Squaring the Circle - Numberphile

[@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

[@GMPStudios]

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

[@Mike-739]

So much kraft

[@mohammadumair3108]

Sharpies too...

[@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!

[@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...

[@glenn2687]

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

[@hannahkan0622]

+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

[@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.

[@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.

[@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.

[@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!

[@michael-h95]

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

[@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

[@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.

[@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.

[@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!

[@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).

[@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.

[@bowl1820]

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

[@thomashan4963]

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

[@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

[@Crunchymixx]

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

[@SnakesAndApes]

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

[@Zalemones1]

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

[@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.

[@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)

[@timothyclements2290]

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

[@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!

[@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.

[@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.

[@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.

[@casplant]

This is friends talking about cool stuff! Loving it!

[@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!

[@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

[@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.

[@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

[@christiaanbalke]

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

[@prometheusxo6013]

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

[@RigoBuitrago]

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

[@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 :)

[@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.

[@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

[@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

[@eossakira2653]

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

[@thesimulacre]

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

[@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.

[@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

[@ollomont830]

4 years later still watching, again. Numberphile

[@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.

[@SimchaWaldman]

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

[@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.

[@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..

[@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.

[@barenuffsafe]

Quite happy to be strung along by these two!

[@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.

[@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.

[@johnbeene3117]

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

[@trallalala100]

2:32 - mind blown.

[@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!

[@gfetco]

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

[@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...

[@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.

[@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

[@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.

[@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.

[@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.

[@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

[@monkeyfoetus]

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

[@Babluuyt]

He is addicted to mathamphetamine

[@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

[@stealthwolf1]

This channel is severely underrated

[@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.

[@Pumbear]

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

[@harryw4802]

yup

[@Megaman-ws8ik]

Okay! I have a solution! I’m not certain if this counts, but here goes. You take your compass and draw a circle with radius 1. Then you take a piece of paper and roll it up so that it exactly lines up with the circle. Cut of any excess. Unravel the paper. Find the length of the paper. (This is the circumference). Divide circumference by 2(diameter) and there is your pi. We know how to square root so that is not a problem, and to construct a square is just a matter of 4 perpendicular bisectors.

[@MuffinsAPlenty]

"Then you take a piece of paper and roll it up so that it exactly lines up with the circle. Cut of any excess. Unravel the paper. Find the length of the paper." None of these steps are allowed. You are only allowed to use a compass and unmarked straightedge.

[@MikeRosoftJH]

And to be precise, the only allowed operations are to draw a line connecting two existing points, or a circle centered on one existing point and going through another, and then adding all points where the new line/circle intersects the existing lines and circles. (This assumes a "collapsing" compass. It has been shown that a "non-collapsing" compass, which allows taking a distance between two points and drawing a circle with that radius centered on the third point, doesn't allow any more constructions - any point that can be constructed with a non-collapsing compass can also be constructed with a collapsing compass.)

[@dx8pi6o48]

6:22 you forgot .org

[@nzmfpv]

Lol

[@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.

[@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.

[@pauldogon2578]

Here something that has always bugged me, maybe you numberphiles can help. the sum of the product of 9x anything = 9. eg 9x1 =9. 9x2 =18 the sum of the product = 9 (1+8=9) This works for 9 x anything. Why

[@AlsteinLe]

it's cause it's always missing 1 from 10. u can think of it being +1 instead of +9/-1. so if it counting +1 for each number u got. it's the same as that number . ex 5=+5

[@bjornsahlin]

+Paul Dogon Look up modulo calculation and/or the proof of why a number is divisible with 9 if the digit sum of that number is divisible by 9. :)

[@user-zh3sn6fo5o]

+AlsteinLe Can i sue u? U just made me brain wrinkle.

[@AlsteinLe]

+ʎɯɯıɾ ɔ haha...

[@coopergates9680]

+8070alejandro What's your preferred base then?

[@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

[@djw11111]

I like Dr Math's answer to this: These are just the rules the Greeks chose to play by - they could have used other instruments e.g. origami. The straightedge and compass are the tools that relate to Euclid's geometric axioms and so they wanted to know what was possible in this axiomatically defined universe.

[@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.

[@MatteoBlooner]

What if pi^^5 is rational

[@abd.137]

it isnt. no power of pi is rational

[@MuffinsAPlenty]

Also, things like a := ln(2)/ln(π). Then π^a = 2. But it is certainly correct to say that if c is a nonzero integer, then π^c is irrational. This may have been what Muhammad Abdullah meant.

[@michaelbauers8800]

impossible; no power of an irrational number is rational and it's so obvious I would not need to see a proof

[@aweseman]

No power of an irrational number is rational? as in, (√2)^2 or 2^(√2) Because the former is a rational number by definition

[@michaelbauers8800]

Agreed :)

[@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.

[@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?

[@TheMinecrafterBoyzz]

Tau is better

[@snowfloofcathug]

Pi times 2

[@AgglomeratiProduzioni]

After centuries of useless discussions and argumentations, we finally got this: "Tau is better." -cit. TheMinecraftBoyz

[@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.

[@3kbschannel288]

"Algebra is an amazing, powerful tool in mathematics." Thanks Muslims

[@TheInfallibleChase]

Chinese did this too. Both civilizations found it to be spiritual

[@pinksphaghetti]

I absolutely love this channel its marvelous

[@robertwilsoniii2048]

I'd prefer learning from a wordy proof based "greek" text book.

[@nowandaround312]

+Robert Wilson III Then his comments are not directed at you. I assume you did not whine in school "but why do we have to learn _algebra_, when am I ever gonna neeeed it?"

[@gupta-pw5xb]

*pi is constructible* Let you have a wheel with circumference pi Now take that wheel and mark any point A Now put the wheel on the surface and rotate the wheel till point A reaches So, on the surface,point A to point A is equal to PI...........

[@MuffinsAPlenty]

A wheel is neither a compass nor a straightedge.

[@hanniffydinn6019]

Ashish Gupta yep. Simple really.

[@hanniffydinn6019]

Goku17yen obviously is an approximation Idiot! Plants, and all of physics is full or pi, e, phi.

[@Qardo]

You know...I randomly clicked on one video and soon watched this. I have to admit. This is far more interesting than what my math classes could teach. Yet...also could be that your British makes it more interesting lol.

[@MrGingerpupes]

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

[@styk0n]

every time I watch a numberphile video, I wish I had dedicated more of my time at high school to appreciating maths.

[@Brainiacs0]

Make a right-angled triangle with vertices at the centre of the semicircle, and the top and bottom of that root (a) line.

[@athletico3548]

hes the only guy that makes me entertained

[@howie5697]

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

[@MadiLush]

I am terrible at math. Geometry I get, any kind of practical application type math. But, I find these videos fascinating.

[@roderik1990]

Yes that would be a solution, but also changes the properties of your operations. See, the interesting thing about ruler and compass constructions is that if you look at the set of points you can construct as complex numbers, the set of points is exactly those you can generate by combinations of addition, substraction, multiplication, division, complex conjugation, and square roots. The string addition, would implicitly add multiplication by pi as an operation.

[@falconbaker8553]

Bro how does 2:25 work that absolutely blows my mind

[@MuffinsAPlenty]

If you add some line segments in, you can get a triangle with the dashed line as an altitude and the diameter of the semicircle as an edge. By Thales' Theorem, this triangle is a right triangle (hypotenuse being the diameter of the circle). You now have three right triangles - the big one already described, the "left" one (with a base of "a"), and the "right" one (with a base of "1"). Now, all that's left to do is use the Pythagorean theorem and do some algebra to solve for the altitude's length.

[@O-Kyklop]

@@MuffinsAPlenty Two German guys did a geometric derivation of the π value, but working on Areas and not on straight lines. (Das Geometrische π) the Geometrical π . Well. Guess what did they get as result? => 3.1446. They think it is an approximation because they didn't have 4b=π as this other guy who solved the problem, working on areas as well.