Squaring the Circle - Numberphile

Why squaring the circle - the old-fashioned way - was found to be impossible? Numblr: / numberphile
More links & stuff in full description below ↓↓↓
This video featuring Dr James Grime: / jamesgrime
The paper from this video on ebay - bit.ly/brownpapers
NUMBERPHILE
Website: www.numberphile.com/
Numberphile on Facebook: / numberphile
Numberphile tweets: / numberphile
Subscribe: bit.ly/Numberphile_Sub
Videos by Brady Haran
Patreon: / numberphile
Brady's videos subreddit: / bradyharan
Brady's latest videos across all channels: www.bradyharanblog.com/
Sign up for (occasional) emails: eepurl.com/YdjL9
Numberphile T-Shirts: teespring.com/stores/numberphile
Other merchandise: store.dftba.com/collections/n...

[@TheZenytram]

omg he said pi on 3:14, i can die now.

[@aimeecortez5899]

😮😮😮😮

[@ellinaras4566]

+Zenytram Searom he said "Pie"

[@Danielkaas94]

OMG!

[@LightningJackFlash]

All's hidden in numbers ;)

[@johnplays9654]

illuminati confirmed

[@keithwilson6060]

I love how obviously excited you get about math. That more teachers would have such zeal.

[@thanatosdaughter6298]

I completely agree! He's so obviously passionate and it's great. If my teachers were like this, I'm pretty sure I would have a lot more fun in my classes.

[@Schobbish]

The thing is that I probably learned more from this channel than my math teachers. (Sorry math teacher...)

[@seanp4644]

+Nathan Adam (SchobbishBot3000) don't apologize. These guys do it better.

[@supersohig3671]

Aragorn Stellar by v.

[@Tom-vu1wr]

Bruh my pure teacher is this excited about maths

[@renjiai]

But can you cube a sphere?

[@adant9536]

Yea

[@MatteoBlooner]

No

[@zachmanifold]

I gave it a try: This is for surface area, and I will do volume after. So, let's say 'Sa' = sphere area, and 'Ca' = cube area. Let's give the sphere a radius of five. Therefore, Sa = 4pi(5^2) = 314.16 units^2. Now we have Ca which is an unknown. The formula for the area of a cube is 6a^2, so to get rid of the 6, I divided the area of Sa by six, which gives us (314.16 / 6) = 52.36. Now we're left with a^2 = 52.36, so I took the square root: sqrt(52.36) = 7.236021. So a = 7.236021, now let's plug it into the formula for the surface area of a cube: Ca = 6(7.236021)^2 = ~314.16. Seems like we got surface area, now let's do volume: A sphere with a radius of five (just like the sphere above) = (4/3)pi(5^3) = 523.6 units^3. The formula for the volume of a cube is a^3. We already solved for a when a sphere has a radius of five, so let's plug it in: (7.236021)^3 = 378.88 units^3. The cube appears to have a lesser volume than the sphere. ((523.6 / 378.88) * 100) - 100 = 38.2%. The sphere's volume is about 38.2% larger than the cube. Thanks for taking the time to read, I hope my maths is all correct. (:

[@unicockboy1666]

same system

[@unicockboy1666]

Figgy Winks Clear NO: you multiply the radius by an infinite number, so that you cant take the 3rd root (or any root in fact) out of it...

[@stripeysoup]

His skin is brighter than my future.

[@ObsidianShadowHawk]

+stripeysoup Making me laugh at 3am... Thank you, sir!

[@reizayin]

이강민 Vantablack is brighter than mine.

[@vijayshejal4322]

ha ha :)

[@onyxgod777]

you almost made me choke laughing loll

[@clayz1]

and too close.

[@nh-a6713]

make a circle out of playdoh, then mold it into the shape of a square, wheres my nobel prize?

[@michaelbauers8800]

You would have to keep the playdough perfectly flat and the same height it originally was.

[@aiden359]

were talking about two dimensions though lol

[@ingolfura.4327]

watch it from above :)

[@jimbobago]

a) There's no Nobel Prize for Mathematics b) No one's saying you can't solve the problem with Play-Doh. It's only impossible under the rule that you have to do it with nothing but a compass and unmarked straightedge.

[@Edgard422]

That's a compressible material, no nobel prize for you.

[@GMPStudios]

*Greeks:* Straight edge and compass *Numberphile:* Straight edge, Ccompass and loads of brown paper.

[@Mike-739]

So much kraft

[@mohammadumair3108]

Sharpies too...

[@glenn2687]

soooo.... who's watching this after the 'pi nearly became 3.2' vid

[@yiuyeungkan157]

+Mica Santos me

[@RizqieL]

me

[@kimberlychin1996]

+Mica Santos We can also say pi bearly becomes 3.15

[@AnteP-dx4my]

me 3

[@infinitegamer308]

+Mica Santos me

[@swipenet]

In case anyone is wondering about the square root thing at 2:15, it's pretty simple. The ratio between the dotted line and 1 has to be the same as the ratio between a and the dotted line, because if you draw lines from the ends of the diameter to the top of the dotted line, the resultant triangles have the same angles. It would be a lot better if I could draw this out, but hopefully you can visualize it. In other words, call x the length of the dotted line, and you have x/1 = x = a/x. Therefore, a = x^2, so x = sqrt(a).

[@philipk4475]

Neat

[@idzudinsaffuan9095]

@U.S. Paper Games exactly. the ratio couldnt be the same

[@johnnye87]

@U.S. Paper Games Maybe your description is unclear but it doesn't sound like you're doing what the video demonstrated. You need a semicircle of *diameter* A+1, with a line segmenting it 1 unit from the perimeter. If our radius is 15, then A (the number we're going to find the sqrt of) is 29. So our dotted line is 14 units from the centre, and forms a right angled triangle with the radius such that its height is the sqrt of (15 squared minus 14 squared), ie root (225-196), ie root 29. Which shows you what's happening in algebraic terms: the length of A (diameter minus 1) is 2r-1, and the Pythagorean formula gives you the sqrt of (r sq minus r-1 sq)... which simplifies to the sqrt of 2r-1. Neat!

[@E1craZ4life]

If you draw a rectangle and then draw diagonals connecting opposite vertices, the diagonals would bisect each other. So drawing a circle with a center at the intersection point between the diagonals would pass through all four vertices of the rectangle if it passes through one of them. What that means is that any triangle with points on a circle is a right triangle if the hypotenuse is the same length as the circle's diameter. If a line is drawn perpendicular to the hypotenuse passing through the point opposite the hypotenuse, then this will produce two smaller right triangles. Since the sum of a triangle's angles has to be 180 degrees, the smaller triangles will be similar since the original right angle was split into two smaller angles. By that logic, the smaller leg of the smallest triangle would have to be scaled up by a factor of the longer leg to match the size of the other triangle. And that in turn, means the longer leg of the larger triangle has a length equal to the shared leg's length squared.

[@franciscohamlin7544]

Beautiful!

[@adityakhanna113]

At time 3:14 he said "Pi"

[@njood96]

Aditya Khanna and now your comment likes are 314 XD i want to like it but i don't want to ruin it XD

[@zashtozaboga]

comment something else please

[@thefremddingeguy6058]

+Aditya Khanna You're right....

[@rongliu3339]

+Aditya Khanna creepy

[@coopergates9680]

+Стилиян Петров I think the Zeno's paradox video doesn't say how you could really "make" a square with side Sqrt(pi).

[@jonasbindslev9894]

Why is he so shiny?

[@galek75]

Battle typhoon truuuuuuuuuuuuuuuuu

[@frtard]

Battle typhoon Too much maths. It's coming out his pores.

[@Toimi]

Battle typhoon He's a robot. His skin is actually plastic.

[@nourse]

Battle typhoon He's shiny and chrome to go to valhalla.

[@castleblack6941]

Cause he's brilliant. Duh!

[@burnsy96]

Guys I found the solution to this so called 'unsolvable problem' and I will patent it so you have to pay me when you math it out except for my home state Minnesota as a gift to them.

[@fullyverified7491]

thats funny

[@burnsy96]

Tsavorite Prince Yes, I'll get the Nobel prize for this one

[@General12th]

+burnsy96 I think you meant Fields medal.

[@LivingChords]

no i'm pretty sure he meant the nobel prize.

[@Carter040404]

burnsy96 I also live in Minnesota

[@michael-h95]

10 years later and I still come back to these videos videos 😅

[@dante224real1]

you cannot theoretically square a circle, but realistically you can. in realist terms we are left with approximations determining the effectiveness of theorems in geometry, physics, etc. if you can find me a perfect circle in real life that has exactly an area of x^2xpi, and you can prove it to any digit within pi with no room for error, i'd eat my house.

[@coopergates9680]

+Daniel Williams (Invents arbitrary unit such that x = 1)

[@TheChangingWays]

I once ingested an e. It was truly a transcendental experience. ‪#‎MathJokes‬

[@SpaceGuru5]

Hopefully you had pi for dessert.

[@Intel1502]

+The Changing Ways Meth Jokes.

[@losthor1zon]

+The Changing Ways - Hope it didn't require a transcendentist.

[@qclod]

+SpaceGuru5 I can eat a whole pi, but a tau is too much to handle.

[@SpaceGuru5]

levizna Either would be just as irrational.

[@KpxUrz5745]

I love this channel and return to it often. Not only fascinating and educational, but the sheer excitement and clarity by Numberphile is a joy to behold!

[@swinki33]

I love Dr James Grim's enthusiasm when he tries to explain such not so easy math problems. I wish I had such a math teacher. Or all my teachers. Fantastic!!! Thank you.

[@moonblink]

Algebra rocks. I've been explaining that to people since high school. Algebra is there to make sense of everything. Algebra is like the ABC's of math.

[@TehKorwinMikke]

+moonblink Algebra is THE alphabet, words, and sentences of math, yo.

[@carbon13]

+moonblink Cough, Calculus is more fun, cough

[@carbon13]

***** But the fundamentals of Calculus differentiate from every other form of Algebra.

[@carbon13]

***** Really depends on what you're doing with your programs.

[@moonblink]

Tsavorite Prince a = c - b

[@Zalemones1]

Dr Grime sure is a bright man, no pun intended!

[@JesseRaylabrancaro]

I really wish I had had y'alls videos when I was a kid. I think I would've liked math A LOT more.

[@alexeysaranchev6118]

What sort of videos could've made you love the English language enough not to use "y'alls"?

[@nickwilson3499]

@@alexeysaranchev6118 yaull’ses

[@puppergump4117]

@@alexeysaranchev6118 It's about as improper as your use of "could've". Sieg grammar I guess.

[@alexeysaranchev6118]

@@puppergump4117 what's the correct way then?

[@puppergump4117]

@@alexeysaranchev6118 It's only correct if you stick to one standard. Either accept contractions or don't. Since contractions are accepted by the vast majority, with the exception of some college teachers, the use of both "y'alls" and "could've" are grammatically correct. Of course, not in the technical sense. However, if half of our country accepts a form of a word, who cares if some college's dictionary accepts it? Language is meant to express meaning, not to be restricted by redundant rules.

[@blacknwhitestripe]

You people are wonderful wonderful people. I've never been great at math but it's really fun to watch your videos and enjoy it without worrying about skill

[@michaelchen5575]

It is possible to use materials that the Greeks had at their disposal to "square the circle": Draw circle, radius 1 (area=π) Outline circumference with string, straighten out the string, then draw line (this has a length of 2π) Divide length by 2, use triangle scaling method Use the square root finding method thing with the semicircle (to get √π) Side for square has been found Of course, there will be some error due to the elasticity of the string and the human impossibility of perfectly measuring where the string coincides with itself after one rotation among other factors, but theoretically and statistically speaking it is possible

[@kevinoduor9841]

use a ruler and a compass only, that's the rule.

[@KnakuanaRka]

The Greek problem only permitted compass and straightedge; there is no way to emulate your “straighten out the string” bit under these rules.

[@hanniffydinn6019]

Yeah, simple really, it's called string theory !!!

[@pbierre]

You're allowed to use the compass as a caliper to copy distances, right? So break up an arc length into a series of piecewise line segments, and copy them out to a straight line length. If you solve for the half-width of the square , sqrt(pi/4), you only need to "linearize" 1/8th of the unit circle arc.

[@KnakuanaRka]

Pierre Bierre It wouldn’t be possible to exactly replicate the length of the arc unless you used an infinite number of line segments, which is not allowed, as the construction must be finite.

[@SaveSoilSaveSoil]

Awesome presentation! Thank you! I hated straight edge and compass problems back in junior high (esp. the "is it possible" type, which are way harder than the "construct..." type). I always wondered what the point was. I wish this video had been my introduction to straight edge and compass.

[@bowl1820]

Thank you for not having distracting background music, like so many others! Like given.

[@RigoBuitrago]

Awesome videos, man, I watch a few of them every day and re-watch them every few days. Fantastic!

[@thomashan4963]

This channel made me like maths and now I'm an educator sharing problem solvings based on calculations ❤️

[@SnakesAndApes]

I'm not really into math, but so far I'm enjoying these videos, Good job!

[@dnnstalks]

Im so hypnotized by him, thats the stunning thing in these Numberphile clips, these people have a passion with their theme, its so fun to watch.

[@TheJtyork420]

I wish u were 1 of my teachers in school. I hated math class but seeing someone who not only actually enjoys it but is also passionate about it brings a lot of excitement to the subject.

[@Crunchymixx]

I really love how passionate he is about mathematics :D it's amazing.

[@timothyclements2290]

lol I love this guy. Great smile and he absolutely enjoys his field.

[@thesimulacre]

The fact that we can't just change our units to solve this also points to something transcendental

[@barenuffsafe]

Quite happy to be strung along by these two!

[@casplant]

This is friends talking about cool stuff! Loving it!

[@IaFsI]

Wish I saw enough videos of numberphile before finishing high school. I would have been more interested in maths, not that I wasn't interested at all.

[@christiaanbalke]

Numberphile still going strong during the corona-lockdown! Fabulous!

[@ollomont830]

4 years later still watching, again. Numberphile

[@htomerif]

This isn't how I've heard of "squaring the circle" I'm thinking of something different I guess but I thought it was a (possibly equivalent) problem of dicing up a circle in such a way that you could construct a square from its pieces. And I think this was solved relatively recently, but using some not very satisfying feeling rules.

[@steffenjensen422]

No, the problem your describing is trivial. Just look at the curved parts, you're not gonna get rid of them

[@nikhilnagaria2672]

@@steffenjensen422 you can actually :)

[@firstnamelastname-oy7es]

Squaring the circle? If you think that's difficult, try Cubing the Sphere! I've been trying to do that for the last 141 years!

[@pinksphaghetti]

I absolutely love this channel its marvelous

[@jakobygames]

i lovehate this channel so much. its so interesting that i end up clicking video after video in my recommended late into the night and i cant sleep because i need to ABSORB ALL THE KNOWLEDGE IN THE UNIVERSE

[@DudeGlenn]

Algebra is brilliant. I knew it!

[@gfetco]

+Glenn Beeson (BeesonatotX) You don't say.

[@DudeGlenn]

+Enlightenment I did say. And you replied.

[@gfetco]

Glenn Beeson Do you know who I am?

[@DudeGlenn]

+Enlightenment You know that I don't hence the question. I assume this is going some where correct?

[@gfetco]

Glenn Beeson I am Ronnie Pickering! Don't you forget! :D

[@gwendance]

In case you didn't get it: √2 is an algebraic number because is is the square root of a rational number. Although there is an n where √n = π, there would have to be another number (let's call it 'm') where √m = n, and (let's call the next one 'p') where √p = m, and so on to infinity, That's why π is not an algebraic number.

[@steffenjensen422]

You left out the crucial point that none of those numbers are rational

[@howie5697]

You guys are freaking great. Thank you very much!!!

[@OrianneCorman]

You are amazing. I feel that I could love maths with you enthusiastic presentation. Thank you!

[@sameash3153]

I wish I had math teachers that were excited about math and could rub it off on their students. Well, I did have a few, and their classes were the ones I passed. But other teachers I had, especially my college teachers... Well, I didn't take anything away from them. Now I'm going on a self teaching spree with math.

[@eossakira2653]

I have never been so interested in math in my whole life.

[@vedangratnaparkhi]

James, love you!

[@HowdyStranger12312]

I love this channel!

[@prometheusxo6013]

I wonder what goes through your head when you solve a problem like this

[@Appolyon]

While trying the squaring of the circle, Is it allowed to use a thin string or twine? I mean: If i draw a circle with radius 1, i can messure the lenght of the semi circle with the twine. Now i have the lenght pi and can draw a line of this lenght + 1. Then i can draw the semi circle over this line and can messure the square root of pi like the square root of a in the video. And now i have the length to draw the sides of the square. Or am i making any mistake here?

[@raykent3211]

I was thinking along similar lines in the video about an attempt to legislate that pi = 3.2. Here, the prof emphasises that they were playing by certain rules. You've stepped outside the rules that are considered pure mathematics. But I bet ancient greek engineers didn't rely entirely on the mathematicians. Archimedes invented a simple machine (trammel) which draws ellipses. If it could be made perfectly, they'd be perfect ellipses (proven by mathematicians). But it's less "pure" than just straight-edge and compasses. Who makes the rules?

[@siekensou77]

i think they would have access to string or twine..

[@gfetco]

That you speak about maths with such enthusiasm it makes me so happy.

[@johnbeene3117]

I love James' skill at explanation but can I just say how CUTE he is too?! :D

[@BrickfilmMan]

At 2:26, why is the length equal to √(a)?

[@BrickfilmMan]

Thanks for your reply, but I still don't quite understand. What does that have to do with the length?

[@jeymsie2474]

This is also new for me so I tried searching for proof but sadly there was'nt any in the net so I made my own proof. Bear with me please. From that semi-circle, make a line from the upper part of the line measuring √(a) and connect it to the center to make a radius. So now we have a right triangle and we can make use of Pythagorean's theorem. The diameter measures (a+1) so we can say that the radius is (a+1)/2, so... HYPOTHENUSE = (a+1)/2 LEG 1 = √(a) Now, leg 2 is just the radius minus 1 right? So that means, LEG 2 = ((a+1)/2) - 1 OR (a-1)/2 Now, using pythagorean's theorem, √(a)^2 + ((a-1)/2)^2 = ((a+1)/2)^2 a + (a^2 - 2a + 1)/4 = (a^2 + 2a + 1)/4 4a + a^2 - 2a + 1 = a^2 + 2a + 1 4a - 2a = 2a 2a = 2a So that's it, hooray or something

[@Sonny_McMacsson]

If the arc's diameter (a+1) is labeled A_B, put a point C where a and 1 meet then move up perpendicular to A_B until it touches the arc at D. Triangle ABD is a right triangle therefore triangles ACD and BCD are similar. The relationship exists: B_C / C_D = C_D / A_C (1) The lengths are: B_C = 1 (2) A_C = a C_D = b Substitute lengths (2) into (1) to get: b/a = 1/b Therefore: b^2 = a b = √(a)

[@BrickfilmMan]

embustero71 Thank you very much for your proof! :D It works very well, and I understand it! Just one quick question, why is the value of angle ADB a right angle?

[@Sonny_McMacsson]

Brickfilm Man Draw two intersecting diameters in a circle (they'll cross at the center of course). Take care to notice that the outer hull of the four points where the diameters meet the circle just happen to make a rectangle with the diameter segments being its diagonals.

[@McDanny420]

What about circling the square?

[@olli343]

+McDanny420 If you can find a circle with the area of a square, you have square with the area of a circle, sooooo...?

[@cclupu]

+McDanny420 Same way

[@seanp4644]

Walking around a square is easy...

[@chlover5853]

McDanny420 you got em there

[@Theo_Caro]

We are given a square with side length "s." We need to construct a segment with length "r" so that s^2=pi*r^2. Since s is a constructible number, pi*r^2 is constructible. However, we know that pi is transcendental and not constructible so that pi*r^2=s^2 is not constructible, a contradiction. Thus, we cannot construct a circle with an area equal to a given square. Squaring the circle and circling the square are logically equivalent in fact. "Squaring" was a word for what we know call integration. So the problem is really one in just being able to talk about the area of circles in terms of how we normally measure area (i.e. with rectangles). The problem fundamentally is about the nature of pi. And the solution is ehm... really cool.

[@dcsnunes]

Use the last way to construct a number. Draw a line sized pi, add 1, make a circle with pi + 1 and the height will be sqrt(pi). Get this dimension with a compass and draw the square.

[@christosmourouzidis9420]

Thanks a lot for the wonderful videos over the years. Just to highlight that the fact that you can approximate the side of a square that has the same area with a given circle using algebra, doesn't mean that it can actually be done. Since you can only approximate it and not really find it (pi is transcendental), it doesn't exist, no matter the intermediate tools you are using, computers or otherwise. The only tool we have in any case is our mind. Thanks again!

[@Twinrehz]

How does calculating Pi with a calculator work? I did a simple experiment once, I typed in 3.14 instead of using Pi on the calculator, then afterwards I did the same formula again, this time using Pi, and as some of you probably already have guessed, the numbers where quite different. My question is then if the button for Pi on my calculator, is defined with a very long row of numbers, or if there's another method used in the calculator's programming to define Pi?

[@Aerxis]

Pi digits can be calculated using taylor series, among other methods, but your calculator is only using a fixed set of digits (10 or 12), most likely.

[@Aerxis]

Slimzie Maygen Not all of what you said is true, and I fail to see why is it relevant in connection to my reply.

[@drearyplane8259]

Twinrehz My calculator has a verify mode, and, using this, I found it uses 13 digits of pi.

[@unicockboy1666]

Its using a lot of numbers (depending on your calculator), but not quite pi. It only comes so close to it, that for us and our practical universe, it doesn't matter anymore. In fact you cant even form a perfect cirle of sphere in real life...

[@pedrosaenzsantamaria2358]

Pi is burned in the prom

[@trallalala100]

2:32 - mind blown.

[@hollo500]

Love these vids!

[@stealthwolf1]

This channel is severely underrated

[@tapwater424]

Still watching in 2015

[@samkollmeier753]

watching in 2016

[@AoSCow]

+Desmond Dishwater watching in 2016.02716895

[@Alliloux]

Still watching in 1996.

[@AoSCow]

***** The video was made in 2013 March. So it's closer to pi years.

[@fiona9891]

Step 1: Make a circle with the radius 1 Step 2: Cut a wire the same size as the circle's circunference Step 3: Wire equals Pi Step 4: Make a line the size of the wire, add the 1 which we used for the radius Step 5: Take the square root of pi Step 6: Cut a wire of that size Step 7: Use wire to draw a square with the sides equal to the square root of pi Done.

[@fiona9891]

+( ͡° ͜ʖ ͡° )TheNoobyGamer *Looks at comments* Oh, this has been said before? Anyways, can someone figure out sqrt(π) ?

[@Lastrevio]

+( ͡° ͜ʖ ͡° )TheNoobyGamer 1.77245385090551...

[@fiona9891]

Lastrevio There you go.

[@enderman6777]

but the wire's length would not be exactly equal because of physical limitations (atoms; material decay; acuracy and all that). You'd get, for the length of the square, and approximation of the length "root of pi".

[@gilbertonogueira3481]

Assuming it would possibly work, the lenght of the wire would equal 2Pi, not Pi.

[@monkeyfoetus]

He looks so excited at 4:35 talking about transcendental numbers. It's adorable.

[@MrGingerpupes]

this is the best channel.! Video every 60 squared second.

[@AuddityHipHop]

You can only ever approximate the area of a circle.

[@cclupu]

As lenght of a segment too

[@harinandanrnair6768]

Fleegsta no and yes ....actually Area of a circle is exactly pi times r^2, but as u said it can only be approximated because pi can only be approximated

[@cclupu]

For Harinadan Nair : But if you put r=Pi the area becomes r^3. Isn't so weird if you use the fact in physics...

[@simonruszczak5563]

Because a polygon of infinite sides can't really exist.

[@Usammityduzntafraidofanythin]

Algebra is a tool of convenience. Makes sense to me. A lot of what the arabs did was taking greek texts that came from all over the place and just consolidate it into something more interpretable.

[@GreenMeansGOF]

One thing that I would like to point out is that there are ways of solving this problem as well as the other two famous impossible problems of Euclidean Geometry. The three problems are 1)Squaring the Circle, 2)Doubling the Cube, and 3)Trisecting an Angle. However, it requires us to move away from Euclid's axioms. 1) & 3) can be solved using the Spiral of Archimedes and 2) can be solved using parabolas. Perhaps Numberphile can make a video about those constructions in the future.

[@pythania]

I've watched this video for years now and I don't understand one thing. Until last week, I couldn't find any other reference of geometric constructions of arithmetic. I don't understand how multiplication/division works. Do I use an arbitrary angle? What about the unlabeled sides to the right? Is it an isosceles right triangle? Thanks to the person who clears this up to me.

[@benjaminbrady2385]

Easily! You can make a square with holes in a fractal pattern to get it, that might not count as a square though, so...

[@SimchaWaldman]

04:22 I thought Algebraic numbers are numbers which solve "rational coefficient equations" - not necessarily "constructable numbers". Like ³√2.

[@jabara83]

doggonit numberphile. I'm trying to do math homework; I take a study break, and I decide to watch a silly 4 minute video. Instead of being 5 minutes you string me along for a half hour. errrggg

[@SmellsLikeEMinor]

James Grime...the man who made me love math

[@JorgetePanete]

So, a circle with radius 1 is just a pie with π area

[@Marcelo-yp9uz]

@Fester Blats No, a circle with a diameter of 1 has an CIRCUMFERENCE of pi

[@egs_mythicgamer4013]

Anifco67 No they’re right the area formula is pi times r^2 so if r is 1 then the area would just be pi.

[@fifafutbeast]

Can someone explain why the sqrt(a) part of the semi circle is sqrt(a)? or just explain the steps for finding the measurements of the semi circle? thanks!

[@Titurel]

+Angel Urbina Draw a triangle by connecting the ends of the diameter to where the line sqrt(a) (call this line "h") meets the circumference. This larger triangle is a right triangle. The two smaller triangles are also right triangles. All are Similar (check by adding up angles) in two smaller triangles ratio of a/h is equal to h/1. so h^2 equals a*1 so h equals sqrt (a*1)

[@fifafutbeast]

+Titurel ohhhhhh... that makes sense. thanks!

[@cukka99]

Archimedes merely found one of a long series of approximations. As mentioned in the video, Ramanujan found a very close one too. What happened in 1882 was that it was finally proven that the circle in fact CANNOT be squared using just a straightedge and a compass. When they say the problem was "solved", this is what they mean.

[@athletico3548]

hes the only guy that makes me entertained

[@franz14ade]

You guys need a board or something. Papyrus has been used too much...

[@AiZeno]

+Oh Kazi but those are recycled paper aren't it? (not papyrus, but the paper used in their videos)

[@askingstuff]

NYEHEHEH...HEH

[@hetakusoda2977]

That's corier new. (I think)

[@pe3akpe3et99]

you mean..THE GREAT PAPYRUS

[@Mike-739]

That is Kraft paper

[@dandanthebabyman]

π=3.2

[@Molly_Bloom_]

+Clorox Bleach 3.2*

[@ryanbright2696]

na because 3.14 does not round up

[@DerUbermonke]

Clorox Bleach r/wooosh

[@006bartdiebrak7]

π~3.2

[@006bartdiebrak7]

π~~3.2

[@Schinshikss]

That's the major point of Zeno's paradox: a quantity that is infinitely divisible doesn't mean it is infinite. Numbers are merely symbols to communicate the quantity of something. Something that can be represented with infinite digits (such as 1/3) are not necessarily infinite. The infinite digits of π only means that it can always be divisible in a smaller order of magnitude, but it still has a finite amount (length). The perimeter of the 1-inch-radius circle will be ,and always be, 2π.

[@foldingcircles9521]

Brilliant Video.

[@BillySugger1965]

Numberphile At 0:13 James says that squaring the circle was solved in 1882. Please show us how...

[@BillySugger1965]

George Sorrell Thank you for that. :-)

[@Scy]

Solved as in proven impossible.

[@RotcivOcnarb8000]

what about strings? you cant put a string around a circle of radius 1, and then divide by 2? this would be pi with no doubt

[@ilyatoporgilka]

You would not be able to calculate it further after millimeters,microns,atoms,etc.

[@harryw4802]

you can't use strings.

[@Kraniumbrud]

his passion is infectious..,

[@NavalKishoreBarthwal]

using trigonometry cosine (angle[theta]) = base / hypotenuse : that triangle thing for multiplication and division really changed my perspective...

[@MultiXGamer1]

Why is he so shiny xD

[@TigerXeN]

Rare Pokemon

[@ryanlira7194]

why are you so shiny

[@Ghost____Rider]

When a reply gets more likes than the original comment

[@rajens1]

i don't understand how you get the root 'a' part by adding 1?

[@polpat]

Between the diameter and any point on the circle you get a straight triangle. When you add the vertical line he added you get 3 similar triangles. Similar means their ratios are the same. write down the equality between the ratios in the triangles having this vertical line in common. As you will see it shows that the unknown length squared is a.

[@thekrautist]

Okay, I can't watch this full screen. The close-ups kinda freak me out. PERSONAL SPACE, MAN.

[@joshuarohla2358]

Don't worry Brady has nearly 10 of these channels with years of footage

[@Pumbear]

So the greeks didnt have numbers or algebra but they did have square roots?!

[@harryw4802]

yup

[@DrScrubbington]

If we had no algebra there would be no cities. There probably wouldn't be any computers either, but that's all I'm saying.

[@gavinwightman4038]

Andrew S We wouldn't know the distance of roads with curves.

[@unicockboy1666]

Dr Scrubbington There is an explanation below a comment about the same question

[@AlexKing-tg9hl]

You always have that Rubik’s cube. Can you solve it in a video sometime?

[@PlatonicPluto]

He said Pie, on 3:14, on March 14, I am complete now, thank you Numberphile for activating the heehoo neurons in my brain.

[@otherssingpuree1779]

That unsolved rubik's cube was driving me crazy. Anyone else?

[@mr.j_krr_80]

Adarsh Singpuri ow yeah

[@ilyatoporgilka]

Search "Radio cube 3".It is a shape mod of another difficult puzzle "Eitan's star".Basically,an icosahedral variant of a Rubik's cube. In my channel you can watch hundreds of videos about that kind of puzzles.Go and do so.

[@zerospin876]

Take a tube with a radius of 0.5. Wrap a sheet of paper around it. Draw a line around the perimeter. Unfold the paper. You now have a line with a length of PI. Done. You just need to use warped space. Next problem?

[@BetaDude40]

This problem only works in Euclidean space, you can't use a third dimension.

[@AsBi1]

Why i find each n every video of this channel extremely easy to understand ? What's recipe

[@jean-stefankoskivirta8526]

Great video. I think the definition of algebraic numbers is slightly imprecise, as defined in the video you only get abelian extensions of Q, I think.

[@dx8pi6o48]

6:22 you forgot .org

[@nzmfpv]

Lol

[@SecretlyStarscream]

Just because you don't know all the digits of pi doesn't mean that a square cannot have an area of exactly pi.

[@angelmendez-rivera351]

Starscream That's not what he claimed. Did you watch the video?

[@richardsilva-spokane3436]

I’m not a math brain, however I was wondering the following: • with only circle/line knowledge, how would you determine the actual area of a circle in the first place? • ...in order to create a corresponding square with the same area?

[@utherpendragon6955]

Thus why it's unsolvable... Because it's unprovable

[@theonetrueignus]

For my undergraduate seminar, I researched and presented on constructable numbers and their role in building polygons, and I myself briefly touched on this ancient problem (as well as another called "Doubling the Cube")

[@cclupu]

+John Petters If you can construct a circle, you can construct transcendental numbers. But not algebrically. So, Daniel Williams must begin to eat his house.

[@theonetrueignus]

+Cãtãlin Pomparãu well, of course there would be transcendental numbers on the circle, but how would you know where they are? You can only find them at intersections with other lines and circles.

[@cclupu]

+John Petters It's a little missunderstanding. The only one transcendental number is the length of circle. Otherwise all points on circle are real. In my view over numbers, you can never draw a whole number, as 1 or any integer. You can only approximate the length of one unit. Integers in mathematics are approximations of a nonlinear phisycal operations, depending on PI. How can you describe in mathematical formalism term of rotation ? You must describe the final result or relations for intermediary steps. That's why the phisycs is so different. Anyway, in mathematics is something missing, and I know what is missing and how to integrate the missing element. I don't know yet why, but is only a problem of hard verifications because there are a lot of ways leading me at the same result, but there must be only 5 ways of transformation. I must verify the similitude of some of them

[@theonetrueignus]

You might be operating with a different form of construction than the one I used (and what the Greeks used, which is what he refers to in the video). In that method, the numbers you can construct are represented by the distance between points, and you can use existing points to form lines (with a ruler) and circles (with a compass). You are also given a unit length to start with. You can find new points (and thus new numbers) where these lines and circles intersect. Using this, you can build the set of constructible numbers to include all rational numbers and some of the algebraic irrationals. But the transcendental numbers (like pi) remain elusive, hence the impossibility of the problem. Also, judging from your comment, you may have a misunderstanding about transcendental numbers themselves (unless you misspoke). The set of real numbers includes numbers that are transcendental: in fact, "most" of the real numbers are transcendental. The only numbers in the complex system that are not real are the ones that have a nonzero imaginary part.

[@cclupu]

+John Petters Is a different way to construct geometric figures. The result is obvious the same, but with reasonable accuracy, using simple tools as Euclid did. Coming back to numbers : is a general missunderstang in the way we thought about numbers. Every number is composed of 3 parts : real, imaginary and virtual. That kind of thinking extends the mathematics with a new concept (is a work in progress), for better application in phisycs. We must understand that numbers are only reflexions of reality, abstract objects, that reflects phisycal world. I can't describe here shortly the entire logic of the idea. In that way we slowly slip in number theory, wich is one leg missing. I have not enough time (I'm too old for that) to full grow all aspects, but instead, I create a simple and substantial frame for all further developments needed.