The moment when you add pi

6 min read

The moment, that is, when he got a chance to add pi to a bill and produce a nice round figure. Not given to a lot of us.

Then again, was it really pi? Then again, was it really a nice round figure—or, shall we say, nicer and rounder than other figures?

Those are actually questions I find fascinating. To begin with, it wasn’t really pi he added as a tip. After all, pi is an “irrational” number. That means it cannot be expressed as a fraction. Another way of saying that is that its decimal expansion never ends. In this bill, what the man added as a tip was $3.14—and 3.14 is, of course, the fraction 314/100. So it is, as you no doubt know, close to but not actually pi. Accurate to the first 10 places after the decimal point, pi is 3.1415926536. Accurate to 18 places, it would be 3.141592653589793238. And, of course, I could go on forever. So, 3.14 is indeed an approximation—a reasonably good one, but an approximation all the same.

In passing, you would not call any of those approximations to pi a “nice round figure”, I’m sure. Why not, though?

I’ll return to that. Before I do, a reminder of the great Srinivasa Ramanujan. If you know anything about him, you know that through his short life, he churned out endless exotic formulae that mathematicians are trying to understand even today, a century after he died.

Here’s a description of one of them that you can tap out on your nearest calculator: Add the square roots of 72, 90 and 80. Add 10 to that total. Of the result, take the natural logarithm (“ln”—never mind what it means, but your calculator will have a button, so press it). Multiply by 12. Divide by the square root of 190.

What you see on your calculator is a number astonishingly close to pi. In fact, if you now press the button with the symbol for pi on your calculator, I bet none of the digits you see will change. Incredibly, Ramanujan’s formula gives us a number that isn’t pi, but matches the first 18 decimal digits of pi. It’s only from the 19th digit that it differs. Cue the question Ramanujan-watchers have been asking for a century: how did the man come up with this? With his other remarkable formulae?

For one more example, there’s this approximation: divide 9,801 by 1,103. Divide the result by the square root of 8.

This produces a number that matches pi to 6 decimal digits—good enough for almost any calculation you might need. This is actually a special case of a more complex Ramanujan formulation that calculates pi to any accuracy you might want. How did he create it?

Still, this column is not really about the genius of Ramanujan. Instead, it is about how we see numbers. After all, the reason we find these formulae so fascinating is the existence of pi as a number that is of some serious significance to our lives, mathematical or not. If this wasn’t so, if there was no such thing as pi—well, our lives would be much the poorer. But also, nobody would pay attention to these formulae; in fact, Ramanujan would not have worked them out at all.

But I also want to suggest that the same lack of interest would prevail if pi was a “nice round figure”. If it was exactly 3, for example, who would be interested in a formula to generate it? Who would choose to work out such a formula? No, it’s because pi is irrational, because we find ways to match it to six, or 18, or more digits, because there are people who can rattle off a thousand of those digits from memory—these things contribute to the mystique of pi.

And yet, round figures can be intriguing in other situations. You’ve heard of the Hemachandra or Fibonacci numbers (see my column), in which each number is the sum of the previous two. Well, in the 1980s, a mathematician called Michael Somos came up with some variations on that theme. The simplest of these Somos sequences starts with four successive “1″s. After that, any given term is formed like this: multiply the previous term by the term three places behind. Add the square of the term two places behind. Divide that sum by the term four places behind.

Do this repeatedly and you generate the Somos-4 (named thus because it starts with four “1″s) sequence: 1; 1; 1; 1; 2; 3; 7; 23; 59; 314; 1,529; 8,209; 83,313; 620,297; 7,869,898; 126,742,987…

You notice that all those are round figures, called integers. No fractions there. Think of it: calculating the last term listed above, 126,742,987, involved a division by 8,209. It must come as a surprise that there’s no remainder to that division, and thus no fraction. In fact, the Somos-4 sequence comprises only integers. Even more startling, the Somos-5, Somos-6, and Somos-7 sequences also have the same property—they are “integral” as well, containing no fractions. And Somos-8? Its first 17 terms are integers, the 18th is fractional. (Note: Odd-numbered Somos sequences are defined slightly differently than what’s above.)

What accounts for this bizarre feature of these sequences? Does this integrality, or do these sequences, carry any deep mathematical significance?

That’s an area for mathematical research. For now, I’ll leave you with this thought: the reason these sequences are interesting is precisely that they are made up of whole numbers, round figures, integers, whatever you choose to call them.

If instead, Somos-4 looked like this—1, 1, 1, 1, 2.8, 3.14, 7.22503, 22.0980118, 59.416457…— admit it, you’d pay zero attention to this sequence. You’d be in good company too—absent something else that tied those numbers together, mathematicians wouldn’t be interested either. In fact, Michael Somos would have ignored this Somos-4 sequence and moved on.

So, there are times when round figures are of no interest, and times when they stimulate great curiosity. Times when the digits after the decimal point grab our attention, times when they don’t.

Numbers all, though. Endlessly fascinating.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun.

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Published: 08 Dec 2023, 12:57 AM IST

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