Counting and a fun harmonic series fact

Saw this tweet (and embeded comic) from Evelyn Lamb last week:

I remembered the comic this morning the boys and I were talking about section 2.3 of Art of Problem Solving’s Introduction to Counting and Probablilty – complimentary counting. Though the divergence of the harmonic series wasn’t something that they were likely to remember, showing them why most large integers had the digit 9 seemed like a fun complimentary counting project.

First I just showed them the problem and we had a fun little discussion about infinity. They didn’t remember seeing the harmonic series before, but, of course, they remember the Numberphile 1 + 2 + 3 + . . . = -1/12 video 🙂


After that quick introduction, we moved on to discussing why most integers will have 9 as one of their digits. In this part we talk about 1 and 2 digit integers as well as introducing a way to count the numbers with 9 as a digit using complimentary counting.


Next we moved on to 3 digit numbers. The boys counted the number of three digit numbers with 9 as a digit using complimentary counting. After that I showed them how you could count all of the number from 0 to 999 with at least one 9 all at once:


Now we moved on to six digit numbers. The original goal was to see if more than half of the six digit numbers had 9’s. Almost!!

In the middle of the video my son wondered why the number of integers with a digit of 9 kept increasing. I tried to give him a quick explanation. This part also provided a sneaky way to talk about / review powers.


For the last part of this project I wanted to explain in a more intuitive way why it makes sense that almost all really numbers have a digit of 9. For this part we talked a little bit about basic probability. I’m not sure if this approach was the right way to explain this idea to kids – my younger son was confused about adding vs multiplying probabilities for example – but it was fun to try. Also some neat ideas about infinity right at the end 🙂


So a nice little project inspired by a fun little math comic from last week. Quite a happy coincidence that the problem had a little bit of a relation to the counting tecnique we were talking about today.



3 Comments so far. Leave a comment below.
  1. ok, I’ll be the slow kid in class who asks the stupid question: this isn’t specific to 9, is it? I mean, if you pick 4’s or 7’s or whatever, the 8/9 *(9/10)^n is still going to be the same, isn’t it?

    (I found a “not specific to 9” on the SMBC board – in fact, apparently it’s apparently not specific to 1-digits, you can take out everything with 9284 in it and it still converges, but I’m guessing it does so more slowly), but I don’t know who that is, so I trust you more.)

  2. The slow divergence of the harmonic series is pretty cool. Your boys might be interested in the sum of the reciprocals of the primes.Contrary to the result about dropping terms with a 9 in the digits, or some other fixed sequence, the reciprocals of primes does diverge.

    It seems highly counterintuitive that there are more numbers with a 9 in the digits than there are composite numbers (in this divergence/convergence sense). No?

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