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.

4 thoughts on “Does 1 – 1 + 1 – 1 + 1 . . . . . = 1/2

  1. This evening, our two boys were talking about appropriate language for different contexts and audiences. Coming from that conversation, I was thinking about the role of language in signaling group membership and establishing group identity. I believe it is not controversial that we feel a strong pressure to use language to fit into desired social groups and have a cognitive bias to pretend to understand group in-terms, including strong self-deception.

    This links to Grandi’s series because of the way it is usually presented. As in your videos, we see familiar symbols: 1, +, -, =. We think we know what these mean. It takes a very iconoclastic (or heavily trained) personality to recognize that asking about definitions of familiar objects might be the right path forward to help clarify what is really happening.

    One form of training we have found really helpful is listening to and playing with lateral thinking puzzles. In many of these puzzles, the answer often hinges on an unusual meaning for one of the “normal” words in the scenario, so you get used to questioning every definition and assumption. If you want to a source for puzzles, the Futility Closet podcast, which is excellent for many reasons, has one at the end of each episode.

    1. Great timing – was just looking for a new podcast to add to the list. Slate Money is only once a week, Relatively Prime stopped, and Quanta Magazine could publish 10x as often as they do and I’d still want more!

  2. I think it’s important to frame questions like “What is 1-1+1-1+…?” in a way that permits the answer “The question does not make sense” or “We need more information to answer the question.”

    1. I’ll by trying to figure out just the right way to share this one with kids for a while!

      It all got started with the -1/12 video from Numberphile a few years ago. My youngest son, who was in 2nd grade at the time, was screaming “no no no” all the way through the video. I really hit him hard – and you can see it, too, in the video in the “Algebraic Intimidation” project.

      He’s in 5th grade now and I think still too young to understand some (or probably “most”) of the subtleties, or even what questions to ask.

      I want him to know, though, that it is ok not to believe something even when he is shown a “proof.”

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