Aryabhata · Mathematics
Aryabhata’s strikingly accurate value of pi - 3.1416, correct to four places - and the single word <em>asanna</em>, ‘approaching,’ with which he may have signalled that the true ratio can never be exactly caught.
Buried in the mathematics chapter of the Aryabhatiya is one of the most quietly astonishing sentences in the history of number. Aryabhata gives a rule for the ratio of a circle’s circumference to its diameter - what we call pi - and he gives it as a small arithmetic recipe. Add four to one hundred; multiply the result by eight; add sixty-two thousand. That yields 62,832. This, he says, is approximately the circumference of a circle whose diameter is twenty thousand. Divide 62,832 by 20,000 and you get 3.1416 - the value of pi correct to four decimal places, closer than any value known in the West for another thousand years. What makes it more remarkable is that Aryabhata does not present it as a mystical or god-given number but as a plain computational rule, a piece of working mathematics offered for use. He had, by some method now lost to us, pinned down one of the deepest constants in nature to an accuracy his contemporaries could scarcely have checked.
In the verse, just before naming his result, Aryabhata uses the Sanskrit word asanna - ‘approached,’ ‘approximate,’ ‘near.’ It is easy to pass over, but it may be the most sophisticated word in the sentence. To call a value merely ‘approximate’ could just mean ‘this is a handy rounding.’ But the great Kerala astronomer Nilakantha Somayaji, reading the verse a thousand years later, took it to mean something deeper: that Aryabhata was flagging pi as a quantity that cannot be expressed exactly, a ratio that any finite value can only approach. On that reading, asanna is not an apology for imprecision but a claim about the nature of the number - that it is, in our terms, irrational. Whether or not Aryabhata fully intended that meaning, the word shows a mind alert to the difference between a value that is close and a value that is exact, and unwilling to pretend the first is the second.
Why should the humble ratio of a circle’s edge to its width be so hard to pin down? Every circle, large or small, has exactly the same ratio of circumference to diameter; it is one of the most universal facts in geometry. And yet that ratio refuses to be any tidy fraction. You can approach it - 22/7 is close, 355/113 closer, Aryabhata’s 3.1416 closer still - but each is a fraction that misses, and there is no last, perfect fraction waiting at the end of the chase, because pi is irrational: its decimal expansion runs forever without repeating. This is why the history of pi is a history of ever-better approximations rather than a single triumphant answer. Aryabhata’s contribution to that endless chase was twofold: a value good to four places, and, in the word asanna, an apparent acknowledgment that the chase has no end - that the circle keeps a small, permanent secret from the straight line.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned that Aryabhata computed the circumference-to-diameter ratio as 62,832 to 20,000, that is 3.1416, accurate to four decimal places, and that he described it with the word asanna (‘approaching’), which some scholars read as a hint that pi is irrational. Explain the achievement and the meaning of that word.
Leads to Archimedes.
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