@Alkis05]

As anyone can imagine, calculus as a word already existed. Actually, originally the field was called infinitesimal calculus, calculus for short nowadays, since we don't speak latin anymore and have the word calculation. Calculus just means calculations with infinitesimals.

[@vampiresquid]

It's not an etymological discussion. "Calculus" in modern times is used to refer to the branch of mathematics based on limits, including differential and integral calculus. What the word meant 2000 years ago is no more relevant to the discussion than the fact that the same word describes the crap the dentist scrapes off your teeth.

[@rv706]

@@vampiresquid: yes but, if I remember well this video that I watched ten months ago, she was talking about the reasons why the word "calculus" is used to refer to infinitesimal analysis and she said "because it's a Latin word which means 'to break down into smaller pieces' " . But that's not the meaning, so...

[@delaware137]

Once you reached the first 1/2 Stade mark, you've already traveled an infinite number of fractional stades to get there.

[@elihu6037]

o.0

[@elihu6037]

nice

[@scottfineshriber5051]

Oooooh! That’s deep. 😁👍

[@NullPointerException7]

you need more subscribers!! your videos are always so fun to watch!

[@MemphiStig]

Zeno's mistake, of course, was that *steps* *don't* *work* *that* *way* even so, he changed everything just by asking a simple question

[@definesigint2823]

What usually bothers me about this paradox isn't the steps, but the omission of time.

[@noconaroubideaux9423]

@@definesigint2823 I'm more fucked up that he doesn't seem to know how mass works.

[@iamtrash288]

@@noconaroubideaux9423 Do you? I'm pretty sure that humanity itself didn't even have an idea of what mass (i.e. the numerical measurement of a body's inertia) was until a rather short period of time before

[@iamtrash288]

Well, you move your foot, at some point in time it moves over all those halves and quarters etc, so they actually do work that way though?

[@noconaroubideaux9423]

@@iamtrash288 I don't mean mass in the sense physics describes. I mean that, eventually, if you keep breaking distance in half, your foot is gonna end up being at two of those different locations at the same time and it will continue to step on an expanding number of distances as those distances become smaller until it crosses the point of measurement. The only way this thought experiment could work is if you were actually shrinking continuously with every step like a Mandelbrot Set to compensate for shrinking half distances.

[@Skibbityboo0580]

Oh my god. You're the first person that made it click for me!! Now I actually want to go learn how to do some of this!

[@joshuaadamstithakayoutubel2490]

3Blue1Brown also explains Calculus.

[@hoptanglishalive4156]

This is like how I approach women but never actually reach them. I guess I’m just limited.

[@David_Last_Name]

I too enjoy studying the area under their curves.

[@HHHHHH-kj1dg]

@@David_Last_Name lol

[@NormalLunk]

@@David_Last_Name r/cursed comments

[@HHHHHH-kj1dg]

@@NormalLunk Not cursed. Hot.

[@jg-reis]

According to Zeno, each time you try to prove yourself with a lady, you're putting in half the effort you put in last time... this way, *of course* you won't get there. Just give it your all. Do it man. Infinity awaits you. Be happy! ^___^

[@workinalday4351]

Zippy Greek Dude in the bed sheet had this backwards "AKA Zeno's dichotomy paradox", He made it complicated. Always look at this problem with a Beginning and a End. Always a Start and a Finish point. Take one unit of measurement from start to finish, Label it whatever you want to call it. Then when you start to "half the problem" you do it with a defined set each time. Hence the Limit. When doing this you "Don't" look at the problem as 1 endless or continuously chopped up set of numbers getting larger. You look at it as "Sets of Fractions" that get applied in a "Predefined Measurement", and Each new "Set of Fractions" gets smaller. Example - Set of fractions 1 takes you from 1 to 100 or from Start to Finish. Set of fractions 2 smaller by 1 half takes you from 1 to 100 or Start to Finish also, however they are smaller by 1 half so you need to use more of them. You repeat this process with however many new sets of fractions you would like to use in this "predefined measurement". You always go from 1 to 100 or Start to Finish, and your measurement never increases. This can be applied to all manor of measurements not just distance and is the proper way to address this. So where would Zippy Greek Dude in the Bed Sheet's Paradox come into play? We would apply this if building a model to make sure we were building the model correctly, and we would use Zippy Greek Dude in the Bed Sheet's Paradox as a method to test for correctness.

[@CristobalHernandez]

This is the best video I've seen in calculus. When I saw these subjects, I always asked how they came up or thought of these solutions. Very good.

[@HoD999x]

how about this: the smaller the steps get, the faster you traverse them. so, you become infinitely fast and can get to the end of infinity this way in finite time.

[@adamlea6339]

Yes, if you halve the steps, you halve the time taken to make those steps, and the infinite sum of those times converges to the time taken to cover the full distance. Since time moves at a constant rate, you cover the distance in the time equivalent to the infinite sum of times taken to cover all half segments. If it took the same time to traverse each half segment i.e. your speed halved each time, it would take infinitely long to traverse the distance. The point is the paradox would be complex to get your head around if the concept of mathematical limits has not yet been established.

[@definesigint2823]

@@darbyl3872 The animator apparently suspects this. 🤪

[@olas9400]

@@darbyl3872 yet you can't explain it without needing a "definition fix". Let's see you explain it in those terms in order to get to the Truth. That's the issue. Logic can't get you there, approximations must be made. Hence, the birth of conventions *instead* of truth.

[@olas9400]

And no, your speed doesn't change if how you divide space changes. Proportions however... (Your foot would be 10ⁿ compared to the miniscule divisions) But your foot would also be divisable... Again, the confusion comes from the half-baked portrayal of the philosophical thought. Zeno's aim is to say: everything is one. (That's his main premise/theory.) Division, happenings, etc, are therefore just facets of the whole. Taken in isolation, they produce paradoxes, are distractions if you wish. A more crafty, statistical, scientific way to say this is to say scale and dimensionality of interactions matters. Studying x in isolation from the rest, yields a considerably different model than what we observe in nature.

[@leotamer5]

The reason Zeno's paradox does not work is firstly, because we are not moving half-way to any given point, we are moving towards the point and as we are, we are walking past arbitrary points between our original point and our destination. While Zeno points out we move through the 1/2 and 1/4 points, we are also moving through the 1/3th and 1/6th points. Does the fact that we now account for walking through all of the 1/3x points now increase the destination we travelled? So the first thing he does is redefine movement using arbitrary points. But lets grant him this, but rephrase it as the turtle in a room moving from one side to the other in this manner. It would take it infinity steps to reach the other side, if we ignore that the turtle has a length itself. If we place the movement at the furthest point back to see how far he needs to travel, then the turtle will only need as many steps to reduce the distance to the further wall to being less than its own length. If the marker if half-way up the turtle, then it needs to travel until the distance is half its length. And if you want to get the marker as far up one on the turtle as possible, then you run into a zeno's paradox within a zeno's paradox. Zeno's paradox raises interesting questions about infinity that can be explored mathematically, but do not raise interesting questions about reality because it one preposes that objects don't have length and that there isn't any interesting in saying that if you walk X units, you have also walked 1/3 + 1/3 + 1/3 units, which is what he is effectively saying but making the expanded form of X infinitely long.

[@josephcafariello365]

Very well explained. Well done. Let me consider the first paradox covered, Zeno's. The fault with Zeno's logic is that he assumes all we move is 1/2 of one segment each time, stopping at each 1/2 segment. If we did that, then of course we would never reach the end, because we would be shortening our step each time. Toward the end, our paces would shrink down to 1 cm per step, then the next step would be 0.5 cm, then our next step would be 0.25 cm, etc. If we did that (shorten our step to cover just 1/2 of the remaining distance), then yes, we would never reach the finish line. But in reality, we do not keep shrinking our steps by 1/2 each step we take. We simply take as many steps as we need to cover the total distance. When we add up the distance of each step, we end up covering the entire distance to the line and even cross over it. Why do we reach the finish line? Because the sum of the lengths of all steps equal more than the distance to the finish line. We are not shrinking our steps and stopping at each 1/2 point. But why is Zeno even proposing this? He presents a ridiculous problem. Ultimately, all he is saying is, "If you reduce your stride by 1/2 each step of the way, you will never reach the finish line." We would then say to Zeno, "Of course, you weirdo. But why would you reduce your stride by 1/2 each step of the way?" In truth, we reach the finish line every time because we don't reduce our strides by 1/2 each step of the way. We simply add up the distance of each stride, and after enough steps we surpass the distance to the finish line and cross over it. To Zeno I say, "Zeno, motion is no illusion. Your sense of superior intellect is."

[@solapowsj25]

Sums up as: the greater the number of steps 👣tending to infinite, the smaller the error tending to zero 0️⃣.

[@wolfy1597]

I've been waiting for ages now jade. But ur vids r worth the wait hehe😄😄😁😁

[@upandatom]

Yay! Thank you!

[@swayamprakashkar9664]

@@upandatom You deserve it! 😊

[@Trisha_an_Artist]

In school this year we learn integrals, and last year started derivatives, and no teacher could explain those concepts better than you. Thanks

[@slateslavens]

2:51 Once the distance to the next halfway point is smaller than a human step, the remaining half-way points are skipped.

[@darklink539]

A lovely lecture from a lovely teacher. Thank you!

[@dibenp]

Why can’t I like this video twice??? 😭 It’s so good!

[@fredfrancium]

Create other Google account, then like it :)

[@hugodiniz6814]

As a high schooler that wants to be a physicist, this helped me A LOT.

[@HHHHHH-kj1dg]

Me too buddy. Which grade are you in?

[@hugodiniz6814]

@@HHHHHH-kj1dg (Bad english) Last year, according to the educational system of Brazil. Converting, it’s probably 12th grade.

[@cookcodingalliance786]

This is one of the best description of the basic principals of Calculus I have seen. Since it is a public video, I am sharing it with my students. Thank you.

[@thecarman3693]

I believe Zeno's 'Arrow Paradox' is not about the arrow's speed at a given instant, but rather what is it that keeps the arrow in motion at an instant in time as compared to one at rest. In other words, what's different about the Nerf bullet at time 9:15 compared to one just sitting on a table? One might think it's a crude introduction to the concept of momentum, but actually it can better be explained by Special Relativity and Lorentz transformations. The two bullets actually ARE physically different. Zeno was onto something after all.

[@gauravpallod4768]

"I aspire to be her one day" YES! Hannay Fry is such an inspiration!!

[@1_glucose_biscuit_lifetime564]

This is why I love KZbin 🔥

[@cerjmedia]

Ah yes, the memories of College from last year and high school back when calculus was simple Calculus 3+ can be rather difficult

[@inorite4553]

Calc 3 was easy. Calc 2 sucked when taking it but is rather simple to explain now because it's all experience based. You just have to do the work and get exposed to all the different techniques. Same goes for Differential Equations. Calc 1, unfortunately, is rarely ever taught appropriately as there is too much time focusing on limits when the rules of derivation and the concepts behind not "how to derivate but why we derivate" need more focus.

[@cerjmedia]

@@inorite4553 I thought Calc 4 was easier than Calc 3 to be honest

[@MrLucasbear]

This is probably the most concise, least confusing way I've seen calculus described, ever. Thank you for the video, and thank you for the time it must have taken you to make it. Side note: I have a little tool I use for tutoring middle/high schooler's, traditional college freshmen, or adult learners picking up basic algebra to get them to be less afraid/intimidated of actually trying calculus. Maybe it'll help other people. I start with x^5 at the top of the paper, then I write 5x^4 underneath that, and 20x^3 underneath that. I say I'm building a ladder, and I'm missing a rung, so I need their help trying to figure out what my next step down would be. Most people can figure out what 20*3 is in their head, but I normally have a little dinky dollar store Casio handy for them. When they say 60x^2 I always encourage them and tell them it's correct WITHOUT GIVING IT AWAY YET. Then I flip the paper over and tell them "I need help going up the ladder now". I'll write 24x on the bottom of the paper, 12x^2 above that, then 4x^3 above that and ask for the same help figuring out what the top rung is. When they tell me x^4 I completely lose my shit and excitedly howl something along the lines of "YES THIS IS CALCULUS YOU'RE DOING CALCULUS YOU KNEW HOW TO FUCKING DO IT THE WHOLE TIME AND YOU DIDN'T EVEN REALIZE IT BWAHAHAHAHA!" ((Note: I will occasionally omit the curse word when I'm forced to interact with actual children))

[@erictheodorecartman9869]

10:45 That is not quite right, but often repeated. True is that you get as close as you want to the "actual speed" by shrinking the interval further, but you don't get necassary closer to the actual speed by shrinking the invervall. You can even have a large interval and yet the slope of the secant matches the slope of the tangent exactly.

[@ankandatta6683]

this is the best video for understanding the derivatives. When you said 'TAKE THE LIMIT' there was a serious Dora the explorer vibes.

[@inorite4553]

I've always found limits to be something you need to understand but quite useless for applied calculus. Teaching people "Why" Calculus is used in certain circumstances is more important.

[@wizzz8073]

You are so underated!

[@RockHudrock]

Or is she? 🧐 (She’s rated ∞)

[@Lucky10279]

I wish this video was out when I was learning infinite series in calc 2.

[@Alkis05]

I will go in a wild guess here and say you didn't do so well in calc 1.

[@MrArtVandelay]

In 1969 my Calculus teacher taught us Xenos paradox by describing the erasing of the black board, which always could be halved.

[@rdmcampus2545]

Very well built up explanation of calculus paradoxes, nice video animation and a enthousiast and inspiring presentation. Wonderful.

[@HoD999x]

zeno's nerf gun: i imagine everything as teleporting very small distances. problem solved.

[@juzoli]

But how often is it teleportig how small distances?

[@martinklopnof5284]

@@juzoli Planck units per unit of Planck time, I'm guessing.

[@HoD999x]

@@juzoli does it really matter? think about it... whatever you pick as the minimum distance will be our plank size

[@inorite4553]

It's easy to explain if you don't care about exactness and are ok with "ehh....close enough." The answer would be, there is no zero. Zero is a human concept used to power our invented language of Mathematics. Aaaaaaaaand we're done here. The natural world doesn't care about our concepts. It just is.

[@juzoli]

@@inorite4553 I have zero elephants in my garden. Here is a natural example of zero for you.

[@albertliu1068]

I would have been proficient also with my maths if I had a maths teacher like her in my high school years!

[@UhOhTheStoveIsOn]

Just travel 2 stades and only go to the half way point. Take that Zeno

[@RadeticDaniel]

You do realize his argument also enjoys scaling the other way around, right? His main argument could be reverted to before getting to the midpoint, you have to walk half way there instead of the presented version of after walking half way there is still another mid point . The "before you get there" version is the one yielding the conclusion that you could never move anywhere presented in the video

[@UhOhTheStoveIsOn]

@@RadeticDaniel Bruh

[@johnmcclain3887]

I'm more than a bit dyslexic, and every time I see this, I go back to how I did it in high school, putting triangles in the curves, and doing the individual area from them, and arguing with the teacher I got a more accurate total. I always lost, but always got the steel to the right shape and size. I make sense of it in my head, it's rational, logical, I get lost twisting the numbers, doing it. Very well demonstrated, and with Greek, that I saw at about five, in Greece. Thanks very much.

[@vandresv]

I have been always fascinated by the first paradox. On my mind, the limit is just an extremely useful tool, but in reality the reason we do reach the end point is because the quantum nature of the world, like energy, space is also divide in small segments, so when we move towards the end point, there is a moment where there is no way to split the remaining gap in two segments anymore, there is nothing to be divided.

[@jimnonya9180]

Not that I agree with this theory but your comment is flawed. No matter how small a fraction is, it can always be divided again, In reality, the last step you take would cover an infinite amount of those HALF fractions. So the whole idea/theory is flawed.

[@officialspock]

"Small, really small" - that's what she said 😥

[@pvic6959]

i didnt need to be attacked like this today

[@toaster4693]

And then she broke up with you by saying "I'm making a you-substitution."

[@milkmanconspiracy4346]

These animations never fail to make me laugh, I love their crudeness lol.

[@MoempfLP]

i was today years old when i found out that "lim" in mathematics stands for limit.

[@fresnik]

Well, we can't all know everything all the time. Consider yourself one of the 10,000 lucky today (google if you don't get the reference)

[@franciscofarias6385]

lol

[@_okedata]

it's really sad how school teaches math(s) in a that this can happen. limits are literally what infitesimal calculus is built-on, but school focuses on computation so 'lim' is just the thing you have to write there so its right.

[@matthewluecke3704]

When I take notes by hand, I abbreviate limit as lim (like speed limit) and I still draw the arrow under the lim.

[@inorite4553]

And now I"m about to blow your mind, if you have no intentions of getting a PhD or going into education, you'll never use it ever again.

[@aouerfelli]

How about this definition of motion: An object is said to be moving at the instant t if and only if any given neighbourhood* of t contains another instant t' at which the object is located in another location than it is at the instant t. And conversely, an object is said to be still (not moving) at an instant t if there is a neighbourhood* of t in which the object is in the same location all along. *A neighbourhood of an instant t is an interval of time containing t such that t is not one of its bounds. (This is not the topological definition of neighbourhood but it is similar)

[@yash1152]

yeah, well, this kinda is indeed the definition of velocity ≠ 0, or = 0.

[@saliso470]

For Zeno's Paradox you can also just draw the lines with 1/2t + 1 and x which cross at t=2. Doesn't even need a limit.

[@nithyadavuluri7287]

Both kurzgesagt and up&atom uploaded today.. lottery!

[@jackhandma1011]

Math is both an invention and a discovery. Proof: Terence Tao says so, QED.

[@VidhathShetty]

It got discovered right before it's invention?

[@c4stmiranda902]

Lets just say we're using it before notations were invented

[@tevildo9383]

More precisely, it’s an abstract, arbitrary system we use to quantify and conceptualize real, observed facts.

[@RobbofromCronulla]

It would be easier to turn back the tide with a tea cup than for the human mind to comprehend infinity. Oh, and way to rock that Flanno.

[@MakeDataUseful]

Thank you! I have been learning neural networks and was so happy to see you explain the derivative so clearly.

[@traildude7538]

When I first took calculus it took me and the whole class a while to get the humor of our professor who had background music playing when class started: the whole first week he had "Take it to the limit" as the theme piece.

[@nickromo8195]

Was it always actual songs like that or did he ever just play like piano music/instrumentals..? I would actually really likethat but I feel like actual songs with words would get distracting quick

[@Niv0505]

Literally just explained calculus in 10 minutes...

[@Renato404]

Lol, Zeno's interlocutores be like: - yeah, you lost me there around one of those half ways...

[@VK-sp4gv]

"Ghosts of departed quantities"

[@katczar]

Seriously Jade. I love your education ways

[@Roberto-REME]

Great video and you are a excellent narrator. You explained the subject matter with expertise.

[@patrickfreeman9094]

Zeno did not consider the size of his feet in relation to 1 over X...

[@inorite4553]

Spoken like an Engineer and not a Mathematician. ;-)

[@patrickfreeman9094]

Thanks for calling me an engineer. I call it logic.

[@noconaroubideaux9423]

@@patrickfreeman9094 Yea, I came to the same conclusion but I think the issue here is that Zeno assumes motion is relative to distance when motion is only relative to the energy used to move a certain distance. Since he is not taking energy into consideration, he isn't taking mass into consideration either [E=MC(2)]. When applied to mass, this concept is actually really useful and I think the best example is a Mandelbrot Set.

[@RadeticDaniel]

His original argument used a turtle in a confined space to get time out of peoples mind and step size somewhat dismissable. It is also part of his train of thought an inclinations to take always smaller things like bugs and the likings. Dude was trying to play abstract physics before symbolic algebra was invented and found an infinite process with finite outcome. Yet he had no ideia how to explain the case or how to work it out to match other models of the same movement. Any size is dismissable if you go down all the way to a grain of sand and beyond... his foot didn't matter =)

[@ang_gml]

Never closed snapchat that fast

[@PlayTheMind]

42, -1/12, 2020

[@SilhSe]

🤣🤣

[@Dragrath1]

the answer to the unknown question and summing those infinite series huh you thinking you got the question by the end of the year?

[@chrisschene8301]

Wonderful job explaining Calculus. I am an electrical engineer and I speak calculus now,. but I wish I had a teacher like you when I was first learning

[@chiragadwani1875]

In my undergraduate math, while we did talk about sequences “in the limit”, the main focus there was on arbitrariness. For instance, it would be claimed that Sn, as in the first example of this video, limits to 1 if we can get a tail of the sequence arbitrarily close to 1. In other words, given any arbitrary distance e between Sn and 1, there is a number M such that |Sn-1| M. Though this way of putting it obviously has been designed keeping in mind the large N limit, it manages to escape ANY mention of N->inf or anything like that. Which I find to be pretty neat. Perhaps the reason I find it cool is because N->inf never made 100% sense to me. I understood what it was trying to say but I couldn’t ever think of being able to tame it, just like it’s possible for the above definition.

[@RockHudrock]

She’s even beautiful in flannel?! How is that possible?! 🥰

[@x_abyss]

4:20 The limit police approachs 😂

[@samarpanbiswas7474]

Absolutely Gorgeous!

[@upandatom]

no you :)

[@leif1075]

@@upandatom Hey Jade, I hope you can respond to my question about quantum tunneling when you get a chance. Thanks very much.

[@Warhawk76]

You are absolutely correct. And she is brilliant too, which is even more awesome. I think I am finally starting to understand this math!

[@fireheart072]

By far the most elegant and comprehensible explanation of calculus. It was utterly clear and made complete sense. You have taught me in 12 minutes what a teacher couldn't in one year. I am amazed. Well done. Thank you.

[@antonnym214]

OR , for Zeno... Say your goal is a distance of one stade. Set a point beyond that (at any arbitrary distance beyond, such as 1/20th of a stade) as a "pseudogoal". Then make your continued fractions until you pass the original, actual goal. All good wishes!