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The symbol ∞ is so familiar that it is easy to forget someone chose it. That person was John Wallis, an English clergyman, codebreaker and Oxford professor who also found one of the strangest formulas for π. This post covers his life, his product, and a worked calculation that shows how slowly it closes in.
Who was John Wallis?
Wallis was born in Ashford, Kent, on 23 November 1616 and died in Oxford on 28 October 1703, according to his MacTutor biography. Mathematics was not taught at his schools; his brother showed him the rules of arithmetic over the Christmas holidays of 1631, and he kept at it as "a pleasing diversion at spare hours."
He studied at Emmanuel College, Cambridge, and became a clergyman. In the Civil War he decoded a cipher letter about the capture of Chichester in two hours, and he went on breaking Royalist codes for Parliament. In London he joined the weekly meetings of the group that would become the Royal Society. In 1647 he read William Oughtred's *Clavis Mathematicae*, and MacTutor says he mastered it in a couple of weeks. In 1649 he was appointed Savilian Professor of Geometry at Oxford and held the chair for over 50 years.
Where did the ∞ symbol come from?
The MacTutor page on the earliest symbols of calculus quotes Wallis's *De sectionibus conicis* (1655). He supposes, following Cavalieri, that a plane figure is made of infinitely many thin parallelograms, each with height 1/∞ of the whole, and adds that ∞ is to stand for an infinite number. He used it again in *Arithmetica infinitorum*. The slicing idea came straight from our post on Bonaventura Cavalieri and the principle of slices.
What is Wallis's product for π?
In *Arithmetica infinitorum* (1656) Wallis wanted the area of a quarter circle of radius 1, which is the integral of √(1 − x²) from 0 to 1 and equals π/4. MacTutor explains that he could integrate (1 − x²)ⁿ for whole-number n but not for n = ½, so he guessed the missing value by interpolation, a word he introduced. The result was:
π/2 = (2·2·4·4·6·6·8·8…) ÷ (1·3·3·5·5·7·7·9…)
MacTutor adds that Huygens refused to believe it until he was shown that it gave numerically correct approximations to π. MacTutor says Newton later used Wallis's methods in his work on the binomial theorem.
Worked example: how fast does it converge?
Group the factors in pairs: (2·2)/(1·3), then (4·4)/(3·5), then (6·6)/(5·7), and so on. Multiply the pairs and double the result to estimate π.
One pair. 4/3 = 1.3333, so π ≈ 2.667.
Two pairs. 4/3 × 16/15 = 64/45 = 1.4222, so π ≈ 2.844.
Three pairs. × 36/35 gives 1.4629, so π ≈ 2.926.
Ten pairs. π ≈ 3.068, still about 2.4% too low.
Now take the factors one at a time: 2/1, 2/3, 4/3, 4/5, 6/5, 6/7… Doubling each partial product gives 4, 2.667, 3.556, 2.844, 3.413, 2.926, 3.344, 2.972… The estimates jump above and below π, closing in slowly.
For contrast, the infinite product for π that François Viète found in 1593 gives 3.1403 after only five factors. Wallis's product is worth knowing for how it was found, not as a way to compute π.
Common mistakes with infinite products
Mixing the pattern. The numerators repeat each even number twice (2, 2, 4, 4); the denominators are 1, then odd numbers twice (3, 3, 5, 5).
Forgetting to double. The product equals π/2, not π.
Judging from one partial product. Consecutive estimates sit on opposite sides of π, so look at the trend.
Assuming fast convergence. Ten pairs still leave a 2% error.
The integral Wallis started from is a definite integral of the kind listed in Unit 8 of the JEE (Main) 2026 syllabus; check the syllabus for your exam year. Our guide to properties of definite integrals has practice with integrals of this kind.
When an integral like this stalls you, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
What can a student take from Wallis?
Wallis wrote in his autobiography that from childhood he wanted "not only to learn by rote, but to know the grounds or reasons of what I learnt." He also trusted patterns enough to guess, then tested every guess with numbers. Both habits help in any problem set.
Practice routine for this week:
Extend the paired calculation to five pairs by hand and confirm π ≈ 3.002.
Show that ∫₀¹ √(1 − x²) dx = π/4 using the area of a circle.
Write the first eight single-factor estimates and mark which are above π.
Start with step 1: it takes ten minutes and shows exactly why Huygens wanted to see the numbers.
