Explaining how 1 + 2 + 3 + . . . can possibly equal -1/12 to a kid

When I did the my biographies for my kids last week my older son said that the thing in math that he’s see but that he does not believe is this equality:

1 + 2 + 3 + 4 + . . . . = -1/12

This sum was made popular by a Numberphile video a couple of years ago (which now has over 4 million views!):

there have also been several good follow ups. For example this video with Ed Frenkel which was also produced by Numberphile:

and this video by Mathologer which is absolutely excellent:

I spent some time today trying to think about how to discuss this series with my older son. I’m glad that he is bothered by the result – it is obviously very very strange. Obviously I can’t go into the details about the Riemann Zeta function with him, but I still think there’s some what to help him make some sense of the series. So, I spent the day reviewing some ideas in G. H. Hardy’s book “Divergent Series.” Here are a few passages that caught my eye:

(a) Book Cover

I don’t remember where I heard about this book. My best guess is that it was mentioned in Jordan Ellenberg’s “How Not to be Wrong” in the section about Grandi’s series. Unfortunately I only have the audiobook version of “How not to be Wrong” and don’t know how to search it!

(b) first passage

The remark beginning at “It is plain . . . ” caught my attention.  This is right at the beginning of the book – section 1.3.   The statement:

“it does not occur to a modern mathematician that a collection of mathematical symbols should have a ‘meaning’ until one has been assigned to it by definition.”

also felt very powerful to me.

(c) second passage

The continuation of the previous page is also important – the point about Cauchy was definitely mentioned in “How not to be Wrong” as well.

(d) third passage

For the third passage we have to go much later in the book – nearly to the end, in fact.  The passage here – 13.10.11, in particular – shows the strange result.  Not in a Numberphile video, or some other internet video, but in a math textbook by G. H. Hardy:

(e) fourth passage

Finally – and this really is just about the last page of the book – section 13.17 provides a word of caution and an example of what can go wrong playing around with these divergent infinite series.

So, I’m going to spend the next few days and maybe even the next few weeks thinking about how to share some sort of idea about this strange series with my son.  I’ll welcome any suggestions!

Talking through the area of the Koch snowflake with kids

This project is the 2nd of two projects on the Koch snowflake. The reason for the projects was that my younger son wondered how the Koch snowflake could have an infinite perimeter but a finite area.

The first project (about the perimeter) is here:

Exploring the perimeter of the Koch snowflake

Our approach to studying the area was similar to the approach for studying the perimeter. Essentially we looked at the steps in the construction of the Koch Snowflake and then looked for a pattern. Here are the initial thoughts from the kids about the area:

The first step in studying the area was to look at the total area of the first few iterations of the Koch Snowflake.

I decided to avoid the complexity of geometry triangle formulas and just talked about scaling. My younger son also came up with a really nice argument for

Now that we’ve seen a first few cases, can we find the pattern?

The amount of area that we add each time has a fairly simple pattern – it is just multiplication by 4 and division by 9. The only time that doesn’t happen is in the first step.

Can we connect the numbers with the geometry?

Now that we’ve seen and understood the pattern, how can we figure out the sum? I love that the boys saw that the main sum we were looking at here was less than 2.

I didn’t want to derive the geometric sum formula, so I just gave it to them. We can talk about it another time. That formula seems to be the easiest way to find the exact value of the sum, though.

Finally we wrapped up and discussed the process we used to study the area and perimeter. I don’t really believe that my younger son now understands every detail of what we talked about, but I hope that he’s a little bit less confused about the area and perimeter of the Koch snowflake.

I think the math here is something that all kids would find interesting.

Exploring the perimeter of the Koch Snowflake

Last week we have a fun talk about the boys “math biographies”:

Math Biographies for my kids

When I asked my younger son to tell me about a math idea that he’s see but that he doesn’t believe to be true, he brought up the area and perimeter of the Koch snowflake. The perimeter is infinite while the area is finite, and he does not believe that these two facts can go together.

Today I thought it would be fun to talk about the perimeter of the Koch snowflake – no need to tackle both ideas at once. Here’s the introduction to the Koch snowflake and some thoughts from my younger son on what he finds confusing about the shape:

After that introduction we began to tackle the problem of finding the perimeter. We began by looking at the first couple of iterations in the construction of the snowflake to try to find a pattern. At this point in the project the boys didn’t quite see the pattern:

As a way to help the boys see the pattern in the perimeter, I asked my younger son to calculate the perimeter of the 4th iteration. My older son had been doing most of the calculating up to this point, and I hoped that my younger son working though the details here would shed a bit more light on what was going on as you move from one step to the next.

The counting project we reference at the end of this video is here:

John Golden’s visual pattern problem

Finally, we looked at how we could use math to describe the pattern that we found in the last video. We also discuss what it means mathematically for the perimeter to be infinite.

We need fairly precise language to describe the situation here, so this part of the project also gives the kids a nice way to learn the language of math.

Does 1 – 1 + 1 – 1 + 1 . . . . . = 1/2

This morning, for a little first day of school fun, we played with Grandi’s Series.

I’ve seen the series pop up in a few places in the last few days – first in part of a little note I wrote up inspired by a Gary Rubenstein talk:

A Talk I’d live to give to calculus students

and then a day or two later in this tweet from (the twitter account formerly known as) Five Triangles:

So, what’s going on with this series? What would the boys think?

Here’s their initial reaction:

And here’s their reaction when I showed them what happens when we assume that the series does sum to some value x:

We have touched a little bit on this series (and my favorite math term “Algebraic Intimidation”) previously:

Jordan Ellenberg’s “Algebraic Intimidation”

It is fun to hear the boys struggle to try to explain / reconcile the strange ideas in Grandi’s series. I’m also glad that they are learning to think through what’s going on rather than just believing the algebra.

A talk I’d love to give to Calc students

I saw a neat video from Gary Rubenstein recently:

In the video he presents a neat Theorem about partitions due to Euler.

Simon Gregg, by coincidence, was looking at partitions recently, too, and has written up a nice post which includes some ideas from Rubenstein’s video:

The part that struck me in Rubenstein’s video wasn’t about partitions, though, it was about the manipulation of the infinite product. It all works out just fine, which is pretty neat, but sometimes manipulating infinite quantities produces strange results. See this famous video from Numberphile, for example:

Just as an aside, here’s a longer and more detailed explanation of the same result:

The fascinating thing to me is that Euler’s proof in Rubenstein’s video is easy to believe, but the sum in the Numberphile video is not easy to believe at all. Both are examples, I think, of what Jordan Ellenberg called “algebraic intimidation” in his book How not to be Wrong. I used Ellenberg’s idea when I talked about the -1/12 sum with my kids:

Jordan Ellenberg’s “Algebraic Intimidation”

The talk I’d like to give to calculus students would start with the theorem presented in Rubenstein’s video. Once the students were comfortable with the ideas about the infinite products and the ideas about partitions, I’d move on to the idea in the Numberphile video. It would be a fun way to show students that infinite sums and products can be strange and you can sometimes stumble on really strange results.

A fun coincidence with our “1/3 in binary” project

Saw this tweet from Matt Henderson (via a Steven Strogatz retweet) today:

It first reminded me of one of Patrick Honner’s blog posts from a few years ago:

Honner’s post plus a lucky coincidence with a Numberphile video inspired a fun project with the boys:

Numberphile’s Pebbling the Chessboard game and Mr. Honner’s square

It wasn’t until later in the day that a different thought about this graphic hit me – we did a project about this shape LAST WEEK!!

Revisiting 1/3 in Binary

Since I was looking for a quick little something to do with the boys tonight, I head each one of them take a look at the tweet and tell me what they thought.

My younger son went first – he had lots of neat thoughts about the shape and we eventually found our way to the connection between this shape and writing 1/3 in binary:

My older son went next – he didn’t see the connection right away, but we eventually got there, too (oh, and ugh, just listening to this video I realize that I misunderstood my son when he was talking about a geometric series – whoops, he did say the right definition).

Thank you internet – what a fun coincidence!

Revisiting 1/3 in binary

Last night we talked about writing $pi$ in base 3.   A long long long time ago we talked about writing 1/3 in binary.  Here are those two projects:

Pi in base 3

Writing 1/3 in Binary

I suspected that the boys wouldn’t remember the project about writing 1/3 in binary, so I thought it would make a good follow up to last night’s project.

I started by just posing the question and seeing where things went. They boys had lots of ideas and we eventually got most of the way there:

At the end of the last video they boys figured out that if our number was indeed 1/3, if we multiplied it by 3 we should get 1. That reminded them of the proof that 0.9999…. (repeating forever) = 1.

We reviewed that proof and applied it to the situation we had now.

Just one little problem . . . what if we apply the idea in this proof to a different series, say 1 + 2 + 4 + 8 + 16 + . . . . ?

We’ve looked at the idea in this video before:

Jordan Ellenberg’s “Algebraic Intimidation”

We felt pretty comfortable believing that 0.9999…. = 1 and that we’d found the correct series for 1/3 in binary, but do we believe the results when we apply the exact same ideas to a new series?

I love projects like this one 🙂

A nice series problem for kids from Five Triangles

Back in 2013 we did a neat problem on Numberphile’s “Pebbling the Chessboard” video:

That video also reminded me of a neat “proof without words” that Patrick Honner had written about:

Our project is here:

Numberphile’s Pebbling the Chessboard game and Mr. Honner’s Square

and Patrick Honner’s blog post is here:

Proof Without Words: Two Dimensional Geometric Series

Tonight I saw a neat tweet from Five Triangles that reminded me of the prior project:

I thought it would be a fun one to try out with my older son, though I didn’t quite know how to introduce the problem. I started with a slightly easier series as a trial: 1/2 + 2/4 + 3/8 + 4 / 16 + . . .

Since things seemed to go pretty well with the first problem I decided to go ahead and try out the series posted by Five Triangles:

So, a neat problem for kids building off of a the “simple” infinite series 1 + 1/2 + 1/4 + . . . . As our project from 2013 shows, the more complicated versions can have interesting geometric interpretations, but I’ll leave those for another time. Tonight it was just fun to see some neat arithmetic with infinite series.

Building arithmetic and number sense by talking about geometric series

I’ve been a little busy both at home and at work this week and as of this morning hadn’t given any thought to our Family Math projects for this weekend. More or less on a whim I decided to return to an old favorite topic for today’s project – infinite series, and specifically geometric series.

Two of our prior talks on infinite series are linked below, you can find others on the blog under the tag “infinite series”:

Just for Fun: Some Infinite sums

Talking about Infinite Series

The first part of the talk today was introducing the concept of a geometric series. The main idea I’m trying to get at today is showing how we can extend a common way of showing why 0.9999… = 1 to the problem of summing a geometric series. We talk through some of the basic ideas using the series 1 + 1/2 + 1/4 + 1/8 + . . . as our example.

The next thing that we looked at was the series 1 + 1/3 + 1/9 + 1/27 + . . . . My older son initially believes that this series will also sum to 2 because it goes on forever. My younger son’s initial guess is that it will sum to 1.5. His reason is that (except for the first term) the terms are smaller than the series 1 + 1/2 + 1/4 + 1/8 + . . . .

One theme that shows up here that will continue for the rest of today’s project is that subtracting two infinite series is a little confusing to the boys. I should have found a better way, or at least an alternate way, to explain this idea to them.

In the next talk I wanted to have the boys pick their own series to sum. Unfortunately, I wasn’t clear with them that I wanted to look only at series where the terms went to zero. That lack of clarity caused a small problem at the start of this part of the project.

Once we got on the right path, we worked through the series 1 + 1/5 + 1/25 + 1/125 + . . . without too much difficulty. But the next series caused a little bit of trouble:

1 – 1/3 + 1/9 – 1/27 + . . . .

The subtraction and the negative signs were big stumbling blocks here. I really needed to provide a better way to help them see what was going on when we subtract one series from another.

In the next part of our talk we moved on to talking about an general geometric series. This discussion is a big step up in abstraction. I think this abstraction was not as difficult for my older son as it was for my younger son, which isn’t a huge surprise. Subtracting the individual terms in each series still presented a little bit of difficulty. We did manage to get to find a fairly simple formula for our sum, though. Even with the difficulty we had, I think the discussion here are a nice example of how you can take an idea from a specific setting and use it in a slightly more abstract setting.

The last part of today’s project involved using the formula we found in the last video in the situations that we’d already considered. A few examples showed that our formula seemed to match the prior results. Yay!

We then wrapped up by looking at a few situation where the terms in the series do not go to 0. Here the formula produces some results that seem strange. For now I’m leaving these odd results as fun little paradoxes for the boys to ponder.

Watching these talks as I put this blog together makes me wish I’d done a better job with this project. I think that the important mathematical ideas here can be made accessible to kids if you present them in the right way. The results are neat and some seem strange (you’ll hear my son reference Numberphile’s video about the result 1 + 2 + 3 + . . . = -1/12 in the last video – so these strange results can really make kids think).

Hopefully the next time we return to this topic I’ll remember the lessons from this one and present some ideas in a slightly different (and hopefully slightly better!) way.

Jordan Ellenberg’s “Algebraic Intimidation”

One of the ideas that seems to have stuck in my mind from reading “How not to be Wrong” a couple of times is the concept of “algebraic intimidation.”   Ellenberg uses this phrase to describe one of the standard ways to “prove”  that 0.9999….. = 1.   I go through the proof that he’s talking about in the first video below if you’ve not seen it before.

The idea of algebraic intimidation is, I suppose, pretty simple:  the math all looks right, therefore the result must be right because you **better** believe the math!

This concept obviously generalizes to all sorts of situations.   As the maybe useful / maybe harmful (depending on what year it is) quantatitive ideas seem to be creeping back into the financial markets, I feel like I’m seeing the old algebraic intimidation hammer at work a lot more frequently these days.  But, hey, we all miss 2008, right?

While a post about martingales might more more relevant to the attempts at using math to intimidate in the financial markets, I think Ellenberg’s example is infinitely more interesting.  Particularly for students, and I’d love to use the examples below in a room full of kids who are interested in math.

The idea of talking about algebraic intimidation once again came up this past weekend in our Family Math project.  I asked the boys what they wanted to talk about  and they gave me a surprising answer – “Infinite Series.”   The entire set of talks from this weekend is here:

The two conversations relevant to algebraic intimidation  are below and came when one of the examples that they wanted to talk about was “the -1/12 series.”  Say what you want about that old Numberphile video, but the ideas in it sure stuck with my kids!

I led off this part of our project with the standard proof of why 0.999…. = 1 and then, following some examples in Ellenberg’s book, extended the ideas in that proof to a few other areas where you get some rather odd results.  We then moved on to the “-1/2 series” and followed the ideas in the original Numberphile video.

You’ll see that both kids are quite skeptical of the results.  My younger son in particular is almost physically upset.  That’s good.  I want them to learn to question results rather than just blindly trusting the math, and I especially want them to feel free to question results that seem odd.  You certainly won’t find many results that seem more goofy than the ones below 🙂