![]() 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. That’s an area for mathematical research. What accounts for this bizarre feature of these sequences? Does this integrality, or do these sequences, carry any deep mathematical significance? (Note: Odd-numbered Somos sequences are defined slightly differently than what’s above.) And Somos-8? Its first 17 terms are integers, the 18th is fractional. ![]() 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. In fact, the Somos-4 sequence comprises only integers. It must come as a surprise that there’s no remainder to that division, and thus no fraction. Think of it: calculating the last term listed above, 126,742,987, involved a division by 8,209. You notice that all those are round figures, called integers. Divide that sum by the term four places behind.ĭo 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. Add the square of the term two places behind. After that, any given term is formed like this: multiply the previous term by the term three places behind. The simplest of these Somos sequences starts with four successive “1"s. Well, in the 1980s, a mathematician called Michael Somos came up with some variations on that theme. You’ve heard of the Hemachandra or Fibonacci numbers ( see my column), in which each number is the sum of the previous two. 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.Īnd yet, round figures can be intriguing in other situations. But also, nobody would pay attention to these formulae in fact, Ramanujan would not have worked them out at all.īut I also want to suggest that the same lack of interest would prevail if pi was a “nice round figure". If this wasn’t so, if there was no such thing as pi-well, our lives would be much the poorer. 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. Still, this column is not really about the genius of Ramanujan. This is actually a special case of a more complex Ramanujan formulation that calculates pi to any accuracy you might want. This produces a number that matches pi to 6 decimal digits-good enough for almost any calculation you might need. Divide the result by the square root of 8. Cue the question Ramanujan-watchers have been asking for a century: how did the man come up with this? With his other remarkable formulae?įor one more example, there’s this approximation: divide 9,801 by 1,103. It’s only from the 19th digit that it differs. Incredibly, Ramanujan’s formula gives us a number that isn’t pi, but matches the first 18 decimal digits of 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. What you see on your calculator is a number astonishingly close to pi. Of the result, take the natural logarithm (“ln"-never mind what it means, but your calculator will have a button, so press it). 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. 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. Before I do, a reminder of the great Srinivasa Ramanujan. In passing, you would not call any of those approximations to pi a “nice round figure", I’m sure. So, 3.14 is indeed an approximation-a reasonably good one, but an approximation all the same. Accurate to the first 10 places after the decimal point, pi is 3.1415926536. So it is, as you no doubt know, close to but not actually pi. In this bill, what the man added as a tip was $3.14-and 3.14 is, of course, the fraction 314/100. Another way of saying that is that its decimal expansion never ends. That means it cannot be expressed as a fraction. To begin with, it wasn’t really pi he added as a tip. Those are actually questions I find fascinating.
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