Change and calculus
c. 1400
Infinite sums in Kerala
Madhava of Sangamagrama and the Kerala school
Can you add up infinitely many numbers and get a finite answer?
In Kerala, India, Madhava found that π, and the sine of an angle, can be written as sums that go on forever, each term smaller than the last. His results were recorded by his students in later texts. Europe found the same series about 250 years later.
First, make a guess
Adding up 1 − 1/3 + 1/5 − 1/7 + … for 100 terms, how close do you get to π/4?
- Sum so far
- 3.041840
- Error
- 9.98e-2
Madhava’s series adds and subtracts ever-smaller fractions. It does home in on π, but slowly: after 100 terms it is still only right to two decimal places.
How they figured it out
Madhava lived in Sangamagrama, near today’s Thrissur in Kerala, around 1400, and founded what we call the Kerala school. Astronomers and navigators needed very accurate sine tables, and Madhava found that sines, cosines and π could be written as endless sums of ever-smaller terms.
His own writings are mostly lost. We know his results through students and their students: Nilakantha’s Tantrasangraha (1501) and Jyesthadeva’s Yuktibhasa (around 1530), written in Malayalam rather than Sanskrit, which gives proofs. Using clever correction terms, Madhava’s methods gave π correct to 11 decimal places.
The equation
An endless list of ever-smaller fractions adds up to one exact number.
See it in real life
Walking to the wall
Walk half the way to a wall, then half of what is left, then half again. Infinitely many steps, yet you arrive: . When your calculator works out sin 30°, it adds a few terms of a series very like Madhava’s.
Where engineers use it
How calculators work
Sines, cosines and logarithms in software are computed from series like these, with tricks to make them converge fast.
Approximation
Engineers keep the first one or two terms of a series to make hard equations solvable.
Where it came from
- The first trigonometry table c. 150 BC
Built on
Madhava wrote the sine, the function astronomers had tabulated since Hipparchus, as an endless sum.
What it led to
- The invention of calculus 1665–1684
Same technique
Madhava’s infinite series anticipate the power series at the heart of calculus, about 250 years earlier.
Where the comparison stops: There is no evidence Newton or Leibniz knew the Kerala work; the ideas were found independently, and Madhava did not have the general machinery of derivatives and integrals.
- Taylor series 1715
Same technique
Madhava’s series for sine and cosine are exactly what Taylor’s recipe gives for those functions.
Where the comparison stops: Madhava found series for particular functions; Taylor gave a general method for any smooth function.