Imagine teaching fractions in elementary school and a kid says "I'm not doing addition wrong, I'm computing the median" and then explains everything from this video.

@cleats727

7 ай бұрын

That's some Terry Tao shit

@alonelyphoenix8942

7 ай бұрын

Bro's believes hes Gauss

@canyoupoop

4 ай бұрын

​@@alonelyphoenix8942 “By the time Gauss was 7 years old, his schoolmasters admitted that there was nothing more they could teach the boy.” ~David Burton, Elementary Number Theory

Even if the other entries in SoME3 are incredible, this deserves at the very least an honourable mention.

@snowfloofcathug

7 ай бұрын

Good news!

i was the apprenticeship instructor for the roofing program in calgary for 13 yrs... my most fulfilling memory was being able to teach fractions and the metric system to 40 yr old roofers with learning disabilities, addiction problems and 'incomplete' education scenarios... seeing the look on someone's face when they actually get it and feel good about themselves...i was so blessed to help...

Simpson's paradox seemingly exists because of the disconnect between percentage and absolute values. To make the paradox more intuitive you can use an analogy: you can state that a cup of vodka is 50% alcohol and wins over 5,000,000 barrels of 18% alcohol wine, but if you add a gallon of water with .0001% alcohol to each the wine would "win." Basically the absolute value of how many objects you are working with gets 'lost' or 'obfuscated' or 'untracked' when you mix the two data sets, if stated in a slightly better way it would become obvious that small amounts of data are more susceptible to outliers than large amounts of data.

@RibusPQR

10 ай бұрын

You can even add a gallon of "winning" water with 2% alcohol to the vodka, and a gallon of "losing" water with 1% alcohol to the wine.

@chillbro1010

10 ай бұрын

@@RibusPQR I hesitated to split the second set into two due to making the "simple" explanation too hard to read but the major take away is that the SIZE of all the different sets is a large reason why this works. But you do make a good point that the actual "paradox" comes from adding winners vs adding losers. ---- On a side node, in my example if you added 1 trillion barrels of 2% alcohol and 1% alcohol the vodka would go back to winning. Gosh this entire area of math just doesnt want to cooperate.

@peceed

7 ай бұрын

And, as abstinent, I can always redefine the"win" and the "lost" :P

@monhi64

7 ай бұрын

Not to try and roast ya but I was unfamiliar with simpsons paradox until this vid and it took me so gd long to figure out what you meant. Honestly still not even there, I thought you meant that the vodka wins because it’s a higher percentage of alcohol. And that it changes when you add the gallon of .0001% water to the wine because maybe there’s more gross alcohol in a gallon of low alcohol content than one cup high. But that still doesn’t make any sense because one hand is measuring gross and the other percentage. How exactly are you determining who wins, and why is the vodka just a cup and the wine a barrel (55 gallons I guess)? Funnily enough I understand simpsons paradox (looked into it separately) but I can not for the life of me figure out how it relates to your example lol

@apotatoman4862

6 ай бұрын

@@monhi64 the wine wins in alcohol percentage because the wine is 5,000,000 barrels so adding a gallon of water doesnt change the percentage of alcohol in the wine much but the vodka is a cup so adding a gallon of water dilutes it more so the percentage of alcohol in the vodka ends up being lower than the wine even though the vodka begins with higher percentage

If ad-bc=1, you can use Pick's Theorem to prove a bunch of properties of the mediant. The parallelogram is a lattice polygon with area 1, and we know 4 points on its border. Since the area is the number of interior points plus half the border points minus 1, there must not be any points in the interior and no more points on the border, so there cannot be any rational numbers that would fall in this region or on its border.

@philipppremium5191

7 ай бұрын

Strongly agree

@anon1963

6 ай бұрын

what a load of bs. is this what most PhDs in meth do?

@kg29486

6 ай бұрын

Yes yes definitely agreed

Finally someone talked about this ! Last year I rediscovered most of this, in a attempt to find an algorithm that converts computer floating point numbers into a ratio, without suffering from the precision loss of floating point arithmetic. I couldn't find anything about this on the internet, until a friend of mine did. I'm happy more people learn about this simple but very interesting maths concept !

@siddanthvenkatesh2744

10 ай бұрын

Just wondering, did you do project Euler? If so then for that problem I think what I did was if the |ratio - floating point|

@Nolord_

10 ай бұрын

@@siddanthvenkatesh2744 I don't know about project Euler, but yes you avoid any operations on floating numbers to avoid losing precision, and just use comparisons in order to know if your ratio is close.

@Hextator

10 ай бұрын

That was the first thing I thought of when thinking of a fun application for these concepts. A friend of mine actually fiddled around with similar ways of representing floats after I told them that the only way to never lose any precision was to always preserve the full chain of operations that occur when some variable becomes imprecise, then recalculate values on the fly with the desired precision when retrieving them in the future. I believe they did end up representing floats as ratios of VLIs, but I'm not sure how far they got in terms of being able to achieve any desired precision.

@__christopher__

10 ай бұрын

Actually floating point numbers are alreaady exact rational numbers; imprecision comes in only when you try to calculate a number that cannot be exactly represented as floating point value. However when converting to a fraction, that loss alredy happened, and if you convert to anything but the exact value of that float, you are going to adda second imprecision on top of it. That second imprecision *may* cancel out the first, but unless you take into account knowledge on how you arrived at it, generally it will not. Indeed, every floating point number (except for infinities and NaN, of course) can be written exactly as m*2^n where m and n are integers where m can be derived from the sign and mantissa bits, and n can be derived from the exponent bits (with some special handling for denormalized numbers; those however are also of the form above, just the formulas for m and n are slightly different). Of course in that representation you lose the signed zero (unless you store m in a signed-magnitude or 1-complement integer representation, which also has signed zero), but then, mathematically 0 = -0 anyway..

@Nolord_

10 ай бұрын

@@__christopher__ Floating point numbers aren't exact rational numbers because of infinite (binary) decimals that can't be represented. And in my algorithm for the conversion, you define the converted number as the one that if you convert back to a float, you get the original, so in a way there are no loss of precision during conversion. Also you can represent infinities, nan and -0 in a well-designed library with 1/0, -1/0, 0/0, 0/-1.

That's part of the beauty of mathematics! You do something considered "wrong", investigate that "wrong", and a whole world opens up where this "wrong" becomes a rule, and explains things in those fields perfectly!

It is just nice when u start from very simple things and sort of play around with it to discover interesting observations. It feels like going in the reverse direction when u are reading a theorem. Instead of having a very complicated unintuitive statement thrown at you and then having to use every single brain cell to figure out why that even works in the first place, this just feels very satisfying. It feels like the thought process flows naturally, without resistance.

It was lovely getting an intuition on where the mystifying square root of 5 comes from in Hurwitz’s theorem!

I'm in high school. Clicked with great curiosity to watch and expand my knowledge but it slowly kept getting more and more complex until my brain couldn't understand

@zhulimath

9 ай бұрын

That's ok! It's amazing that you have curiosity about math. The video can get a little advanced at places, it's not a video meant for everyone. Don't worry too much about it, just try your best and take things at your own pace, maybe come to it in the future when you have stronger foundations. Some of my other videos might be a little simpler and easier, if you want to give them a try, like this one: kzread.info/dash/bejne/ZXZhppWwYpyYhKw.html

@canyoupoop

2 ай бұрын

I am in highschool too, few months ago I saw it understood not very much, today rewatching understood more then that time(not everything obviously) so maybe in my next re watch I will understand more

Wow, I saw the thumbnail and thought that the video would be something really simple, but then saw it was nearly 30 minutes long! Unexpectedly turned out to be an absolutely banger! Good job man!

@TymexComputing

9 ай бұрын

Me too - i recall stern brocot series' "name" from some quantum or other foo-bar but only the name - its the first video somebody draw it for me/ made a picture of it :)

I love it when different bits of math come together so sensibly and beautifully. Excellent video.

Damn criminally underrated video. Good work man. Keep em coming

Excellent video. I felt like every time you introduced a new concept, I had some questions pop into my head and thought "I'll have to google this after..." but then you answered the question in the video!

The intro riff sounded vaguely familiar to me, so I checked the description and it’s Kapustin! Great taste in music and great video

I can’t believe this channel isn’t bigger, you’re doing amazing!

Great video, I like it! It's interesting and yet incredibly calming. It draws you in and immerses yourself into the topic. I didn't think that a bit of classical piano could do that to a video. I can imagine myself calming down to it in the evening or something. A portion of Maths before bed. Your calm voice goes with it.

I ran into these structures while solving some project Euler problem. This explanation was just so perfect. Thank you.

The piano is so relaxing, thank you!

Frick yeah, new zhuli video just dropped!

This was really neat. I did see the graph visualisation coming, but seeing a visual example of Simpsons Paradox was also pretty cool. And the irrational approximations was also pretty cool

would love a ford circle video! love the way you break things down :3

This is a great video! I love this graph in the outro, it looks cool and really helps to rewind the video in the head. The theme is very cool too, I love when the video just starts investigating and playing with some concept just to see what happens. That's really my favorite part of math. Last time I caught that feeling when watching that video about hackenbush and surreal numbers. This was like my 2nd favorite math video of all time, and yours is really high up there! It's a shame that you couldn't fit everything in it, on 18:00 the audio quality changes for a moment meaning that this part was recorded after everything else, so you probably didn't have enough time. So please make part 2 to cover it!

Spectacular video and ending message. Keep exploring, gathering adjacrent math topics, and learn something new!

This was very reminiscent of a first year linear algebra course where you just jump from one result to the next, it was actually a very nice style which I enjoyed. I imagine a lot of people will need to skip back and rewatch some proofs (I usually watch on x3 speed, but here I had to take it all the way down to 1.5), but I don't think this is a bad thing, it just meant it was dense with information. The results were certainly better motivated and appeared more naturally than in a university course, and I especially liked how you drew from many areas to show results. This is a perfect mix of technical writing, recreational mathematics, and use of the video medium. Well done!

@1.4142

9 ай бұрын

You have 3x speed?!

@henryginn7490

9 ай бұрын

@@1.4142 I've got a firefox extension that allows me to change the playback speed. Gives me much more resolution in choice of speed, lower and higher speeds, and keyboard shortcuts to change speed. Works on pretty much any video as well, not just on youtube.

Very good video. Criminally underrated.

This is the first video I’ve ever seen from this channel. I love the choice of music for the intro lol - the end of Kapustin‘s 3rd Concert Etude.

Great video, watched the whole thing, hope you get more views 😄

jajaja. I was wondering why you weren’t just using the determinant at the beginning, good that I stayed. Great video! Interesting as well.

This video gives me a brand new perspective of how to look at math. Thanks

The part with the slopes reminded me of "a natural construction for the real numbers" by "Norbert A'Campo", where a real number is defined as an equivalence class of objects called slopes, which are "almost linear" functions on the integers.

جميل جداً، عمل متقن ومثير للاهتمام شكراً على الجهود المبذولة في هذا الفيديو

Great video, I'm subscribing!

I planned on covering this topic as an interactive website for SoME1 for use as estimations. You came at it from a much better angle and covered more than I would have

Your Stern-Brocot (mediant/parallelogram) tree is interesting. Perhaps it has some utility in combinatoric questions, such as the open no-three-in-line problem. In any event construction of the tree is simpler than attempting to enumerate unique slopes on a grid from scratch.

I like how my thought upon seeing the thumbnail was, "So like vectors?" Yep.

this channel is so underrated!

i kid you not, THE DAY after i watched this i had a math competition, and one of the problems was to find the fraction will the smallest denominator between 2 fraction, i tried trial and error and then remembered this video and immediately got the answer

Fun exercise: Find the product of 2*sin(πx) where x ranges over all fractions in the Farey sequence of order n, excluding the endpoints 0/1 and 1/1.

This is a banger!let me share it! Eye opener

This is a killer math video will be recommending to all my friends! 👍👍

Can we do something with this?! Mediants are how almost every US teacher grades their students. a/b and c/d being grades on b and c "point" test/assignments are merged (a+c)/(b+d). The mediants provide weighted averages of tests/assignments. Mediants are how almost every statistical study is done. You send out dozens of collectors to take small samplings and count all the positive events (sum of numerators) and divide by the total samples (sum of denominators) Two samples of 1 out of 2, sum to one sample of 2 out of 4. But a sample of 8/9 and a sample of 1/1 sum to 9/10.

Besides the content of the video, the music of the video is amazing.

Great video. Thank you

Amazing video. Came in there without any expectations, and it fulfilled my interests much more than I expected.

for those still confused about Simpsons paradox, it's because the pair of vectors which have lower slopes has most of its magnitude distributed into its higher slope vector, whereas the higher slope pair of vectors doesn't. basically it's the weighting. 5/5 is larger than 499/500 and 1/4 is larger than 1/5, but 6/9 is smaller than 500/505

@TymexComputing

9 ай бұрын

Hi - i am sorry but i cant imagine your explanation - i think it can happen but not always so the "always iff explanation wont make always working examples" ? regarding the first part of video i simply (as a phycisist) thought about normalized vectors instead of circles or spheres :) and found out there are also some lower + lower pairs that wont make higher sum. in 2D continuus space there is the whole complex theorems space that can happen :)

Very cool video, thank you!

I was not prepared for this wild ride

I took a break from a math practice test to watch a half hour long video about math.

@hairold5680

10 ай бұрын

Truly a chad

You can think of the mediant as translation of vectors. a/b (+) c/d which should be (a + c)/(b + d). given that the nominator is the x component and the denominator is the y component, you can compare a+c to c+a. ‘a’ vector translating by +c in the x axis, or ‘c’ vector translating by +a in the x axis

Zhu Li, you did the thing! (Sorry, couldn't resist that one.) I stumbled across Hurwitz's theorem a few years ago, but this is the first time I had ever seen a proof of it. I'll have to take some time to digest that to see if I really understood it -- some of those steps really flew by. But it's neat to see that you can do it in such an elegant, geometric way.

Hey, I have an interesting question: could there be an isomorphism to study this median such that the group of simplest form fractions (which is an equivalence class) with the operation median is like the group of integers with the sum (understanding the integers like equivalent classes, e.g. (2,1)=(3,2)=(1,0), like in zermelo Franklin axioms)

Overall good concept. Honestly, there's something flawed in the flow of your arguments, as far as presentation is concerned. I kept going blank because I didn't know "what this is leading to", even though every single concept (Farey sequences, Simpson's paradox, mediants) are all familiar to me. Consider mapping out the steps you're leading in advance, because this video's main points are dependent on catching VERBAL content, and not mathematical content.

Thanks a lot for this video! I've recently been learning about continued fractions and best rational approximations, and this idea of treating fractions as vectors really demistifies a lot of these concepts.

It's like a random walk from flower to flower in a garden - then at the end you realise you know the shape of the whole garden.

Very cool way of visualization

we have 3blue1brown at home meanwhile at home: no im just kidding, great video dude, your editing chops are amazing

Great video, found this interesting. Curious about what you use to make the animations?

@APaleDot

10 ай бұрын

Manim, a python library by Grant Sanderson of 3Blue1Brown fame.

Here because the thumbnail interested (and confused) me. 0:19 The thumbnail actually supports this. It had 1/2+1/2=1 being wrong and 1/2+1/2=2/4 written in as being "right." 0:30 Yes, but the thumbnail had the addition operator, not the mediant operator.

@doigt6590

10 ай бұрын

clickbait at its finest right?

4:10 I'm getting Chopin Nocturne vibes from the background music and kind of digging it.

Truly delightful

I never thought something arising from this much of abstractness would attract me to this extent

Nice vid! Considering this uses vectors and addition, and is between zero and one, could it be used in a GPU algorithm for inverse square roots? Not my area, but fun thought!

Great video

What a beautiful video. I should do more mathematics.

I accidentally did something like this, and got the right answer. Much later I needed to take the same code/math and update it to account for multiple fractions being added together, and it all fell apart. Took me a while to realize what the correct answer was, but it was a fun journey. I think it had something to do with accidentally solving for the reciprocal of the fraction, then forgetting that step occurred. Pretty simple with two fractions, much more complex and obvious with three.

School teachers: "You can't divide by 0" Mediant: "Hold my beer"

Great video. Ty ty ty

Amazing video, but not for when I'm delaying going to sleep and only half conscious. Gonna mark the video to watch tomorrow and will give my opinion then, when I can understand anything.

Hmm it seemed like junior high school math, but that vector approach was simply brilliant

Great video.

The title makes it look like it's just vectors, now I am starting to watch whether what I guessed was correct!

I now wonder how to design built-in datatype for rational numbers for n bytes, such that the values cover some level of mediant approximations

> we should learn to embrace this exploration, and let our curiosity take us where we want. if we end up where we expected, great. if we end up somewhere completely different, that's also great. this is one of the most important parts of math, and missing it might make math seem boring when it's really the opposite (although i wouldn't say that all cases of being bored by math are caused by this). maybe not even math specifically, maybe it's way more general, but i don't know a lot about that. thank you very much for formulating it, i'm glad i watched this video. the reasons why i love math have been just vague feelings and intuition for a very long time, and it's really nice to understand them and maybe even be able to explain them to others. although to fully comprehend these reasons, i would probably need to make a list of them, and it would be a loooong list :D

Zhuli:D I would love to see the skipped proofs as i am not knowledgeable enough to prove them solo. Wonderful video! Thanks a tonne🙂

Is it SoME time already?? Heck yeah!

Amazing video. Subscribing.

what software is used to animate these? 3b1b and this channel and many others all seem to have the same designs 13:05

@zhulimath

10 ай бұрын

I used Manim, the Python library 3b1b initially developed!

@0xlogn

10 ай бұрын

@zhulimath Oh! That explains a lot! Thank you!!

Wow I was not expecting Kapustin when clicking on this video, thank you for that

Algorithm bump. I don't follow math well. Butt you did an excellent job. And the recap was really valuable. Wow

This video wasn't meant to be a out tf2, but I think I now better know how trimping with demoknight works. Granted it's mostly eyeballing how much you need to turn to gain an adequate amount of speed, but if you know how much is too much then I feel like estimating what the median of that is would help improve your trimping skills a ton!

@BryanLu0

10 ай бұрын

What is this comment lol

@wun_zee3599

10 ай бұрын

Heho vectors go brrrrrrrrr

Very nice video.

What a nice video, I thought I understood math lol. Such a nice topic.

intro is fire

REALLY cool, thanks for sharing!!!

This video is literally my mind mid-exam "So x divided by y = z." My other part of my consciousness: "hello vesauce here, what if we calculated it differently? What if we change it? And is this question have similarities to questions 5 in page 3?" 5 minutes fly by that moment 😂

Music makes this video perfect

using the mediant operator would be a fun way to try to calculate digits of pi, though you'd need a precise way of generating circle's diameter and circumference. at that point, you'd already be able to compute pi lol

hi , can someone answer this question please proove that for each n>1 there is a and b two integers such as 3/n =1/a +1/b

I don't understand the argument starting at 15:13. What is to prevent, when honing in on the supposedly missing fraction, the mediant of the bounding fractions (to the left and the right) to miss the denominator of the missing fraction ? E.g., in your example, to jump straight to irreducible fractions with denominator >= 7 ? So that no other mediant from that point on could achieve 3/5 (given that the mediant will continue to produce fractions in lowest terms and the denominator of a mediant of two fractions is > either denominator.)

@zhulimath

10 ай бұрын

At 11:10, we explained and showed that when you take the mediant of two fractions, you must obtain the fraction in between with the lowest denominator (if ad-bc=1). If 3/5 is between your two fractions, and so is 4/7, your mediant cannot possibly be 4/7, because 3/5 has a smaller denominator.

@olivierbegassat851

10 ай бұрын

@@zhulimath thanks, that clears it up!

The video could be subtitled to include an answer: "You discover the beauty of math!"

I have a probabilistic argument that the denominator in Hurwitz's theorem should be q^2, which I find pretty neat. It goes like this: Suppose you have an arbitrary irrational number, x. How good can you expect the "best" rational approximation with denominator < q to be? Well, since fractions with denominator q form a lattice with spacing 1/q, the distance from x to the nearest such fraction, p/q, can't be more than 1/(2q). So it makes sense to take the number h = 2 |qx-p| (which is always between 0 and 1) as a measure of how "good" the rational number p/q is as an approximation of x. If x is "randomly" chosen, we can say that h is uniformly distributed between 0 and 1. If you test all denominators between 1 and q to find the one with minimum h, you essentially have q independent tries. The expected value of the minimum of q independent numbers chosen uniformly at random between 0 and 1 is 1/(q+1), so we should expect the "best" rational approximation to x with denominator less than q to differ from x by about 1/(2q(q+1)), or, asymptotically for large q, something of order 1/q^2, which meshes very nicely with Hurwitz's theorem. From this perspective, it becomes really interesting that there are some simple-ish approximations for π (for example, 355/113) that beat this bound by quite a lot. Most other irrational numbers you might come up with (say, e, or √2), don't have such exceptionally good approximations.

@zhulimath

9 ай бұрын

Cool ideas! I haven't looked into this space very deeply or rigorously, but I suspect the reason why you can beat the bound significantly with pi, but not so significantly with some other irrational numbers, as a lot to do with the continued fraction representation. If I had to conjecture, I think there are some ways to metricize how close these rational approximations can get, and there's probably a metric in which the golden ratio is the furthest away from its rational approximations.

@rossjennings4755

9 ай бұрын

​@@zhulimath For sure with the pi thing, you can see it in terms of the continued fraction expansion, which is a really tidy way to get the best rational approximations (that I didn't know about when I was thinking about this the first time). But if you ask me, that just pushes the question back a step. If you start computing the continued fraction expansion of pi, you get 3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, .... That 292 corresponds 355/133 being such a weirdly good approximation, and it's surprising that it shows up so soon. Why does that happen? Unlike for quadratic irrational numbers or for e, for example, there's no clear pattern to the continued fraction expansion for pi. And maybe there's no satisfying answer, but it is interesting.

@zhulimath

9 ай бұрын

I'm sure there are good reasons for it, but I'm not too well-versed on this (yet).

@jameshulse1642

8 ай бұрын

@@rossjennings4755 I think the lack of pattern is because pi is transcendental (cannot be formed from a non infinite equation) and if a continued fraction has a repeated pattern then the result can be written as the solution to an finite equation e.g. sqrt(2) has denominators 2,2,2,2,2,2... and so (2+2/x)=x has roots sqrt(x). e should be the same as it is transcendental but I think it just tends to have smaller denominators than pi (I have no idea why).

I'll make a mental note to rewatch this video. I think it's going to need multiple watches to absorb and understand everything.

If you have a/b + c/d reduced to its simplest form, then say 2*(a/b), would it be correct to state that that would doubble the effect of a/b in comparison to c/d ?

@zhulimath

10 ай бұрын

Sort of. You should use the linear transformation visualization to intuit how it impacts the mediant.

The Stern-Brocot tree is wonderful. All the enumerations of the rationals are wonderful. It feels like they tie so much together. You don't want to use fractions to represent rationals when you know about the Stern-Brocot tree - fractions are ugly, there are more ways to represent the same number! You can just index into the Stern-Brocot tree instead. It may be slightly harder to calculate with, but ...

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0:42 . what does it mean for fraction to be of rational values and in simplest form. To me simplest form makes sense only for fraction with integers, so why say it has to not have irrationnal values?

@zhulimath

10 ай бұрын

You're correct in that there is no well-defined way to represent an irrational value as a fraction in simplest form. I am simply stating that in order for the mediant to be well-defined, not only do the values need to be rational, they **also** need to be in simplest form.

Dang, this is so dense with information that i looked away for 2 seconds and suddenly i have no idea whats going on😂 Math class all over again

13:30 I got very confused here because I thought that this is the determinant of 2 vectors shown on the screen, which should be 2. You should've changed a b c d with actual numbers for a moment to make it a bit clearer.

@zhulimath

10 ай бұрын

Ahhh, that's a very good catch. I think I would have color-coded the ad-bc to be green and red, then substituted them into the matrix to make that clear. Sorry about the confusion. You're right here, and this is going to be a small technical error that will bug me until the end of time!

Someone very wise said : '' Math is you, a paintbrush and an empty board with infinite possibilities. ''