Below is the video of our recent LunchMaths talk. You can comment/correct below and/or at the YouTube link.

A big thanks to Lawrence and Emma-Jane for arranging the talk, and for making the zooming as painless as possible. A couple of aspects that I intended to talk about, and some probably valuable clarification, were only covered in the Q and A. I’ll leave it be except in reply to comments, except for one aspect that I really regret not getting to and which I’ll cover in a separate post ASAP.

It was great! So good to be able to hear you talk Marty.

Thanks, Glen. It was great to “see” you after so long, if only in the idiotic, pandemic sense of the word. It’s not clear to me what the reaction will be, although it’s also not clear to me that I care.

Also, one thing that I don’t want to get lost because it was a bit rushed in the talk and it really is a deep point: the theorem at the end.

The reader, or viewer, or listener, may believe that Marty is making a joke here. He is, but he is also absolutely CORRECT. The only thing happening with these “unconventional summations” is that equality has been redefined. What value should we attribute to the sum of all natural numbers? Well, of course, it has no value. As a sequence of natural numbers, integers, real numbers, the limit does not exist. In the extended reals, we can call it . But it definitely isn’t . Now, is there a NOTION of equality that we can propose such that it *is* equal to ? Yes, there is. That’s it. There is no grand truth, there is no deep and meaningful interpretation, it isn’t a great mystery. It is a definition.

Is it a useful definition? Hmmm, yes, I think it is. But it is very misleading, mathematical clickbait, to try and “explain” or “prove” why that equality is true. It is a definition.

What is equality? At its core, equality as in says that the symbol “a” carries the same semantic meaning as the symbol “b”. What does this mean? Well, if we make a big bag full of all the symbols that are equivalent to , then is in there. And vice versa. We say that defines an *equivalence relation*. You can write this as

and think of the function as returning all the things that is equivalent to. Now, imagine this in an abstract sense. You can (like Marty did) say that for an integer returns all the other integers that differ from by a multiple of 73. Then etc. Of course, you can make the function as complex as you like, and the one for Ramanujan’s sum is complicated indeed. But there isn’t anything more to it than that.

I couldn’t vote in that (-1)^(1/3) = (-1)^(2/6) poll (I was accessing through my browser) – but I was very surprised that such a large majority thought they were unequal.

For those who weren’t there, my memory is that the poll results were about 2/3 for “No” and 1/3 for “Yes”. SRK, I’m surprised you’re surprised. In any case, my informal polls of audiences in my talks has always returned roughly those results, with God knows what proportion answering with any confidence.

Totally speculative, but I suspect that questions like these have a certain amount of bias just as a result of the asking — it looks so obviously one way that some people assume it _must_ be the other, and answer that way to try to look smart. Hence, people see something like this and say they’re not equal because the obvious answer couldn’t possibly be true.

Thanks, nondescript. That is a possibility well worth considering. For various reasons, however, I think the looking-for-a-trick aspect is probably minor.

There is plenty of indication that index laws are commonly misunderstood, treated as Commandments from God, rather than consequences of careful definition. And, there’s plenty of evidence that the understanding of fractions is commonly weak. So, I doubt there is much “obviously” felt about the question I asked. But I agree the responses could be explained in other ways.

It would be interesting to test this in a different way, by asking what equals, or making it multiple choice: 1 or -1 or undefined. I’d be very surprised if the responses for any formation were suggestive of a broad understanding.

That would certainly be interesting. I have historically been shown to have far too optimistic a perspective on these sorts of things.

Believe it or not, amongst my brothers I am the one considered to have too much faith in human nature.

And isn’t that a frightening thought…

Yes. Especially when you consider that my brothers are smarter than me.

Marty, thanks for providing the link to your talk. I really enjoyed it (I couldn’t attend because I was teaching). The only thing missing was a log fire crackling in the background – I really enjoyed the raconteur style.

So I’m genuinely curious – where does this leave the humble a^(mn) = (a^m)^n = (a^n)^m …? Because the first thing a student says is

“But (-1)^(2/6) = ( (-1)^2 )^1/6 = 1^1/6 = 1 …?”

and so you point out that -1 is also a 6th root of 1. Then a Methods student wants to know why the graph of y = (x^2)^(1/6) s/he plotted is not the same as their graphs of y = ( x^(1/6) )^2 and y = x^(1/3). Followed by “And how come my graph of y = (x^2)^(1/6) doesn’t have a range of R?” So then you have to explain how plotting packages plot. And then …

So when a student (or teacher) doesn’t like the “1/3 = 2/6 so suck it up” argument, you have a whole can of worms to explain.

Thanks, JF.

First thing, if a student (or teacher) doesn’t like the “1/3 = 2/6 so suck it up” argument, then what they have to do is suck it up anyway. Lead the horse to water, stick the horse’s head underwater, but if the damn horse doesn’t want to drink then to hell with it. Before anything else, one has to know what the equals sign means. Then, if one accepts that 1/3 = 2/6, which of course it is, there is nowhere to go.

Secondly, I agree that the question then opens up a discussion of what went wrong with trying to apply the index “law”, and what is going is not obvious. But this non-obviousness should not be used to cloud the first point: 1/3 = 2/6 comes first.

Finally, I am not particularly suggesting that teachers need to present this question to their students. My point in presenting the question was to indicate the ways that school material can be thought of as, and thus taught as, dogma, and how this dogma can blind one to obvious truth.

Can we also get around this by saying that is defined for and continue with something to the tune of, “even though we can take the cube-root of a negative, we have to be careful assuming that the index laws will ‘just work’ “?

Sometimes I wonder if it isn’t easier to start with this and deal with the craziness of CAS/Mathematica graph plotting later.

RF, you have to tread carefully. What, precisely, are you trying to define?

Again, I don’t care all that much if teachers and students fight with this problem. What I do care is for teachers and students to realise that you can’t possibly have index laws for indices that are not yet defined.

Yeah, I see your point.

So, unless we define the process of raising to a power very carefully, the rest is simply not going to work (is this the basic idea?)

Defining the squares and cubes of a positive integer are not too difficult, but going beyond this to define, say gets a bit more difficult.

Yes, that’s the point. You first have to define , carefully and step by step.

But failing to do so is much worse than “not going to work”. It’s that it doesn’t make any logical sense. You can’t possibly prove anything about wombats, or even say anything meaningful about them, unless you somehow define what a wombat is.

OK then, at Year 8 level, how would you define ?

How would you define it at Methods Year 12 level?

It’s worth its own separate post, but how would *you* define in year 8? You already answered (in effect) if is a positive integer. So, what do you do for other types of ?

Damn! I thought I might get a straight answer out of you…

But, since you asked… In year 8 I don’t touch negative or fractional indices, I don’t think they are quite ready for them.

At Year 9, I pose the question, “if we now want a number , how can we define this in a way that fits with the index laws we have learned so far?” and proceed from there.

This gets me through negatives and fractions.

By VCE when we get to graphs of I just show them the graph and hope they don’t ask me to justify the individual points!

Hi, RF. Your Year 9 approach is exactly the approach to take: how to define new index things so the index laws we already know continue to hold.

As to what to do in Year 11, you pretty much have to cheat, or take a *lot* of time going down the path Franz suggested. But, you should tell the students that you are cheating (which, of course, the textbooks don’t).

The poll is misleading.

Stating that “x=y” is false does not imply that “x≠y” is true.

There is also the possibility that both statements are meaningless, or both are false.

Case (1): Assume “=” to be an equivalence relation between numbers.

In this case, we need a proper definition of “x^y” for x,y in Q in general. There is no such definition given in the context of the talk, and there is no canonical way to define such powers, and therefore the terms on both sides of the equations are not numbers, and you cannot apply the “=” relation to them. Both statements are meaningless.

Case (2) Interprete “=” as a wide as possible. The terms on both sides are literally different, and both have no numerical value (see above), so in which sense should they be equal? Are all “undefined” objects equal to each other? Or only objects that are “undefined” for the “same” reason? What about (-1)^(1/3)=(-2)^(1/3) then? There is a reason most computer languages define both NaN=NaN and NaN≠NaN to be false.

Um, nope.

hjm, this is way too sophistic. I guess since I didn’t define “=”, one could assume “=” means “not equal to”. But what on earth is the point of going down such roads?

Almost anyone who watched the video would have had one or more standard interpretations of as a real live number. With any such interpretation will equal .

Thanks for the reply!

Yes, I completely agree on the issue itself. And I know from personal experience that this and many similar problems exist in high school math. I’m just arguing that the poll result might be misleading because some participants answered “no” because they are well aware of the problem and therefore consider both expressions to be undefined. It’s only about the implied statistics, not the issue itself.

(Btw, I came across this site because I’m a math teacher in Germany and we obviously struggle against the same kind of madness. At least those of us who notice the madness. It’s interesting to see that this is a global issue.)

Thanks, hjm. I’m happy to concede that some people may have answered “not equal” because they considered the quantities undefined. However, though we can’t know for sure, I have strong reason to suspect the number of such people was pretty low.

I’ve been asking that question in classes and presentations for almost a decade now, ever since I discovered a prominent Victorian textbook got it wrong. Overwhelmingly, people who explained to me the supposed inequality relied upon an invalid application of the index “laws”.

I think the number of people who answered “no” for that reason is negligible. The majority would have voted no because they think but is either undefined or

Of course this is only indicative of a failure in education, nothing personal.

(BTW I guess this is the case because whenever I’ve had anything to do with high schools, they get my version of Marty’s question wrong, and we have a discussion.)

Thanks Marty – quality stuff. One question though, and similar to Glen’s idea raised above: is there a balance (in your opinion) in say early highschool mathematics between giving definitions which are true by virtue of the fact they do not contradict other definitions, and giving definitions which are useful but not completely true with a statement to the effect (there is more to this, you will hopefully learn it later)? For example, when teaching square-roots, is it acceptable to tell students that a negative has no square root or should you tell them there is one but it is, for now, in the “too hard basket”?

Thanks, RF.

Your question is a good, important and tricky one. But the first thing to say is, you shouldn’t think of this as mathematicians, and me in particular, breathing down teachers’/writers’ necks. There are judgment calls to be made, and room for varying opinions. What there is not room for, however, are straight out falsehoods, such as “proving” with index “laws”. Such dishonesty serves no purpose and is actively confusing. That was the kind of thing I was attacking in my talk.

Again, these are judgment calls, but in general I am fine with limited truth in context. So, for example, I have no problem with lower primary teachers presenting multiplication as repeated addition. Eventually, and pretty soon, that notion has to be dropped, but I don’t think that means repeated addition is a bad place to begin. Similarly, I’m fine with lower secondary textbooks and teachers indicating that negatives do not have square roots.

Having said that, it is usually possible to at least hint at deeper truths along the way, and I know many teachers try to do so. It takes very little time and can be tantalising to indicate parenthetically that roots of negatives will eventually make sense. In synch with this, it is worthwhile at least trying to choose one’s language carefully, to make the limited “truth” as true as possible; so, for example, one might indicate negatives don’t have “real” square roots. The significance of this may be lost on most or all students, but I’m not sure it doesn’t have subliminal effect. And, it is a worthwhile exercise in careful thought and expression.

In my experience, even one comment can have a big impact on a student – for better or worse.

This is why one must be careful in what one says as a teacher.

Maybe. I’d say thoughtful rather than careful. If you are overly careful, to the point of never trying to say anything wrong, then you probably won’t say much right either.

I watched the video; very enjoyable; thank you; I note that you used the example as I did; is there something about this multiplication that makes it so difficult?

Thanks, Terry.

The 7s are always left out, and 7 x 7 is a square. So, I usually pick on 7 x 8 or 7 x 9. But 7 x 8 = 56 is not “difficult”, any more than “The capital of Peru is Lima” is difficult.

“A shop window displays x cheeses”. You can’t invent stuff like this. Do you know the title of this book?

Oh, and is 1/(1-1) = 1/(2-2)?

Hi, Franz. I understand the relevance of your 1/(1-1) = 1/(2-2) question, but I’m not sure of your purpose in asking it.

The cheese book is indicated in my and Dave Treeby’s review here. In the talk I deliberately didn’t indicate the publishers/authors, although I also didn’t work to obscure them. The point wasn’t the individual stupidity, but the whole culture of stupidity.

My point is that I am a bit uncomfortable with the equation in absence of a definition of these expressions. For example, what does the equation mean if you define ?

Ah, I see why I was confused: you were intent on defining things properly.

But I don’t think your question dents the simple truth: if then , whether one wants to consider both as single (real)-valued, or multiple valued or undefined. Same thing with : they are equal, possibly in the sense of being equally undefined.

I liked that the second shop always displayed x^2 +2x – 48 cheeses. It obviously has a very bad mouse problem whenever the first shop displays less than 5 cheeses …. (Maybe they’re mathematical mice).

Can you have a negative cheese?

Makes as much sense as anything else in the question.

You can clearly have a negative cheese question.

Like I said – A very bad mouse problem. (If I hadn’t seen it with my own eyes, I’d swear it could only happen in Looney Tunes!)

Yes, a great example. Martin Gardner brings it up here.

Another question about cheese. (I can’t recall where I found it.)

You have a cube of cheese and a knife. How many straight cuts of the knife do you need to divide the cheese into 27 little cubes?

I think a similar question appeared on an AMT competition paper in the 1990s.

My first instinct is that you need 8 cuts, since re-arranging after a cut doesn’t help reduce the number of cuts.

But I can’t find a good argument for why you can’t do it with less than 8 cuts.

Is it 8 though? We can double the number of segments at each step, so after 4 cuts I can have 16 pieces, 5 cuts 32 pieces. If they have to be little perfect cubes, some care is needed. 6 would be enough (cut each face twice). Sorry I dont see the 8.

Yes, 6. I was still in index law mode and thought of instead of

I have had issues for many years about the problems involving probability transition matrices in examinations and text books. Attached is one of two papers on the matter that I have written with colleagues.

The main point is that, usually, unless the Markov condition holds, the problem cannot be solved.

Here is an analogy.

Prove the following result. If f is a real-valued function defined on [0,1], f(0) is negative, and f(1) is positive, then there must be a value of x in [0,1] such that f(x) = 0.

2016-IntJMathEdScTech

You mean, prove the IVT?

Doesn’t that require f to be continuous on [0,1] and differentiable on (0,1)?

Or have I totally missed the point of your question?

I think you hit on the exact point of Terry’s question. We tend to assume functions are continuous, but not all are, and most applications/theorems require it. Terry is saying the same unstated assumption issue comes up with probability questions, on whether a process is Markov or not.

I agree with Marty’s point in his lecture that there are many instances where mathematical problems are stated in contexts that are not necessary (e.g. chess example) and sometimes they are downright distracting from the main point.

Even worse, sometimes they are thinly veiled attempts to make statements about society.

Often they require knowledge of the context. Halsey (2018, p. 32) refers to a NAPLAN question that refers to a “busy train station”, and many Australian students will have no idea of what a busy train station might look like.

Designing questions on applied mathematics (which includes statistics) requires more than superficial window dressing.

Reference

Halsey, J. (2018). Independent review into regional rural and remote education—Final report. Canberra, Australia: Commonwealth of Australia.

cheese (not chess)

My sincere gratitude to Dr Marty.

Such a fabulous online lecture must be recognized by more people.

In the past，my favourite was your “Joy of Gambling”. However I only got the chance to find the podcast… If Marty agreed maybe I could post his old podcast here for more people to notice and appreciate.

Thanks, PN. Perhaps for now it makes more sense to just point to the QEDcat public lectures page. That page links a bunch of (clunky) videos of me and Burkard.

The gambling “podcast” to which you refer is really just the audio of a talk, which used to be on an MAV webpage, and which doesn’t make as much sense without the slides. I’ll try to update the public talks webpage with properly linked materials, if they exist.

Of course the gambling audio not being a proper talk doesn’t excuse the airbrushing dickishness of the MAV taking it down.

So great to see you presenting again after so long Marty.

Thanks, Sabine. Skipping gently over the irony, I’m genuinely pleased that you enjoyed the talk.

Hehe yes indeed… it’s funny but I think I must have been to every one of the talks you mentioned. I even used to attend the Lunchmaths talks at Monash and I certainly remember a lot more than 15 people there for your Joy of Gambling. Between that and Burkard juggling and riding his unicycle in the hallways it made for lots of good memories of my time at Monash.

I understand children brought up in the depression era also have fond memories of the time.

Marti,

Hopefully your message gets out to a wider audience …

QED is a good read

KISS and peer review pay dividends in spades

Steve R

Nah, no one is listening. All teacher groups want is mathematicians as performing monkeys.

Any reason there was no Corollary #2 at the end there? I missed this point the first time through and am now wondering if it was a deliberate joke on your part…

No, no hidden joke there. That part of the talk was taken from the AustMS talk. I decided there was no need to renumber, and anyway wasn’t sure until kick-off which corollaries I would include. Corollary 2 was that Twitter is a cesspool. A corollary to that corollary was that Twitter is worse than Facebook.

For what its worth, I think corollary #3 is right on the mark.

Yes, of course it is. It’s completely obvious, and there’s not an education “expert” in the country who says it. Dumb fucks.