This is way unimportant in comparison to the current idiocy of the neoliberal nutjobs. But, as they say in the theatre, the shitshow must go on.*
We had thought of taking this further whack at Bambi a while back, but had decided against it. Over the week-end, however we were discussing related mathematics with Simon the Likeable, and that has made us reconsider:
Get to work.
*) Mostly Andrew Lloyd Webber productions.
We were working on an update to polish off this one, when Simon the Likeable pointed out to us the video below. It could easily be its own WitCH, but it fits in naturally here (and also with this WitCH).
We’ll give people a chance to digest (?) this second video, and then we’ll round things off.
OK, time to round this one off, although our rounding off may inspire objection and further comment. We will comment on four aspects of the videos, the third of which we regard the most important, and the fourth of which is mostly likely to cause objection.
The first thing to say about these videos is that, as examples of teaching, they are appalling; they are slow and boring and confusing, simultaneously vague and muddyingly detailed. In particular, the “repeated addition” nonsense is excruciating, and entirely unnecessary. You want us to think of division as “how many”, then fine, but don’t deliver a kindergarten-level speech on it.
Eddie Woo’s video has the added charm that at times no one seems to give a damn what anyone else is saying; particularly notable is the 6:00 mark, where the girl suggests “Therefore it’s [i.e. 1/0 is] undefined?”, the very point Eddie wants to make, and Eddie pointedly ignores her so he can get on with his self-aggrandizing I’m-So-Wonderful performance. Dick.
The second thing to say is that the Numberphile video is littered with errors and non sequiturs, the highlights being their dismissing infinity as an “idea” (as if 3 isn’t), and their insane graph of . We’ll go through this in detail when we update this WitCH (scheduled for sometime in 2023).
The third thing to say is that the videos’ discussion of the impossibility of defining 1/0 gives a fundamentally flawed view of mathematical thought. The entire history of mathematics is of mathematicians breaking the rules, of doing the impossible. (John Stillwell has written a beautiful book, in fact two beautiful books, on the history of mathematics from this perspective.) As such, one should be very careful in declaring mathematical ideas to be impossible. So, 1/0 may generally not be defined (at school), but is it, as Eddie declares, “undefinable”?
Of course taken literally, Eddie’s claim is silly; as we suggested in the comments, we can define 1/0 to be 37. The real question is, can one define 1/0 in a meaningful manner? There are reasonable arguments that the answer is “no”, but these arguments should be laid out with significantly more care than was done in the videos.
The first argument for the (practical) undefinability of 1/0 is that we’ll end up with 1/0 = 2/0, leading to 1 = 2. What is really being claimed here? Why is 1/0 = 2/0, and why should it lead to 1 = 2?
The heart of this approach is asking whether 0 can have a multiplicative inverse. That is, is there a number, let’s call it V, with 0 x V = 1? Of course V couldn’t be an everyday real number (not that real numbers are remotely everyday), but that’s neither here nor there. It took a hugely long time, for example, for mathematicians to leave the safety of the world of everyday (?) integers and to discover/create an inverse for 3.
Well, what goes wrong? If we have such a number V then 1/0 stands for 1 x V. Similarly 2/0 stands for 2 x V. So, does it follow that 1 x V = 2 x V? No, it does not. V only has the properties we declare it to have, and all we have declared so far is that V x 0 = 1.
Of course this is cheating a little. After all, we want V to be an infinityish thing, so let’s concede that 1 x V and 2 x V will be equal. Then, if we assume that the normal (field) rules of algebra apply to V, it is not hard to prove that 1 = 2. That assumption is not necessarily unreasonable but it is, nonetheless, an assumption, the consequences of that assumption require proof, and all of this should be clearly spelled out. The videos do bugger all.
The second argument for the undefinability of 1/0, at least as an infinity thing, is the limit argument, that since tiny numbers may be either positive or negative, we end up with 1/0 being both and , which seems a strange and undesirable thing for infinity to do. But, can we avoid this problem and/or is there some value, in a school setting, of considering the two infinities and having them equal? The videos do not even consider the possibilities.
The fourth and final thing to note is that, as we will now argue, we can indeed make sense of 1/0 as an infinity thing. Moreover, we believe this sense is relevant and valuable in the school context. Now, to be clear, even if teachers can introduce infinity and 1/0, that doesn’t imply they necessarily should. Perhaps they should, but it would require further argument; just because something is relevant and useful does not imply it’s wise to give kids access to it. If you’re collecting wood, for example, chain saws are very handy, however …
First, let’s leave 1/0 alone and head straight to infinity. As most readers will know, and as has been raised in the comments, mathematicians make sense of infinity in various ways: there is the notion of cardinality (and ordinality), of countable and uncountable sets; there is the Riemann sphere, adding a point at infinity to the complex plane; there is the real projective line, effectively the set of slopes of lines. Cardinality is not relevant here, but the Riemann sphere and projective line definitely are; they are both capturing 1/0 as an infinity thing, in contexts very close to standard school mathematics. And, in both cases there is a single infinity, without plusses or minuses or whatever. Is this sufficient to argue for introducing these infinities into the classroom? Perhaps not, but not obviously not; infinite slopes for vertical lines, for example, and with no need for a plus or minus, is very natural.
What about the two-pronged infinity, the version that kids naturally try to imagine, with a monster thing at the plus end and another monster thing at the minus end? Can we make sense of that?
Yes, we can. This world is called the Extended Real Line. You can watch a significantly younger, and significantly hairier, Marty discussing the notions here.
The Extended Real Line may be less well known but it is very natural. What is in this world? Take a guess. Or, ? It all works just how one wishes.
But what about when it doesn’t work? You want to throw or or at us? No problem: we simply don’t take the bait, and any such “indeterminate form” we leave undefined. In particular, we make no attempt to have be the multiplicative inverse of 0. And, then, modulo these no-go zones, the algebra of the Extended Real Line works exactly as one would wish.
Can these ideas be introduced in school, and for some purpose? No question. Again, whether one should is a trickier question. But as soon as the teacher, perhaps in hushed and secretive tones, is suggesting or , then maybe they should also think about this in a less Commandments From God manner, and let come properly out of the closet.
Finally, what about 1/0 in the Extended Real Line? Well, the positive or negative thing is definitely an issue. Unless it isn’t.
There are many contexts where we naturally restrict our attention to the nonnegative real numbers. And, in any such context 1/0 is not at all conflicted or ambiguous, and we can happily declare . The exact trig values from 0 to 90 is just such a context: in this context we think it is correct and distinctly helpful to write , rather than resorting to a what-the-hell-does-that-mean “undefined”.
That’s it. That’s a glimpse of the huge world of possibilities for thinking about infinity that Numberphile and Woo dismiss with an arrogant, too-clever-by-half hand. Their videos are not just bad, they are poisonously misleading for their millions of adoring, gullible fans.
156 Replies to “WitCH 41: Zero Understanding”
Here is the bit that annoys me the most (well, in this video…)
“Multiplication is repeated addition”.
“Division is repeated subtraction”
This guy should write for Cambridge.
Everything that annoys me stems from these two sentences. I’ll elaborate once others have had a go.
Thanks, RF. Yes, at that year level, teachers definitely shouldn’t be pushing the “repeated addition” thing. Woo wants it so he can do the “how many” thing, but he could have done that directly. I didn’t understand the “Cambridge” crack.
His logic makes as much sense as what you see in Cambridge textbooks, think index laws for example…
It is painfully over-done, misses the entire point and is factually wrong.
I could go on, but I suddenly have online lessons to prepare. For tomorrow.
Thanks, RF. No question it is over-done. How is factually incorrect? Just the repeated addition thing, or something more central? What do you regard as the point?
“Division is repeated subtraction”.
Even if I accept Subtraction as the opposite of addition (inverse is not the same as opposite) and division as the opposite of multiplication, “repeated” is not a mathematical operation, so to me the argument fails simple logic.
But maybe I’m over thinking it all.
My brain hurts.
Whenever I get that question from a student “but sir why can’t we divide by zero” I just point out that if there was an x such that 1/0 = x, then 0x = 1. I’ve never had any dissatisfaction with this response. I’m waiting for the student who asks “but sir, why is 0x = 0 for all x?”
Trying to more precisely identify the crappiness in the video, apart from the general tediousness, I just felt that it doesn’t quite get to the heart of the issues. If I was a determined defender of division by zero, I’m not sure I’d be persuaded by the (1) argument-from-limits, or (2) the argument-from-2=1. I need to channel my inner crank a bit more, but both of these seemed to burden the division-by-zero-ist with commitments they wouldn’t accept.
Thanks, SRK, and I apologise for hurting your brain. No question the video is tedious.
I’m going to elaborate a bit more, but try to be abstract for now.
I find Mr Woo annoying because I know most of his lessons are really good but these examples are quite possibly doing more harm than good. In which case – why do them? To show off? It isn’t working. To get students to “think”? See WiTCH 40.
So… why do these lessons exist? Genuine question, not sure there is an answer.
Thanks, RF. No question videos such as the one above, which have been watched by millions, are doing more harm than good. As to why the videos, and the lessons, exist? Good question. I don’t think the videos are showing off; It seems that Woo uploads videos of his lessons pretty much religiously (pun intended). The lesson itself smell like showing off.
Are Woo’s more standard lessons better than the one above? When Woo first appeared on the scene I looked briefly at his videos and I liked them, while thinking they were nothing that many good teachers do as a matter of routine. The more I reflect upon them, however, the less I like them. But, I haven’t given them much thought.
My initial thoughts upon first seeing Woo-Tube were along the lines of “any time Mathematics gets more attention is a good thing” but I’ve come to realise there are limits (no pun intended) and when I hear comments from “Mathematics teachers” say things like “I wish I had a teacher like Eddie” I start to realise my original assumptions were quite wrong.
So who watches Marvel’s Agents of S.H.I.E.L.D.?
The scene with Mack in the bar, then Deke appears on stage, claims ownership of Don’t You Forget About Me and then performs it with his band as an original. Agent May’s look as Deke retells the story (50 second mark) and Mack’s eye-roll in the bar (60 second mark) – they’re the exact reactions I have to Woo after his 5 second mark.
(For those who are interested: https://www.youtube.com/watch?v=3eWSPHzPLsc )
(And I confess, I played Woo at warp speed and dropped out of warp for a second or 2 each minute). By the way, what’s wrong with 1×0 = 2×0, divide both sides by zero, 1 = 2? I’ve found this always gets the point across about 8:45 minutes faster than Woo.
On the topic of jokes, I heard a good one today that’s going around … “Did you hear the one about VCAA planning to cancel the 2020 exams?”
I enjoyed the one going around where the GAT is running in term 4.
As for the (1×0)/0 = (2×0)/0 -> 1 = 2 argument, this seems a lot better than the detour into 1/0 = infinity.
Hmm. I think this one might have me beaten up (again).
I also have a problem with but this is a battle I don’t think I can win.
Edit: I should elaborate… many of my former colleagues seem to believe is a number. I believe they are not quite sure what they mean by number.
I think this is an interesting example because Mr Woo is doing something that I think we were encouraged to do in my teacher education – he’s taken something simple (that I would explain the same way as SRK says above) and made it complicated. This is encouraged because it is seen as “creating opportunities”. For instance, it is creating opportunities to engage in “mathematically literate talk”. Look how long this goes. So many opportunities.
Now when education people say “literacy”, I quietly replace it with “bullshiteracy” in my mind. The reason is that it now seems to mean paying attention to form rather than content. To create an argument that looks like a mathematical argument. And I wonder if that’s really something we want to do – to train people to bullshit better? And the answer might be: “yes, because people who bullshit better do better at life – don’t you want your students to succeed?”
Also, I regret maybe sounding a bit too grumpy here. It is a genuine dilemma I see in deciding what skills to focus on in teaching, but maybe not particularly evidenced by Eddie Woo. I had a look at his video about Pythagoras’ Theorem and thought that was pretty nice.
s-t, I think you were just the right level of grumpy. Whatever Eddie thinks he’s doing, it’s a poor lesson, and tedious. I’m willing to believe Eddie has better lessons, although I don’t really care enough to look.
s-t, watch your language! There are statisticians present!
“Creating opportunities” …? Yep, that’s the sort of total bullshit that spews from the mouths of these so-called ‘educators’ all the time. It’s actually called creating bullshit and is done for several reasons, the main ones being to hide a superficial understanding of the content being taught, and to sound knowledgeable (to those who lack the knowledge).
These so-called ‘educators’ know all about it because they do it through the entire teaching courses. As do the DuLL pontificating nitwits at VCAA and at things like Meet the Asses.
As for Woo and Pythagoras, any teacher with an ounce of knowledge, creativity and passion can teach Pythagoras’ Theorem “pretty nice”. Woo has done nothing that any good teacher doesn’t do as par for the course. (But that teacher doesn’t get 20,00 likes from people who don’t know any better, or lucrative advertising deals). Woo has become nothing more than another nitwit self-promoting ‘influencer’ and that video proves it.
(The real problem with teaching Pythagoras’ Theorem is that it’s one of those things that must get taught every single bloody year from Yr 7 onwards but it stays at that Yr 7 level).
His use of the word ‘undefinable’ makes me uneasy – mostly because I’m not convinced I really understand what it means for something to be undefined. In this case, my understanding is that (for example) on the projectively extended real line a/0 is defined as being equal to infinity. So we can define division by 0 if we want to – it is not ‘undefinable’. However, the algebraic structure that we are left with is not a field. Technically, I guess that we could define division by zero to be anything that we choose it to be, it’s just that if we do so then we can’t expect the basic arithmetical operations to keep behaving the way that they normally do. Under the real numbers, when division is defined, we exclude division by zero for this reason. This is why it is undefined – not because it is undefinable.
Thanks, Damo. I think this an important aspect to consider, and I’ll make a further reply in a moment.
Clearly, no one commenting here is impressed by Woo’s video, which is good: it is impressively unimpressive. I want to consider Damo’s angle, on the content of the lesson, but I think there is probably also more to say, just on the video as a model of teaching.
If the video is so poor then why do four million or so viewers buy into it? Yes, yes, people are dumb, or whatever. But what, more specifically, is the disconnect between their view of the video and the views expressed here?
To take a line from Seinfeld:
“Why am I watching it?”
“Because its on TV!”
People watch Woo because others tell them to (or send them links to videos on social media). People watch Khan Academy videos for similar reasons, except I don’t hear the names of those individual teachers thrown around the same way I do with Woo.
Woo is a good teacher, that I’m not going to dispute, I’ve seen a few of his more “mainstream” lessons and they are good teaching, but that is it – you would see the same quality if you knew where to look in 80% of schools.
Where I have a problem is these “special” lessons where Woo is clearly trying to achieve something (as S-T suggested) except I don’t think he is achieving what he intends to and is quite possibly doing a lot of damage in trying.
The only thing Woo is trying to achieve in these “special” lessons is self-promotion. He would make the perfect daytime chat host, where you can fill 1 hour with 59 minutes of nothing.
RF, I love the Seinfeld quote. So true. And taken from a show about nothing. What irony.
No Marty, I think it really is that simple: There are more dumb unthinking people that are easily (and maybe want to be) conned than intelligent thinking people.
They can point to Woo and say “That’s why I was never good at maths, I never had Eddie Woo as my teacher.”
By jumping on the Woo bandwagon they boost their own self-esteem. His success and popularity is theirs – vicariously. It’s more or less how cults work – in this case it’s the Cult Of Woo. Bart Simpson warned us year’s ago when he said “Don’t have a COW.”
The more generic question is why do nitwit ‘influencers’ in general attract so many followers? How many posters here follow some nitwit ‘influencer’ in general? What’s the disconnect in general?
Thanks, JF. Perhaps I’m looking for more than there is. But I’m still curious. I don’t believe people can get much of anything in the way of mathematics out of Eddie’s video above. Assuming that is correct, do people nonetheless think they are getting something, or are they just pretending?
What people are getting is entertainment and the illusion of either learning something or of knowing more than they thought they knew (in the exact sense of what s-t describes below). It makes them feel good and lament that they never had Woo as their teacher. They feel part of a ‘tribe’ (cult).
Oh snap! Yet another moment predicted by the Simpsons…
I’ve heard that a recipe for a successful talk is: one third telling people stuff they know already, one third telling them stuff they don’t know but can understand, and one third telling them stuff that ‘blows them away’, i.e., it’s impressive but not understood. It’s possible he achieves this mixture for a large group of people. It creates the feeling that the stuff you know already is just a short step away from the amazing stuff you don’t understand. I think a lot of pop science works that way too. (I’m just guessing.)
He’s revising basic arithmetic and giving people an opportunity to feel good that they can calculate 15 – 5 – 5 – 5. He maybe gives them a bit of a glimpse of something and helps them understand. And he makes little asides to material that is perhaps beyond what he’s teaching. This is a recipe for an enjoyable talk, not a lesson though.
Thanks, s-t. The distinction between entertainment and teaching is a good point. I’m not convinced Woo’s talk is either good entertainment or good teaching any more than a Big Mac is a good (tasty) meal or a good (nourishing) meal.
Damo points out Eddie’s description of 1/0 as “undefinable”. So, let me explicitly ask some questions.
1) Is 1/0 undefined? Why or why not?
2) Is 1/0 undefinable? Why or why not?
3) Is there any purposeful distinction between 1) and 2)?
4) To what extent has Eddie clarified these notions or proved any truths about them?
Yes, there is a distinction between (1) and (2). is undefined, but you can define it to be one (or zero) in situations where that may be useful.
Of course, Woo seems to think (unless I misunderstood) that is equal to 1.
Thanks, RF. If something is “definable” then why don’t we just go ahead and define it? Is 1/0 definable, defined, both or neither?
I suspect its got to do with how division is defined. Either the number of times a number is contained in another number or the inverse of multiplication. When dividing by zero you get different conclusions for that are not reconcilable (e.g. 0 could be thought of fitting infinitely many times into any number but the inverse definitions gives you answers like 1=2). Zero division can be defined but the definition is not consistent and so it is left as undefined.
OK, lets suppose, for argument sake, that a divides b if and only if there exists a number c such that .
By this definition, any number divides zero, because you can set
If you have then it becomes problematic (unless you are working in modular arithmetic or something similar)
Thanks, Potii. So, thinking of division as “how many”, we get the idea of 1/0 = ∞. Is the inconsistency, in this case, with “multiplicative inverse” a deal breaker?
I think it should be since you want things to work consistently but also thinking about other situations in mathematics inconsistencies have work arounds. So maybe what I said is an insufficient for zero division being undefined. Though maybe it’s only undefined in certain situations (e.g. arithmetic bu maybe not when dealing with limits of functions like 1/x).
Just throwing some thoughts around. I’ll think about it more carefully later.
I’m not 100% convinced that we need to define something just because we can.
In the case of , sure define because if you don’t then some basic combinatorics rules break down. Define because if you don’t the distributive law breaks down.
But I feel there are situations where is a convenient definition and situations where may be a more convenient definition. Both of these make definable, but I’m not pushing for either definition to be accepted as the universal standard.
Similarly, is zero a natural number? I would argue not, but for some situation, it may be more convenient to define the natural numbers as the non negative integers, rather than the positive integers.
Semantics, perhaps. I am prepared to be convinced of my errors if they exist here.
And I would say is not definable. No matter how you define it there will be a way to use that definition to break some other rule (as far as my limited imagination allows at the moment)
Thanks, RF. I think that is at least a good platform for discussing the notions.
We take something as “defined” if the definition is natural, is useful and is generally accepted. Thus 0! is “defined” to be 1.
By comparison, is not “defined” but one might say it is “(usefully) definable (in a limited setting)”. Any definition won’t be consistent in all desired contexts, but in a suitably clear and limited context, a definition could be useful.
But then, accepting this language, how is “definable” in a way that 1/0 is not?
1 is a number, I can accept that. I cannot accept is a number.
And before anyone asks, I cannot accept is a number either.
Why not? Do you use ∞ in your classes? If it’s not a number, what, if anything, is it?
I use in calculus and that is about it.
I use the phrase “undefined” quite a bit in younger levels.
Is that “limit” a number? If not, what is it?
Numbers at the very least obey the commutative laws of addition and multiplication, and except that but because in order to do the second sum, one first needs to find the end of the first number.
So even if I ignore the logical fallacy there remains a fundamental problem.
So what is ? Good question. The best answer I have read comes from Eugenia Cheng:
“Infinity isn’t a natural number, an integer, a rational number or a real number. Infinity is a cardinal number and an ordinal number.”
In highschool, students use cardinals and ordinals, I will concede. Do they need to know what this means? I’m not sure.
Why can’t ∞ + 1 equal ∞? I’m not sure Cheng helps all that much.
Transfinite arithmetic is not like ‘real number’ arithmetic.
(“It’s arithmetic Jim, but not as we know it”)
1) I would say Yes and No. It is undefined in the Real numbers. However it is defined on the projectively extended real line. As Potii covered above, once we define division then we cannot define division by zero in the Real numbers and retain consistency in how we apply the basic operations. However, for example, there are geometrical and topographical motivations for defining a/0=inf, which gives the projectively extended real line. However, defining a/0=inf creates problems with our basic arithmetic operations, which makes it necessary to define some of these operations (such as a + inf) while leaving others undefined. The consequence of this is that our new number system is no longer a field. (it’s been a loooong time since I left University, so my understanding of this stuff is very sketchy – I don’t think that detracts from my point).
2) Is 1/0 undefinable? If you want to retain the consistency of the basic operations, sure. Otherwise, of course it’s not undefinable. But ‘undefinable’ is not a mathematical term – Woo’s use of it to explain why 1/0 is undefined is, well, crap.
3) Is there a purposeful distinction? Yes. One has a specific mathematical meaning. The other does not. There is usually a motivation for leaving something undefined – such as retaining the consistency of the basic operations. I’m not sure what undefinable would or could mean in a mathematical sense.
I’m going to leave 4) for now. I’ve spent too long on this and my kids are getting hungry and grumpy. Apologies for the length and if I’ve replicated what other people have said.
Of course “definable” taken literally is vacuously true: we can obviously define 1/0 to be 37, for example. Doesn’t do a lot, but we can do it.
But, as I suggested in reply to RF, let’s take “definable” to be “usefully definable in a limited setting”. Can that be done for 1/0? And, not in a weird projective world manner, but in the world of school mathematics.
The “undefinable” vs “undefined” bit is weird. If I were more of a modal logician I might have some sharper thoughts about it.
My naive reaction is that we say that 1/0 is undefined because if 1/0 were a real number, then contradiction follows. Contradictions are necessarily false, hence the undefined-ness of 1/0 is necessary, hence it is undefinable.
As Damo noted above, it is possible to say things like “1/0 = infinity in the projectively extended real line”, but I’m not persuaded that this refutes my view. My response would be that “1/0” is being used in two different ways, depending on whether one is talking about the real numbers or the projectively extended real numbers.
Thanks, SRK. Obviously defining 1/0 as a real number doesn’t work well, although I’m campaigning for 37. But who says we have to stay within the world of real numbers? After all, √(-1) leads us to …
So, I’ll ask the same question I asked Damo. We obviously don’t want to get into projective worlds or Riemann spheres, or whatnot; no superhero cheating. But, in the context (or some context) of school mathematics, can we somehow make useful sense of 1/0?
What about gradients? If we consider the gradient of a straight line to be rise/run then the gradient of a vertical line can be considered to be 1/0. Defining this to be inf could make more sense than leaving it undefined?
As well as making sense of the gradient of x^1/3 when x = 0…
Except that if you define the line to have a gradient of and the line to have a gradient of zero then you get
Which is amusing, but not much else.
Why do you get ∞ x 0 = -1?
The lines are perpendicular; the product of their gradients is therefore .
Again, may amuse a good Year 7 student or a teacher with minimal experience but no one else.
RF, are you sure that the product of the gradient of perpendicular lines is -1?
If the gradients are non zero and you use the formula then yes, you can show the product of the gradients of two perpendicular lines is
I’m guessing it is the if statement you were looking for here?
I guess it also depends on how you define perpendicular…
Yes, RF, the point is that you m x n = -1 formula depends upon the gradients being non-zero. So, I don’t see that it tells us anything about vertical lines and infinite slopes.
That’s an interesting thought, but I’m inclined to say that doesn’t accomplish much. In that context, stating that the gradient of a vertical line is infinity seems no more than verbal shorthand for saying that the difference quotient increases (or decreases, as it may be) without bound as h -> 0.
SRK, doesn’t there feel to be something infinityish about a cliff?
Not when I hit the bottom.
Doesn’t there feel to be something infinityish about the slope of a cliff?
OK… here is where I start to have a problem.
If then, by logic . I’m fine with this.
I’m less fine with: if then
Putting these together, if we define as something then do we define as the same something or a different something? If we define them as the same, then which is clearly nonsense. So they must (by school-level logic) either be different things, or our concept of numbers needs to be changed.
At school level, I think it is easier to not define
To be clear, I am not advocating teaching this at school level. Getting back to the original video, I don’t know what Woo is thinking. You can easily give some examples of how division by 0 breaks down to illustrate why we leave it undefined. However, Woo goes down a different path and ultimately justifies it by saying that it is ‘undefinable’. I think that this is (clearly and unnecessarily) wrong.
Marty asked if we could make useful sense of 1/0. I think that we can, as the gradient of a vertical line. However, you would have to be very careful in doing so, in particular you would need to be upfront about the fact that this has implications for arithmetic (without necessarily going into those implications). Would I ever try to help my school level kids make sense of 1/0 in this way? I don’t know. Probably not.
Why does 2/0 = 1/0 imply 2 = 1?
It doesn’t. But IF
RF, you wrote that if we define 2/0 to be the same thing as 1/0 then it follows that 2 = 1. I don’t see that.
I teach my (year 7) students that if .
So, by my definition, since it must be true that or my definition of equal fractions is wrong.
I’m guessing it is me who is wrong.
RF, that certainly shows that, whatever 1/0 and 2/0 might mean, they cannot be equal and work like normal fractions. Does that mean that the idea is doomed to failure?
Using your definition, do we not get 1 x 0 = 2 x 0 = 0 and 1/0 = 2/0 = inf?
Yes and no. By my (year 7) definition we get the fractions are equal, but I’m not convinced of what they are equal to.
Yet again, I’m going to have to go away and ponder this for a while.
Despite the constant feeling that I don’t know anywhere near as much as other commenters, I do really enjoy this experience by the way… I wonder if the VIT will let me claim this as PD hours.
RF, Most definitely it counts as PD. Look at the rules.
Hi RF. Your ‘theorem’ is OK, it’s an “if-then” theorem (=>) not an “if and only if” (“iff”) theorem (<=>).
In other words, you’re only saying if then , and that’s OK.
An “if an only if” goes both ways and the other way is if then . That’s not OK but you know it’s not OK, and it’s not what your theorem is saying, so that’s OK. Ok?
But I would state the natural restriction (so that the fractions define a number) and then all is well both ways. The restriction might make for an interesting discussion.
(Truth be told, I don’t think that many teachers are too careful with their definition of equal fractions, and are a bit too free with the phrase “equivalent fractions”)
Your restriction suggests you don’t think is a number. I’m in agreement (of sorts) but sense that this is the bit that Marty sees as… problematic?
Marty might have something in mind but I think he’s mainly looking to nudge people down the rabbit hole and enjoy the ensuing fun. I’m going to hold the rope and see who climbs back up.
Come on in. The water’s fine.
It’s not the water I’m worried about.
I think that amongst primary school kids (certainly) and secondary kids (to a certain extent) it is useful to talk about addition, subtraction, multiplication and division as things that just exist – these are the rules of arithmetic and, let’s face it, they kinda make sense. And dividing by zero doesn’t make sense when we try to understand it using these rules – and it’s easy to show that it doesn’t make sense – so we can’t and don’t do it. For most of these kids, I don’t think that you need to go into any more depth. But Woo has attempted to go into more depth, particularly this concept of ‘undefined’, and I think that he has got it wrong – although I’m aware that I don’t fully understand it myself, so below I am both trying to make sense of it myself, as well as being most likely wrong.
Anyways, first thing to note is that the ‘rules’ of arithmetic are based on rigorous mathematical definitions – starting with the addition of natural numbers, the additive inverse, addition of integers … In other words, the ‘rules’ of arithmetic are a consequence of a series of careful and purposeful definitions. In order to define division, we need to have first defined multiplication. One definition is c/a = b iff a x b = c. Except that if a = 0 then 0 x b = 0, which means that if c is not equal to 0, this definition of division breaks down. This is where the contradiction that SRK refers to occurs and it leaves us with a choice to make – we have to do something about division by zero. We can either define it to be equal to something, or we can exclude it from the definition, leaving it undefined. But we have to do something.
I don’t think that choice is trivial and I think that it gets to the heart of the distinction between ‘undefined’ and ‘undefinable’. Of course at this point in time we can define it to equal something. However, there are consequences to defining it, in particular there will be times when our carefully crafted arithmetic operations will break down. As a result, we should only define it if doing so serves a purpose. And there are clearly times when it does serve a purpose. Yes we are talking about 1/0 in two different ways, but that is a consequence of whether we choose to define it or leave it undefined.
Thanks, Damo. So that is the question: is there a purpose to defining 1/0. The algebra doesn’t appear to be very promising, but are you sure nothing is there? Could there be another purpose?
The way you ask this question suggests you’re pointing at something much bigger, such as defining (or not) ?
Hi, RF. I think √(-1) is a good example to think about.
In early years, kids are taught that it “doesn’t exist” (perhaps more correctly and carefully worded)? Why?
Of course there is a reason, but I think it is worth considering that reason? Why does that reason not remain a barrier when √(-1) is eventually defined? What, if anything, do we have to concede when we do define it?
Then, can the same thinking be applied to 1/0?
OK, I never say doesn’t exist, even with junior students. I say that for the moment we don’t know how to take the square-root of a negative.
That said, after reading all this, I’m not sure I know how to not divide by zero.
Maybe JF’s “don’t go there” was the wiser choice.
I refuse to go down the rabbit hole. At secondary school level dividing by 0 has symbolic meaning – go back and use limits. It may even have contextual meaning (eg. vertical line in the context gradients).
But this is a long, long way from it being defined.
In the world of school mathematics dividing by zero is undefined. Finis. https://www.youtube.com/watch?v=b9434BoGkNQ
JF, it’s fine for you to refuse to go down the rabbit hole. I’m just gonna give you a little push …
I think that it is useful when it comes to the gradient of vertical lines. For example, the derivative function is undefined where there is a discontinuity, but it is also undefined where there is a vertical tangent line, such as for the graph y=x^1/3. If we define 1/0=inf then dy/dx=1/3x^2/3=1/0=inf when x=0. I think this is a ‘useful definition in a limited setting’ that gives a clearer picture of what is happening. It also provides a different (geometrical) perspective on tan(pi/2) (where the line through the origin runs parallel to x=1). One of the things that I’ve struggled with is that cot(x)=1/tan(x)=0 when x=pi/2, even though tan(x) is undefined at that point. If we take 1/0=inf and 1/inf=0 then both tan(pi/2) and cot(pi/2) make sense.
Thanks, Damo. So, in particular, you’re comfortable to have tan and cot (and sec and cosec) take on the value ∞, rather than be undefined?
@Marty: I’ll declare my comfort for tan(x) and cot(x) approaching (not equalling infinity) (and try to save Damo from going down the rabbit hole).
@Damo. Re: One of the things that I’ve struggled with is that cot(x)=1/tan(x)=0 when x=pi/2, even though tan(x) is undefined at that point. If we take 1/0=inf and 1/inf=0 then both tan(pi/2) and cot(pi/2) make sense.
tan(x) = sin(x)/cos(x) so I prefer to consider cot(x) = 1/tan(x) as defining cot(x) = cos(x)/sin(x). Then there’s no ‘problem’.
More generally, I initially found it tricky to teach the sketching of reciprocal graphs until it occurred to me that there wasn’t a ‘symmetry’ between x-intercepts and vertical asymptotes:
If y = f(x) has an x-intercept at x = a then y = 1/f(x) = g(x) has a vertical asymptote x = a ( as ).
BUT if y = f(x) has a vertical asymptote x = a then y = 1/f(x) = g(x) has either an x-intercept OR a ‘hole’ (removable singularity) at x = a. To decide which, you must go to the rule for y = g(x) and see what happens at x = a: 0 (=> x-intercept) or (=> ‘hole’)?
So I have a contextual meaning for but this does NOT define .
So the reciprocal graph of y = tan(x), that is, y = cot(x), has an x-intercept at (pi/2, 0) and the reciprocal graph of y = ln(x) has a ‘hole’ at (0, 0).
Thanks John, you’re a good friend. Unfortunately, in this instance, I think I’m a lost cause, I’m already in too deep. With regards to considering cot(x)=cos(x)/sin(x), this is what I do as well. However, I struggle with it, because I feel it’s almost cheating.
Using a simpler example, if f(x)=1/x^2, then f(x) is undefined at x=0. If g(x)=1/f(x) then surely it must also be undefined at x=0? If f(x)=sqrt(x) and g(x)=(f(x))^2, what would the graph of y=g(x) look like? Going back to the first function, f(x)=1/x^2, if we have extended the real numbers to include inf, then f(0)=inf is defined and thus, so is g(0)=1/f(0)=0. Which is nice.
Well Damo, a friend in need is a friend indeed.
For your first example I’d say f(x) = 1/x^2 has a vertical asymptote at x = 0 therefore 1/f(x) = g(x) = x^2 has either an x-intercept or a ‘hole’ at (0, 0). Looking at g(0) I see it’s equal to 0 therefore it’s an x-intercept.
On the other hand, f(x) = x^2 has an x-intercept at x = 0 therefore 1/f(x) = 1/x^2 has a vertical asymptote x = 0.
I’m not sure where you’re coming from with the second example.
I did have a point, but as I began typing my reply I decided that it was going too off topic, down a different rabbit hole that I really don’t want to go down. I beg your pardon.
Hah! I’m not comfortable with any of this. I see it as being a fairly natural (even necessary?) extension of defining the gradient of a vertical line as inf. The difference between the example of dy/dx=1/3x^2/3 and tan(pi/2) is that the limit of the first exists and is inf, while the limit of the second doesn’t exist as the left limit = inf while the right limit = -inf. Unless we define infinity in such a way that we take the right limit to also equal infinty. (That is, there is no +inf and -inf, just inf).
Am I comfortable with all of this? Nope. It’s been too long and I think that I would need to redo my degree to really feel comfortable discussing this. Do I think that defining 1/0=inf is useful in the above contexts? Yes. Is not being comfortable with it a problem for me? No, it spurs me to look deeper into ideas that I haven’t thought about for years, which can only be a good thing. Would it be useful to teach this to students? It depends. I can see it having a place, in the right class, with the right students and the right teacher, and if it was taught with a great deal more care than Mr Woo took above.
Excellent reply, Damo.
“I think that amongst primary school kids (certainly) and secondary kids (to a certain extent) it is useful to talk about addition, subtraction, multiplication and division as things that just exist.”
I feel really uncomfortable with that, because why would you want to do that? I am a big fan of playing with blocks and water and dirt, and when I think about the question, ‘what is division really?’, I come back to thinking about dividing actual real objects into parts or sharing them into groups – there are different possible interpretations and they could all be the source of a reasonable definition, so long as they make sense. Obviously none of these make sense with 0. How can you divide something in zero equal parts? (It’s still maybe a useful starting point for thinking about it though.)
When the concrete definitions don’t make sense, we use more abstract structures to extend them, so long as it works. There’s no guarantee that it will work. I don’t think we should assume it will always work or that there has to be a way it will work. When it does, it’s kind of amazing.
Trying to define 1/0 in the context of the algebra of fields leads to problems because 0 is defined as having a special property that is contradicted if we set 1/0 equal to anything inside the field.
But we can extend the real line by adding a point (which is identified as plus/minus infinity and joins the real line together in a loop). This works for certain purposes and is useful, but it’s never going to reckon consistently with the field axioms.
Hi, s-t, I agree with the general sense of what you say: we generalise or abstract our notions if it is useful and if it works. I think, however, you may be being a little too legalistic on whether something “works”. Do the quaternions “work”, for example?
I think it depends on your goals. I think quaternions work in the sense that their definition consistently generates enough complexity to be interesting. It’s like deciding the rules of a game. Some rules lead to a trivial game that is not worth playing, and others to all kinds of possibilities.
s-t, the point I am making is that by generalising from the complex numbers to the quaternions you gain something, at least in the contexts (which exist) where quaternions give insight, but you have to be willing to give up something ( a x b = b x a) to do so.
Is it possible, again in the school context, that the same may be true for 1/0?
I’m not sure. I guess when you define , the obvious thing you gain is exactly what you’ve lost, i.e., 0 isn’t special anymore. You can invert and then invert again and get the same thing back, and it swaps and . So I’m thinking maybe there could be something related to geometry of inverting things (inverting around a circle, that sort of thing?). Yes, it’s possible. But there are so so so many things I don’t know anything about, and I would guess the best examples are among them.
Also, it took me a while to remember what book it was, but this reminded me of the book Journey into Geometries by Marta Sved. I read it a long time ago and it has something about dividing by zero and inverting circles. So that is why I thought of that. I think it is aimed at high school level. I think I read it not much after high school (first year uni maybe). I don’t know if I understood much but it was fun!
Thanks, s-t. Of course there is all manner of mathematical exotica that one can, and should, at least try to give a taste of to school students. I’m not really thinking of such infinity exotica here, although of course others are free to do so.
Hi, s-t, I just found a comment of yours in the trash. Was that deliberate? (It seemed like a perfectly reasonable comment to me.)
Thank you! I posted a comment based on my impression and then I worried that it was a silly comment, but I had to attend to parenting things so then I couldn’t think about it. But if you don’t think it is silly I guess it isn’t. 🙂
If only Woo and Numbernuts were half as self-critical as student-teacher. I’ve restored the comment.
I don’t know what you have in mind, Marty – do you have something in mind? Assuming you do have something in mind in the world of school mathematics, is the gain greater than the loss?
(I ask this for the benefit of those who went down the rabbit hole and are now feeling cold, wet, hungry and confused).
A man’s got to know his … limit-ation-s:
Given the interest, and given I’ve been shoving people down the rabbit hole, I’ll try to update this WitCH quickly (give or take coping with the plague.) I won’t preempt that, but a brief response.
I do have something(s) in mind, which are along the lines that people have suggested here. Whether the gain outweighs the loss, I’m honestly not sure. I started pondering this (again) after chatting with Simon the Likeable, but didn’t come to a clear conclusion. But it did seem worth a WitCH, since, however one wants to think about 1/0, Woo’s video sucks.
I understand this is a hard one, and it’s one with no definite answer (except that Woo’s video sucks). But I think it’s a very good one, to think about what the rules of mathematics are, why they are what they are, and to what extent we can present those rules in school in a (mostly) clear and (mostly) honest manner.
I wonder about conditional probabilities where the event conditioned over has measure zero. For example, throw a dart at the interval [0, 1]. (Assume the tip of the dart hits a single real number). What’s the conditional probability of the dart hitting 1/2, given that it hits 1/2? Intuitively, the answer is 1. However, using Pr(A&B)/Pr(B) requires division by zero.
Of course there are other responses. (1) Rule out zero probability events by fiat; (2) Treat the ratio formula as something that can be used whenever the denominator isn’t zero, but not as a definition of probability.
1) means that the dart hitting the board is ruled out. (And yet it hits, depending on the dartist)
2) is an unsatisfactory ad hoc solution.
The resolution is to use limits.
Now, if you’re learning mathematics through a YouTube video, you will never in a trillion years understand the mathematics. You’ll think you have understood it, but you’ll be cheated. It’s such a sadness that you think you’ve learnt mathematics on fucking YouTube. Get real.
Craig. I’m shocked, shocked to hear you suggest such a thing. Does that mean, perhaps, that the Mathologer is truly evil?
It was bad enough that he was doing it without any financial gain, now he’s asking for donations?! He has to be stopped at all costs.
(if anyone didn’t get my “get real” thing https://www.youtube.com/watch?v=wKiIroiCvZ0“
0^0 is another chestnut
As usual Wikapaedia gets it right saying “mathematicians can not agree on this…”
For statisticians combinatorics using 0! , the coefficients of binomial theorem , various Taylor series,validating the derivative power rule etc it is useful to have 0^0 approaching 1 but there are also cases where the value is undefined in the complex domain for example
Thanks Steve, Eddie Woo has a lesson on this very point…
The big problem I have is that as a function is not continuous for x<0 and even if it does approach 1 for x approaching 0 from above, the basic index laws simply don’t agree.
It is more convenient to define for specific purposes, this I will freely admit. Does it mean that I think should be the definition? Absolutely not.
OK… apologies to anyone who has already stated this (there are a lot of comments and not enough time to read them all).
I’ve come to the conclusion that needs to be defined, much like needed to be defined to allow the rule to continue to work for negatives.
I just don’t know how to define it. I’m not satisfied with because 0 is neither negative nor positive but 1 and are positive. Can we get around this with complex infinity? Perhaps, I don’t know yet.
Thanks, RF. I’m not trying to argue that 1/0 needs to be defined. But I’m questioning commenters because I think they’re too quick to dismiss the ideas it can be done (in the school context) and/or that there may be benefits in doing so (in the school context).
EDIT: I’ve changed my mind… if we allow to be neither positive nor negative somehow (much as we allow 0 to have this property) then I think a few of the problems are fixed.
OK, my final thought on this…
If we think geometrically then you could argue that asking “what is ? is a bit like asking how many points (which have width zero) exist on an interval of length 1, such as .
By this definition, I would say that it is most logical to say . When I write I’m usually thinking about .
So… put in other words, I conclude (somewhat un-spectacularly) that .
Now my brain really hurts!
So you’d define 1/0 = ? I’d define 1/0 as the depth of this rabbit hole that’s swallowing everyone up.
Marti et al,
A nice summary of ideas but given your comments about indeterminate forms in the update
Do you prefer to leave functions such as sin x/x as indeterminant at 0 even though their limit approaches 1 as x approaches 0 which can be demonstrated using sandwich theorems or L’Hopitals rule ?
Steve, I was being a little bit tricky with my use of the expression “indeterminate form”. An expression such as 0/0 is usually used to refer to a limit f(x)/g(x) where f –> 0 and g –> 0 at some point; the limit is then “indeterminate” (i.e. we can’t know the existence of or value of the limit) if we don’t have more information about f and g. So, here, the two 0s are standing for functions.
But for me above, the 0s would be standing for the number 0, and the “indeterminate” is stating we can’t assign a meaningful value to 0/0 in our extended real world that keeps the algebraic rules working the way we want. The two uses of “indeterminate” are consistent, but different.
As for sin(x)/x, that is clearly not an indeterminate limit; we have plenty more information than just the numerator and denominator approaching 0, and we can determine the limit to be 1. (Not, for Christ’s sake, using l’Hopital’s rule, which I’ve seen at more than one university.)
Final point. Although sin(x)/x has a limit 1, that doesn’t alter the fact that sin(0)/0 is undefined (or “indeterminate” with my abuse of the term).
I am happy with undefined at 0 and yes I know of some universities which teach L’Hopital’s rule which should be used with care. I prefer the geometric proofs given in the above link
You encouraged a number of interesting responses in this thread
Hi, Steve. The point I was making was that using l’Hopital to find the limit of sin(x)/x is fallacious.
As for the responses on this thread, yes it was/is interesting. As I tried to make clear in my update, I’m not going to hammer a teacher for being suspicious of infinity and reluctant to employ it in their classes. But the ideas should not be dismissed in the flippantly know-it-all manner of Woo and NumberPhile.
RF, I think the countable-uncountable thing is a distraction here. Whatever 1/0 might mean, it’s not a cardinality (or an ordinality) thing.
When we had manual calculators at university in our statistics classes, to do multiplication you would do repeated addition, and to do division, you would do repeated subtraction. Then we got electric calculators. Lads would enjoy getting these electric machines to calculate 1/0 just before the end of class and wander off. These machine often wore out.
After reading Marx’s notes on calculus, Friedrich Engels wrote to Marx as follows on 10 August 1881.
“Yesterday I found the courage at last to study your mathematical manuscripts even without reference books, and I was pleased to find that I did not need them. I compliment you on your work. The thing is as clear as daylight, so that we cannot wonder enough at the way the mathematicians insist on mystifying it. But this comes from the one-sided way these gentlemen think. To put dy/dx = 0/0, firmly and point-blank, does not enter their skulls.”
Click to access TechPaperFahey.pdf
Thanks, Terry. Very funny, and your paper is well worth the read.
That graph of x^x is something else. Who is he?
Yeah, it’s a stunner. I don’t watch Numberphile enough (or at all) to know who the guy is.
It says in the video notes that he is James Grime. I wonder why he would take the discussion to thinking about complex exponentiation (which is so hard) rather than just considering the limit
which is much easier?
A very good question. Or, just simplify to x = 0 and then y= 0, which is how the question arises. None of that nonsense is needed, or helps one bit.
How about this one then? Did Burkard do it justice? I hope so… https://www.youtube.com/watch?v=oc0M1o8tuPo
Hi, Craig. I don’t think it’s a great video, but it is correct and coherent.
Just to be clear though, is there a difference between and ?
I think there is.
If you take as x tends to zero that is a very different graph compared to as x tends to zero. One depends on the direction of approach and the other doesn’t.
Yes, RF, there is definitely a difference. Trying to make any sense of 0/0 is intrinsically more difficult.
It is one of Cauchy’s table of indeterminants from memory. Others (again, from memory, please correct) were , and
I remain skeptical about being able to meaningfully define as every time I try to define it as a mental exercise, I find a contradiction. Maybe I’m not looking hard enough though.
Hi, RF. Yes, 0/0 is fundamentally indeterminate. With 1/0, to avoid the contradictions you have to think about what you might give up in exchange. Then, you can consider whether the exchange is worth it.
OK… In order to define I have to give up the following (perhaps more, perhaps less, I really haven’t had the headspace to fully consider this and suspect a wrap-up is coming soon):
One: If then
Two: If you divide zero by any number the result is zero; only zero has this property. I have to give this up because if then and so .
I’m also not sure all the “rules” we teach for addition of fractions still apply, and all up, perhaps this is too much to give up.
I know there are lots of horrible “definitions” for what mathematics is, but I like to think of mathematics as anything which obeys the laws of logic and division by zero requires some new laws of logic to be developed as far as my (admittedly, limited) understanding goes.
Hi, RF. I don’t think “laws of logic” is the issue here. We’re dealing with arithmetic rules. And yes, you have to give up a/c = b/c implies a = b, but only in the case that c = 0. Similarly, you have to give up 0/c = 0, but only in the case that c = 0. Is that a problem?
Not a problem in and of itself. It is the implications that I haven’t yet considered that I’m worried about…
I notice two things about Woo’s video, not specifically related to the 1/0 thing. The first is the extreme extent to which he babies his audience through his argument. That would pall pretty quickly with me if I were in the audience. The second thing is the amount of continuous chattering coming from the audience. Given that when someone asks a question, their voice is about as loud as the rest of the chatter, it’s reasonable to think that the chatter was quite loud in reality. Is this what teachers have to put up with these days? My audience experience is currently restricted to giving the odd talk to “grown-ups”, and my audiences are always very quiet and focussed. I can’t imagine having to compete with the chatter of Woo’s audience. In fact, I wouldn’t try; I just wouldn’t give the talk.
To address Marty’s comment above of “If the video is so poor then why do four million or so viewers buy into it?”, consider the following video by the space-station astronaut Chris Hadfield, which has been watched by 24 million people so far:
Hadfield sets out to teach us all some physics, but he gets much of his content wrong. Not just wrong, but -really- wrong, or actually “not even wrong”, as the saying goes. He has not the slightest understanding of thermodynamics when he tells us, very incorrectly, that an astronaut who loses his space suit will burn up on one side and freeze on the other. (Specifically, he confuses conducting heat with radiating heat.) He doesn’t understand the meaning of “boil”–he seems to think, incorrectly, that boiling implies high temperatures. When he “explains” relativity in his confident, self-assured way, he completely botches it up; there is not the slightest shred of any relativity in the bogus explanation that he gives.
His comment “The faster you go, the more energy it takes; : it goes up at the square of the speed”, is a bizarre misinterpretation of, well, everything. He is surely one of the most ignorant people ever to present physics to the public. And yet 24 million people have watched that video. Judging by the comments, he has a great many adoring followers who believe his every word. That’s a sad indictment of our times: many, many more people have now been given a completely wrong explanation of thermodynamics and relativity than have ever been given a correct explanation.
So, why do so many people watch these videos? We live in a time of the Cult of the Celebrity. The more views a video has, the more views it generates. I wonder how many of those views are actually generated by people passing these videos around as examples of something -not- to watch?
“The first is the extreme extent to which he babies his audience through his argument. That would pall pretty quickly with me if I were in the audience.”
I agree Don, but this is par for the course in high school, and if you try to assume that your audience is intelligent (or at least aspires to be intelligent), then eventually someone higher up will tap you on the shoulder and let you know that maybe you should tone it down a bit because one of the delicate snowflakes is feeling a bit overwhelmed.
Thanks, Don. On your first point, I agree entirely. Independent of what Woo is trying to teach, and how, the video as a demonstration of teaching is excruciating.
As for why the video gets millions of glowing watches, I’m sure you are correct, that the cult of personality, combined with a general dumbing down, is at the heart of it. But I’d still like to get into their heads to see how these people are conned, or how they con themselves. Maybe, as others here have suggested, there’s nothing there to see. But I’d still like to not see for myself.
marty, another thing I just noticed – if you sort all the videos by “Most Popular” (https://www.youtube.com/c/misterwootube/videos?view=0&sort=p&flow=grid), you end up with the top 3 videos all being about zero – I wonder what the internet finds so fascinating about it…
Well, 0 is a kind of weird number. I’m assuming Eddie doesn’t screw up 0! = 1.
It’s slightly tortured, going by way of rehashing a^0 = 1, but essentially he points out that n! = n*(n – 1)!, so letting n = 1 suggests that we take 0! = 1.
Sigh. Better than his other zero shit, but it sounds pretty dumb.
I’ve always preferred defining 0! = 1 from the number of ways you can choose no objects from 1 object.
Yes, this the clear motivation. Similarly, choosing no objects from N, or N objects from N.
I’ve found that the “how many ways to choose N objects from N” argument is more successful. Quite a few (otherwise intelligent) students seem to have some sort of cognitive resistance to the “how many ways to choose 0 objects from N” argument – they think that’s just a nonsense phrase, and (I’m extrapolating here) perhaps is undefined.
(Of course they amount to the same argument, but the manner of presentation seems to matter…)
Very good point.
(But not so good if you’re trying to Woo an audience with 10 minutes of self-aggrandisement on Youtube … )
My brain exploded when he drew a continuous real-valued curve for y = x^x for x < 0.
Indeed. Let’s forget about x = -1/2, -1/4, …. as well an infinite lot of other negative numbers.
I don’t see how he could possibly think that the left hand limit can be calculated for since x^x behaves badly for x < 0.
Of interest: https://www.wolframalpha.com/input/?i=limit+of+x%5Ex+as+x+approaches+0+from+the+left
Yes, that part’s a real winner.
Fascinating stuff 👍
These videos are just too much horrible awfulness, it makes me too depressed to have even mustered up the courage to write a comment.
Also, I already went on a rant in the previous Woo post.
Marty, your recap is great.
Thanks, Glen. Our motto: “We watch Eddie Woo so you don’t have to.”
How quickly (over the space of 2.5 years) things can turn to crap: https://mathematicalcrap.com/2018/02/23/eddie-versus-the-forces-of-woo/
(This is what happens when ego takes over and it becomes all about being a ‘star’ rather than being a maths teacher. Together with ever-increasing insecurity of being superseded by some new ‘star’).
JF, I’m not sure Eddie’s gotten worse; the videos of Eddie’s I’ve whacked are not new. So, either I was wrong in my earlier assessment of Eddie, or a couple years ago I watched less appalling videos. I think probably a fair bit of both, and a third aspect.
I don’t think Eddie’s straight lesson videos would be as annoying as when he is cosplaying a mathematician. And I do think the basic format of his lessons is fine. On the other hand, I think I overestimated the extent to which he is genuinely engaging with his students, rather than simply performing.
The third aspect, which was the point of that earlier post was that Eddie’s teaching is in no sense innovative. Whether done well or poorly, his videos show a pretty standard version of traditional classroom teaching. In particular, Eddie is using none of the flippin’ gimmicks that the snake-oilers are selling these days. But, somehow, the snake-oilers are more than happy to genuflect to Eddie. It’s ridiculous, and it’s nauseating.