Why is 9² Special Among Squares?

What makes 9² special besides the fact that it is a square number? Well, when you divide 1 by 9², you get the following decimal series:

As you can see, all numbers that belong to the single-digits from 0 through 9, excepting 8, appear in this series in ascending order and repeat infinitely. There are a bunch of questions that arise from this observation. Firstly, is this result trivial? Secondly, why is 8 missing from the decimal series?

We will eventually answer both of these questions. But first, I will demonstrate to you that this result is just one among an even more interesting family of decimal series that are associated with ‘9’.

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The Rabbit Hole Goes Deeper

So far, we have seen what happens if we compute (1/9²). What if we increase the order of magnitude of the base by 1, and compute (1/99²). Let’s do just that:

Well, that’s an interesting result. This time, we have the following repeating 2-digit decimal series: 00, 01, 02, 03, 04, 05, 06, 07, and 09. Again, make a mental note that ‘08’ is missing here. What happens if we increase the order of magnitude further by 1? We get the following result:

This time, we have the following repeating 3-digit decimal series: 000, 001, 002, 003, 004, 005, 006, 007, and 009. Unsurprisingly, ‘008’ is missing here as well. You will find that if you keep going up in order of magnitude with the base, the resulting decimal series goes up by one digit each time as well.

Before we proceed with further analysis of this series, let us look at an alternate way to build this series.

An Alternative Way to Build the Decimal Series

Let us say that we are interested in building the following single-digit series: ‘012345679’. All that we need to do is divide the number comprised of the series by an equivalent number of 9s (a total of nine 9s in this case), and we will replicate the result from before:

(012345679/999999999) = (1/9²) = 0.012345679…

We could replicate the double-digit series as follows:

(000102030405060709/999999999999999999) = (1/99²) = 0.000102030405060709…

Now that we have seen the alternative way to build the same decimal series, let’s get to the question of whether this observation is trivial.

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Triviality of the Decimal Series

Is this one of those cool-sounding facts that doesn’t have much purpose or reasoning beyond what is apparent? In other words, is this a trivial result? Well, it turns out that it is, in fact, a non-trivial result.

The observed result originates from a special case of a Taylor series expansion, which is as follows:

This result will directly help us explain the missing ‘8’, ‘08’, and ‘008’ we’ve noted down from earlier.

Optional Extra — Derivation for the Series

In case you are interested, you can find below, a step-by-step derivation of the above result. If you are not in the mood for mathematical derivations, you can skip right ahead. Consider this an optional extra. This derivation will not affect your understanding of forthcoming sections.

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Where is the Missing 8?

Do you remember the condition we imposed on the above Taylor series result? We imposed that the absolute value of x is lesser than one, that is, |x|< 1. Let us consider x = 1/10 and set this value into the expression we had from before. Consequently, we get the following result:

When we try to add the individual terms on the left, the following transpires:

We can clearly see how beautifully our original decimal series emerges from the Taylor series. But hang on! Our original goal was to track down the missing 8. So, let us go through the entire addition process, and see what happens:

We can see from the addition process that the 8 gets lost due to carrying the sum. Due to the recursive nature of the sum, 8 will never appear in this decimal series.

Final Remarks

Using the knowledge that we have gained from our analysis we would be able to generate any of the members of the entire decimal series family using the Taylor series and just one input (value of x).

At this point, I am not entirely sure about the direct applications of such series, but the immediate thought that comes to my mind is to employ recursive equations to generate cool visual patterns.

If you have better ideas for more practical applications of such series, please do us all a favour and let us know in the comments section.

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Credit: The proof for the missing 8 in this article was inspired by the work done by James Grime.

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Further reading that might interest you: Is Zero Really Even Or Odd? and Why Earning More Leads To Lesser Satisfaction?