The Journey to 3264 - Numberphile

Professor David Eisenbud talks about conics, and visits a few numbers along the way.
More links & stuff in full description below ↓↓↓
David Eisenbud Numberphile Playlist: bit.ly/Eisenbud...
David Eisenbud: math.berkeley....
3264 and All That: A Second Course in Algebraic Geometry: amzn.to/3lQfyKR
David Eisenbud author page on Amazon: amzn.to/3G2VJ9S
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[@numberphile]

David Eisenbud Numberphile Playlist: bit.ly/Eisenbud_Videos David Eisenbud: math.berkeley.edu/people/faculty/david-eisenbud 3264 and All That: A Second Course in Algebraic Geometry: amzn.to/3lQfyKR David Eisenbud author page on Amazon: amzn.to/3G2VJ9S

[@Syncrotron9001]

When you make it above 9000 let me know

[@jenspettersen7837]

Cool video! I am very excited about the recent einstein tiling discovery, I hope you'll manage to make a video about that!

[@averagelizard2489]

Can you please do SSCG(3) next?

[@olivierbegassat851]

I love his D. Eisenbud's cadence and soft spoken-ness. It's always a pleasure when he's on Numberphile : )

[@subnormality5854]

I miss the old days of the Eisenbud 17-gon

[@wheatdaddy_9629]

Woah, phrasing, pal

[@Sad_Bumper_Sticker]

Indeed, his way of speaking creates a truly unique cogni-feast ambience, I could listen to him teaching for hours without losing focus.

[@codycast]

Some of you guys are strange.

[@kostoffj]

Math ASMR

[@MrPictor]

Eisenbud's uncertainty principle: when drawing circles, the location of the pen cannot be known with any accuracy whatsoever.

[@kylegonewild]

Parker Squares and Eisenbud Circles. Can't wait to see what approximate value gets created on Numberphile next.

[@WhereNothingOnceWas]

Numberphile Cinematic Universe lore

[@rif6876]

"The point at infinity" - greatly appreciate the rigor! always surprised that other mathematicians on numberphile dont state it that way.

[@QuantumHistorian]

Lovely topic! I can't help but feel that a little more explanation could be given to statements like _"A circle in the space of line,"_ not everyone watching numberphile is going to immediately see a parameter space as a geometric object.

[@viliml2763]

You mean "a line in the space of circles"? I thought that was a weird choice of phrase too but I guess that just means a one-parameter continuous family of circles

[@QuantumHistorian]

@@viliml2763 Yep, that's what I meant. And yes, that's exactly what it means. But to someone not used to abstracting things via geometry it's not obvious.

[@lonestarr1490]

@@QuantumHistorian And I think this non-obviousness is well placed in that context. It makes people stop and think, "wait. What _is_ a line really?"

[@QuantumHistorian]

@@lonestarr1490Yes, exactly, except that because the speaker *doesn't* stop and instead continues the stream of new ideas, the viewer doesn't have the time to stop and think unless they manually pause (or, if they do, they'll fall behind the rest of the video). That's precisely why the speaker should spend a little bit of time clarifying what he means by that in order to give the viewer time to digest that alternative way of thinking about things.

[@doodlegoat]

Joke reference for those unfamiliar with English-language culture: "1066 And All That" is a parody of History teaching in British primary schools, published in 1930.

[@NoNameAtAll2]

can you give a timestamp as well?

[@jorgechavesfilho]

This 1930 book inspired several other history books with similar titles and in the same sarcastic tone.

[@DukeBG]

Oh, I was definitely not familiar with this fact! Is there something special about the year 1066 in that title?

[@neilmasson3609]

​@@DukeBG That was the year that England was invaded by the Normans. It was supposed to be one of the only two dates taught in history which people actually remembered. I think that the other one was nineteen fourteen eighteen.

[@NoNameAtAll2]

@@DukeBG it's important for english history - french,english and norman leaders battled over who would rule brittain

[@geekjokes8458]

"if you may remember i did a video of the fundamental theorem of algebra", yes it was 8 years ago and one of my favourite numberphile videos ever

[@hufflepuffjoh]

Oh my, it's been so long since we had an Eisenbud video! So great to see him back

[@TippyHippy]

l put my hamster in a sock and slammed it against the furniture.

[@penfelyn]

when you combine complex numbers with conics on Numberphile you get an iConic video

[@Donbros]

I am actually impressed not by the video and theory but by how he drew those circles neatly

[@patrickmckinley8739]

11:50 What about the xy term? I don't understand why we're not dealing with a 6-dimensional space when conics are free to rotate.

[@willnewman9783]

This was a mistake. It is really a 5 dimensional space, but you shold have the xy term: ax^2+bxy+cy^2+dx+ey+f=0 You get that it is only 5 dimensional because, although there are 6 coordinates, multiplying all of the coordinates by a fixed scalar does not change the conic. Example x^2+y^2-1=0 and 2x^2+2y^2-2=0 give the same conic. So the dimension of the space decreases by 1 (from 6 to 5), since we have a 1 dimensional space for this scalar. Quotienting out by the relation of multiplying all coefficients by a scalar gives the famous projective space, with the points at infinity. I think David did not want to go into all of this, and tried a shortcut, which didn't fully work out. A correct way of doing the shortcut would be looking at equations of the form ax^2+xy+cy^2+dx+ey+f=0 where we set b=1. This can be done for most conics by multiplying the equation by 1/b. This only does not work if your conic has b=0, which is a negligable amount of conics, and would not affect the answers for "generically how many conics are tangent to...."

[@pierreabbat6157]

7776 is called "weremeke" in Arammba, also "wärämäká" and "wermeke" in other Yam languages.

[@ryanswindell1936]

Equation of a circle at 8.56 should be (x-a)^2 + (y-b)^2 =r^2

[@redapplefour6223]

i just happened to watch the video on matrix factorization a couple hours ago, i always misread the title as matrix multiplication and thought it wouldn't be particularly interesting since i already understand all that.. lovely to see another one with david eisenbud already! saw a video too, on projective geometry, which is related to this video's topic somewhat too! definitely one of the most odd sub box moments ive had like that before, most i can think of right now

[@RobotProctor]

Collatz Conjecture Guy returns

[@f5673-t1h]

"Collatz Conjecture guy" 🙄 This is David fucking Eisenbud

[@asheep7797]

@@f5673-t1h donald knuth goldbach?

[@RobotProctor]

@@f5673-t1h it's supposed to be tongue and cheek; it's my ignorance not his. I bet he would get a kick out of the comment, not feel anger or anything. If I thought he would be upset by it I wouldn't have made the comment.

[@emirates4321]

​@@f5673-t1h nobody cares

[@NotoriousSRG]

Dang his mom really named him specifically

[@wyattstevens8574]

15:13 I was in a program co-hosted by Dr. Sotille (Dr. "I can make all 3264 conics have real values for a, b, c, d, *and* e") a few years back. I still remember that half an hour into each session of the program, when middle-school and high school students would split up, he'd just yell "HEY!" to catch everyone's attention. Seriously- I can still imagine him doing that, which is the first thing I thought of after you mentioned him! (and after I realized he was it- that made me do a double-take!)

[@nokanol45]

Off topic, but is a video about the singular aperiodic tiling (an "einstein") that was JUST discovered a few weeks ago in the pipelines? If so, I'm really looking forward to it!

[@osmia]

+

[@nosuchthing8]

Here, here! And the trig functions that prove the Pythagorian theorem too.

[@SaveSoilSaveSoil]

I can't wait to see Prof. Kaplan on Numberphile!

[@oz_jones]

@@nosuchthing8 *hear, hear

[@subjectline]

The 1066 and all that reference is very much in keeping with the Eisenbud style.

[@Parekhnish]

8:53 should be (y - b), not (x - b)

[@ravi12346]

Also at 11:40, there should be an xy term as well. That *would* mean there are six parameters instead of five, but really we should think of all of these equations up to a multiplicative scalar (e.g. x^2 - y = 0 is the same conic as 2x^2 - 2y = 0), which drops us back down to five independent parameters.

[@emilianol203]

@@ravi12346 x^2-y=0 is the parabola y=x^2. But without the rectangular term xy u can't make a hyperbola

[@CatInABaseballCap]

I once fell into a rabbithole of 3, 7, 9, 11, 13, and 37. I wasn’t disappointed.

[@junerae]

intriguing, could you say more?

[@zacadoole1]

Y’all should make a video on how those two students proved the Pythagorean theorem using trigonometry, everyone is talking about how cool it is but I haven’t been able to find anyone actually explaining the proof

[@matt_the_musician]

Wow! This is fascinating! As a photographer, I know the long side of an 8 MP picture has 3264 pixels, so that's somewhere else that number shows up! 😀👍🏼

[@sparkie4212]

Man, the Numberphile animations have come a long way...

[@wiserhinoceros]

Loved this - such a clear explanation of ideas in Algebraic Geometry!

[@guessundheit6494]

When will there be a video on the new mathematical proof by two US students, Calcea Johnson and Ne’Kiya Jackson? They proved the Indian sum of squares theory (x^2 + y^2 = z^2) using triogonmetry, the first people to do it. They're in high school.

[@Mikey_AK_12]

This video was incredibly interesting, and I feel like the title and thumbnail don't do it justice - I could care less about a random integer sequence, but a story about quadratics and the fundamental theorem of algebra? That's definitely something I want to watch!

[@xenmaifirebringer552]

Yeah, conic curves and anything with a graphical representation feel more appealing to me than random integers

[@bhardwajsatyam]

At 15:32, "Sottile" turns to "Sotille" for a few frames!

[@mtranchi]

Does that ironing board in his office have a story behind it?

[@asheep7797]

Yes, it was used for ironing.

[@mtranchi]

@@asheep7797 Lol, standard slapstick humor. Kudos :)

[@colinwood9717]

I couldn't stop staring at it!

[@KyleDB150]

He mentions at the end that the "theory of excess intersections" plays a role in physics, anyone know where that is?

[@maxreenoch1661]

8:56 is meant to have (y - b)^2 for anyone wondering

[@topilinkala1594]

All the books in highschool that had conics gave them as solution to ax^2 + by^2 + cxy + dx +ey + f = 0. That is six coefficients. The cross term xy was missing in this and my understanding is that it gives the tilt of the non symmetrical conics. Eg. you can alwas using just rotation of the axis change the co-ordinates so that the cross term disappears. But if you are talking about crossing points of conics I think you need to have it in there. Or can you actually write equations of two conics that have non-parallel axis in a co-ordinates where both equations do not have the cross term?

[@JA-cn6vu]

15:57 "Circles are circles. You know what they are. They're round." Wow! There's my profundity for the night. My brain is full! 🤣

[@realcygnus]

I always enjoy an Eisenbud appearance.

[@the_eternal_student]

I think the questions will stay with me more than the answer: how many places does a line intersect a conic section, how many lines from a point are tangent to a circle,, at how many points does an ellipse intersect a circle, howv many circles are tangent to the inside of intereecting circles, etc.

[@scottrs]

Sometimes I have no idea what you’re talking bout but when that happens I still know more.

[@OlivierMIEL]

Why 5? Where did the xy term go?

[@michaeltajfel]

Yes, the xy term should be included, but you can multiply all six constants by the same number, and you get the same conic. Thus there are really five constants to determine a conic.

[@aceman0000099]

xy isn't a constant, so it can't be altered in the same way

[@EebstertheGreat]

@@michaeltajfel But what if the xy term has a coefficient of 0?

[@ipudisciple]

The right answer is that there are 6 parameters but only up to scaling. The equation is axx+byy+cxy+dx+ey+f=0, but replacing (a,b,c,d,e,f) by (ga,gb,gc,gd,ge,gf) has the same solution set, so we get 6-1=5 parameters. They simplified for presentation and you caught it.

[@EebstertheGreat]

@@ipudisciple That's still only true for nonzero g.

[@nordicexile7378]

Is there a reason that the general equation shown at 12:40 doesn't have an xy term? (ax^2 + by^2 + cx +dy + e) I have a vague memory that the xy term can be eliminated by rotating the coordinate plane, but it has been too long since I last looked at this stuff to remember for sure.

[@landsgevaer]

I think it is a mistake, or perhaps intentionally sweeping it under the rug, although the conclusion that there are 5 degrees of freedom in the parameters is still correct. Generally, ax²+bxy+cy²+dx+ey+f = 0 seems to have six, but if we multiply the entire thing by a (nonzero) number we get a *different* equation for the *same* conic, so that is overcounting one degree of freedom. Your own argument that you could rotate away the xy seems correct, but I would object that you then would get a different (rotated!) conic. Maybe somebody else has a better justification for the choice in the video...

[@diniaadil6154]

xy can be rewritten as 1/4 * [(x+y)^2- (x-y)^2]

[@landsgevaer]

@@diniaadil6154 Yeah, so that is a transformation to variables v=x+y and w=x-y, but if you do that, the x² and y² terms are going to reintroduce v*w terms, so you haven't (generally) lost the product term... Unfortunately, I think it is a bit more tricky than that.

[@stevenmellemans7215]

I also spotted it and I have no clue other than it is a mistake.

[@DukeBG]

It can be eliminated by rotation, yes. Not just v=x+y and w=x-y, but a more generic-looking matrix

[@TheIcy001]

Never clicked on a video so fast in my life! Saw the title and I just knew it would be David Eisenbud speaking, because of his book "3264 and All That" Side note, maybe a very small mistake: I computed the intersection of the hyperbola xy=1 and the degenerate conic xy=0 mentioned in 4:24 in projective space for fun, and I got 2 distinct points: [0:1:0] and [1:0:0]. So I don't think there's a tangency at infinity. Also there is a projective line at infinity, not a single point at infinity. Might have been a momentary confusion with the Riemann sphere which is usually used to compactify C^1, whereas the real projective plane is used to compactify R^2.

[@issoroloap]

Hi! The point is that each of them is a double point (xy=0 and xy=z^2 gives xy=0 and z^2 =0, so the second equation gives twice the line at infinity). After all, as Eisenbud explained, you should expect 4 solutions in total, for the intersection of two conics. This is like a circle and an ellipse meeting at 4 points, or being tangent at 2 double points.

[@TheIcy001]

@@issoroloap You’re right! I neglected the fact that there are 4 points of intersection of two conics counted with multiplicity. Furthermore I realized that I can see the tangency by the fact that the slope of the line joining the origin and the point on the curve (this is y/x) doesn’t change sign as you wrap around. However, I can still make the point that merely having slopes converge is not a sufficient geometric interpretation of tangency. That’s just saying they intersect at infinity. Maybe physical distance approaching 0 on the other hand is sufficient for tangency (for algebraic curves), but I haven’t worked that out yet

[@KalenCarslaw]

I need to go back and finish my Masters in Mathematics

I love this guy.

[@physmanir0]

At 9:00 the video reads (x-a)^2 + (x-b)^2 = r^2, I think this is typo - of its circle it should say (x-a)^2 + (y-b)^2 = r^2, no?

[@christopherellis2663]

Fascinating: like a route map through space in the microcosm and the microcosm

[@SaveSoilSaveSoil]

This professor speaks so melodiously that I wonder whether he has a joint appointment at the music department.

[@GGoAwayy]

The conic stuff will always conjure up Cliff Stoll's bread in my mind. What's going on with that ironing board?

[@SurprisedDivingBoard-vu9rz]

Hyperbole are a system of co-ordinates. Circle and ellipse inverse. Parabola parallel.

[@Doktor_Vem]

I love this guys voice so much, but I do wish he'd articulate his words a little more d:

[@the_eternal_student]

I think the way of classifying the intersecting points has to be more nuanced to make it intuitive. It is not how many times the objects intersect but what intersecting group they belong to even if they are different from the rest of the group. I.e. the line below the circle does not intersect it twice, but it belongs to the group that intersects the circle twice.

[@macronencer]

8:52 It should be y-b, not x-b.

[@SportFury1966]

A voice that is a cross between Tommy Chong and the Ren & Stimpy singer of Happy, Happy, Joy, Joy. 'All the little creatures, maaaan.'

[@adrianf.5847]

Somewhere at about 3:37 the equations become weird. I think the guy means that (1 + a^2)x^2 + 2abx + b^2 -1 = 0 has either two complex solutions or one complex solution of multiplicity two.

[@howardg2010]

Quadratic from Latin for 'square', not Greek for 'two', though.

[@CowboyRocksteady]

More eisenbud videos!!

[@vincehomoki1612]

And another interesting thing about 3264: 32=2^5 64=2^6

[@derderrr7220]

equidecient spheroidal points - love it!

[@moss0964]

"well i wanna tell you about some numbers" fantastic! i love numbers

[@SuperM789]

at 1:44 the ellipse doesn't match with the formula below it. if x² is multiplied by 2, it should get skinnier, not flatten.

[@doubledarefan]

Slice a cone on the diagonal = Oval. Slice a cylinder on the diagonal = Ellipse.

[@p23570]

i'm a simple man, if i see numberphile posting a video of the bob ross of mathematicians, i watch the video and like it.

[@GilesForrester]

... that's the best use of an ironing board I've ever seen ...

[@theimmux3034]

please make a video on the recently discovered aperiodic hat tiling

[@ambassadorkees]

Cut through the top, and you get a triangle, or sharp corner, a special hyperbola

[@Lattamonsteri]

At what point does slicing a cone give me a parabola? :P There's a place where it it is an ellipse and then the other point when there's a clear parabola, but in between those two limits there's an area where the slice looks like an ellipse whose tip has been cut. Is there a name for that position where the slice starts resembling a proper parabola?

[@embryonicsuperfemme]

This would be easier to explain with a picture, but I'll try. He mentioned two degenerate conics: a single point x^2 = 0 and two lines (cut the cone in half). There is a third which is one line. If you take any point but the tip and draw a line to the tip you will have a line that runs along the side of the cone. You can imagine the cutting plane as just touching that line, like you were preparing to wrap the cone with it. Any plane parallel to that plane by pushing inwards will intersect with a parabola. Any deviation from this angle would either tip to ellipse or hyperbola.

[@Lattamonsteri]

@@embryonicsuperfemme i think i got it now, even without the picture :D thank you for your reply/explanation!

[@fierydino9402]

Algebraic geometry looks very difficult. Is it really so? Or if I study step by step, can I smoothly follow the subject?

[@moose9002]

I think "step-by-step" is the key word here. Algebraic geometry requires fairly broad background to really get into, however, this background material is often best understood knowing how it used in algebraic geometry! Commutative algebra is certainly the biggest culprit in this regard. From my experience of learning the subject (which is admittedly not so much, but this is consistent with what more experienced people have told me) what really helps is the willingness to revisit things with the new perspectives you gained. I don't think there is any need to be intimidated, just read what you find interesting, and fill in the background as you need it. Eventually you'll learn a lot!

[@henrikmunch8609]

How come there is no x*y hyperbola term in the general conic equation at 11:48?

[@josephang9927]

It's kind of wild how abstract concepts can make a book and for a family 😊

[@johnchessant3012]

I would love to know if there's a simple explanation for where the number 3,264 comes from? We know 2^3 = 8 and 6^5 = 7,776 but how do we get to 3,264?

[@theflaggeddragon9472]

Intersection theory on moduli spaces. There's not a simple explanation

[@soyoltoi]

​@@theflaggeddragon9472 What is the expression of the number given by that theory?

[@theflaggeddragon9472]

​@@soyoltoi I am no expert in enumerative geometry (I barely know basic algebraic geometry), but after skimming the relevant section, here's what I can say. The space of plane conics (as Eisenbud indicated in the video) is 5-dimensional; in fact it is P^5 (projective 5-space). This means the parameters are unrestricted and have no nontrivial relations (no two equations ax^2 + by^2 + cx+dy + e with different coefficients are isomorphic. Given a plane conic C, it's _dual_ C^* is the set of tangent lines, a smooth conic in the dual projective plane (space of lines in projective plane). The difficulty in narrowing 6^5 = 7776 to 3264 comes down to casting out "degenerations" of conics (double lines and such, as Eisenbud mentioned). This is technical and requires working on a _compactified_ moduli space of conics. This is the closure of the space of usual conics C in P^5. In fact, we take pairs (C,C^*) in P^5 x (P^5)^* and take the closure in there. We call this space X. Fix five general place conics C_,i, i = 1,...,5. The space of tangent conics in X is a hypersurface of degree 6. As mentioned in the video, taking a naive intersection gives a count of 6^5. The issue is the degenerate intersections occurring on the boundary of X. Now there is an object called the "Chow ring" A(X) = direct sum A^i(X), essentially formal integral sums of subvarieties of X modulo an equivalence relation. The A^i(X) encodes varieties of codimension i in X. Algebra in this ring allows us to compute intersections and many other things in algebraic geometry. On the open subset of smooth conics U in X, the hypersurface Z of conics tangent to a given conic has degree 6 (mentioned in the video). Let a,b in A^1(X) be pullbacks to X in P^5 x (P^5)^* of hyperplane classes on P^5 x (P^5)^*, and c,d in A^4(X) be classes of curves that are pulled back from general lines in (P^5)x(P^5)^*. One cna show that A^1(X) is generated by a,b over the integers. Take an equivalence class [Z] containing the hypersurfaces we want to intersect. The degree of its 5th power tells us the number of intersections (this is the point of working with Chow rings). Hypersurfaces lie in A^1(X), and it's free part has rank 2, so you can write [Z] = pa + qb for some p,q in Q and x,y forming a basis for A^1(X) (tensor Q). From basic properties of degree, you can show that [Z] = 2a + 2b in fact. Hence deg[Z]^5 = 32 deg(a+b)^5. So its enough to calculate the degree of a^ib^(5-i) for each i = 0 ,..,5. By symmetry, enough to do for i = 0,1,2. The calculations with explanation are on page 307-308 of 3264 and all that. All in all, you get deg([Z]^5) = 2^5deg(a+b)^5 = 2^5(5C0 + 2(5C1) + 4(5C2) + 4(5C3) + 2(5C4) + 5C5) = 2^5 * 102 = 3264. Reply

[@adamcetinkent]

​@@theflaggeddragon9472 Blimey. That's what we get for asking questions! 😂

[@viliml2763]

​@@theflaggeddragon9472 "no two equations ax^2 + by^2 + cx+dy + e with different coefficients are isomorphic" ax^2 + by^2 + cx+dy + e and k(ax^2 + by^2 + cx+dy + e) have different coefficients but are isomorphic the issue is forgetting the sixth, xy term

[@profdimateonline]

Hi! Beautiful channel 👍

[@curtiswfranks]

Someone should formalize those shaving methods. There is no way that they worked so consistently without something going on there.

[@shruggzdastr8-facedclown]

(@10:07): First, we had Parker Squares -- now, we have Eisenbud Circles!

[@jazzlover06]

3:46 should be x^2 not x

[@karthik_pushparaju]

Finally someone who caught it.

[@Gna-rn7zx]

Fascinating!

[@sumdumbmick]

it's not necessary for the slice to be parallel to the central axis to get a hyperbola. if it were it would mean that there's another type of conic section between the parabola and hyperbola. so it's curious that absolutely nobody postulates the existence of such a thing, and yet most people assert that a hyperbola arises when the cut is taken parallel to the axis of the cone. these are the actual conditions for getting the conic sections: circle - if the slice is perpendicular to the axis of the cone ellipse - if the slice is between perpendicular to the axis and parallel w/ the wall of the cone parabola - if the slice is parallel with the wall hyperbola - if the slice is between parallel with the wall and parallel with the axis you should learn to say things correctly, since it reduces the amount of gaslighting that students have to deal with to figure out wtf you're trying to communicate.

[@sumdumbmick]

also, a proper 'cone' has six lobes. most of the ones you depict have 1, and only for the hyperbola do you finally show the classic 2. but if you use the correct number, 6, then your slices correctly illustrate all of the relationships which occur between conic sections. it's quite nice. and it also demonstrates very elegantly how thoroughly fubar modern philosophy of mathematics really is.

[@sumdumbmick]

@3:54 the 2 imaginary solutions here are on the hyperbola. they're imaginary here because the lobe of the cone that the hyperbola slices is on an axis perpendicular to the lobe sliced by the circle. why wouldn't you just mention that? or did you just not know about this?

[@sumdumbmick]

this isn't controversial, either. it's literally how Special Relativity works. the curve of the relationship between t and t' is circular for vc. this is why tachyons would require energy inputs not to accelerate.

[@sumdumbmick]

the other closure of the parabola occurs at the same exact spot as the vertex you have. it just looks like a mirrored copy of the parabola you drew. there is nothing happening at infinity. that's nonsense. for the hyperbola you showed, there is no contact between the y-axis and the hyperola at y = +inf. the hyperbola reaches a height of +inf when x is the successor to 0, which contrary to Peano, is not 1. but this successor is a value that we use all the time without understanding it, since it's absolutely required for evaluating limits. when you do something like: lim x->0+ 1/x = +inf you obviously can't evaluate at x=0, because division by 0 is undefinable. further, we know this function is discontinuous, since for x0 we get a positive branch that grows in magnitude as we approach 0. so, when we take this limit and say that it gives us positive infinity, what we did is we evaluated it at the successor to 0. which I will notate as L(0). now, you can trivially see that L(0), 0 and -L(0) are completely distinct values, because 1/L(0) = +inf, 1/0 is undefinable, and 1/-L(0) = -inf. your hyperbola reaches L(0), and when it does its height is the largest possible infinity that exists, but it does not reach 0. and thus it is simply nonsense to claim that it touches the vertical asymptote, x=0, at y = infinity.

[@sumdumbmick]

it's hilarious to hear someone speak of rigor in mathematics when it's been known for 92 years that modern mathematics cannot possibly be rigorous. Incompleteness demonstrates that your assumptions about how mathematics works are wrong, and yet you just carry on acting as if that never happened... ok, but when that's the choice you make, you don't get to speak of rigor.

[@JavierSalcedoC]

32 x 102 and 32 x 243

[@Marktewk]

Yes, but why the ironing board?

[@alexandermitrofanov1327]

Employer: shows me the graph of the salary growth Me: but it never reaches the value we agreed on Employer: oh it does, it does! but the point is imaginary

[@animeking5068]

Hey guys can you please update the viewcount of the 301 video

[@phizc]

I'm not a mathematician, and I really don't get this.. How will y=x² meet up again and be tangent at infinity? That would mean that the square root of y would be 0 at high enough values for y, wouldn't it? At least *sqrt(+inf)* = 0..? Thinking about it a bit more, thinking of the parabola as a conic section, if the cone has a bottom plane, they would indeed meet up again - as straight lines, so those would be tangent to the bottom plane, i.e. "infinity", though it would also "work" with a finite sized cone, it just have to be "closed". The way it's drawn at 4:46 would imply the cone has a convexly curved "bottom" though, with a "tangent transition" between the cone and the curve part. Otherwise there would be a "corner".. Anyway, I can't get the math to work with just y=x², but maybe it works with the ax² + by² + cx + dy + e formula..

[@_ilsegugio_]

my grandson's grandson is gonna find this trivial

[@wbfaulk]

Professor Eisenbud's speech patterns remind me of Floyd the barber from _The Andy Griffith Show._

[@jpgsawyer]

Very cool but here is an idea. Are there points in the plane that are not on a tangent to the n conics and is there a way to determine if you are on such a point?

[@leefisher6366]

10:17 - Funny, I always thought they were Parker Circles.

[@Drachenbauer]

is a tilted cut through a cone really a perfect ellipse? I mean, at the higher end of the cut, the curvature of the cone´s surface narrower and at the lower end whider.

[@razielhamalakh9813]

You'd think, right? That's actually a common question. Turns out, because the cut approaches the wider part of the cone at a shallower angle, the section is in fact a perfectly symmetric ellipse. It is counterintuitive, I'll grant you.

[@romainhaym747]

So… What’s the deal with the decorated ironing board???

[@storytimewithunclekumaran5004]

Great video.

[@mofeyTEA]

why there is no xy term in quadratic formula?

[@nickush7512]

Facinating !! Thanks :)

[@justarandomdood]

19:00 bit of a typo/misspoken fact? (x-a)²+(y-b)² not (x-b)², right?

[@michaelfahie4228]

1:37 my brain came to a crashing halt when he showed that graph and said xy =1. The graph is incorrect and the equation is not a quadratic. I’m sure it’s a trivial error. Can someone tell me what the equation is supposed to be?

[@michaelfahie4228]

I kept watching and realize that it’s the graph that was weird, not the equation

[@ravis1577]

Translation to written words scroll is not helping full screen view. Pls do something

[@alan2here]

Does this include x^2 + y^2 = 0?

[@willnewman9783]

Yes, but it should be thought of as being over the complex numbers, so it is more than just the origin.

[@lawrencecalablaster568]

How can he just say that two things which don’t intersect are intersecting, or that one point is two?

[@sloppycee]

When does a parabola curve back on itself? Like, what??

[@DukeEllision329]

Can someone further explain why a line tangent to a circle goes through two points?

[@MeOnStuff]

It's not that it goes through two points (it only intersects the circle at a single point). It's that if you solve the underlying equations you get a repeated root. It's the same as, say, the equation x^2 = 0. This has one solution (x=0), but through the fundamental theorem of algebra we know every degree n polynomial has n roots: we can write it as a*(x-c_1)(x-c_2)...(x-c_n) = 0, where the c_i are complex numbers. So for x^2 = 0 this gives (x - 0)(x - 0) = 0. The c_i are, collectively, the roots, so in this example we have the roots 0 and 0 i.e. a repeated root at x = 0. The solutions are just the roots listed without repeats (without multiplicity, to use the maths term for it). Hope that helps.

[@anon-fo3id]

can't you cut a cone and get a triangle?

[@Nethershaw]

Woof. Is there any way to visualize the complex solutions that makes any sense? It's difficult to imagine tangency between two curves in the real plane that involves the complex plane. Do you need a three-dimensional projection to show such a thing?

[@GGoAwayy]

Turn the paper 90 degrees so you can't see the front or the back and are looking at it edge on... the circle and the line are overlapping in that dimension?

[@lawrencecalablaster568]

I… I don’t understand at all, but I’m along for the ride.

[@adrianf.5847]

By the way: Gauß was awesome, but his "proof" of the fundamental theorem of algebra contained a gap. Only Jean-Robert Argand gave a complete proof. (Gauß did find the Tukey--Cooley algorithm 150 years before Tukey and Cooley. In general, some mathematicians are rightly considered great mathematicians, but their achievements are being confused in rather arbitrary fashion.)

[@adrianf.5847]

Actually, the proof by Argand seems to use an incomplete infinite descent argument, which I believe would need some sort of Weierstrass theorem or ODE method to work. So I don't even know who first proved this theorem.

[@PushyPawn]

So many things went over my head that I now have a new haircut.

[@JavSusLar]

11:39 why don't you invite to the party the term xy?