Srinivasa Ramanujan was an Indian mathematician whose extraordinary intuition changed number theory, infinite series, and the study of special functions. With little formal training, he filled notebooks with thousands of formulas, many of which were later proved and connected to deep areas of modern mathematics. His work matters because it shows how patterns in numbers can reveal hidden structure across algebra, analysis, and geometry.
Ramanujan's story also shows the power of curiosity, persistence, and creative mathematical thinking.
Key Facts
- A partition p(n) counts the number of ways to write n as a sum of positive integers, ignoring order.
- For n = 5, the partitions are 5, 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1, 1 + 1 + 1 + 1 + 1, so p(5) = 7.
- Ramanujan's famous congruence: p(5n + 4) is divisible by 5.
- One Ramanujan series for pi is 1/pi = (2 sqrt(2)/9801) sum from k = 0 to infinity of ((4k)!(1103 + 26390k))/((k!)^4 396^(4k)).
- The Hardy-Ramanujan asymptotic formula is p(n) approximately 1/(4n sqrt(3)) e^(pi sqrt(2n/3)).
- Ramanujan and Hardy studied the taxicab number 1729, where 1729 = 1^3 + 12^3 = 9^3 + 10^3.
Vocabulary
- Number theory
- Number theory is the branch of mathematics that studies integers and the patterns, properties, and relationships among them.
- Partition
- A partition of a positive integer is a way to write it as a sum of positive integers without caring about the order of the terms.
- Infinite series
- An infinite series is a sum with infinitely many terms, often used to approximate constants or functions.
- Modular form
- A modular form is a highly symmetric function that follows special transformation rules and connects number theory, geometry, and physics.
- Congruence
- A congruence is a statement that two numbers have the same remainder when divided by a chosen integer, written a ≡ b mod m.
Common Mistakes to Avoid
- Counting different orders as different partitions is wrong because partitions ignore order. For example, 3 + 1 and 1 + 3 are the same partition of 4.
- Assuming every infinite series gives an exact useful value is wrong because some series diverge or converge too slowly. Ramanujan's pi series is special because it converges extremely fast.
- Using p(5n + 4) divisibility with the wrong input is wrong because the formula applies only to numbers of the form 5n + 4. For example, it applies to p(9), p(14), and p(19), but not p(10).
- Treating Ramanujan's formulas as guesses without proof is wrong because many began as inspired discoveries but later required rigorous mathematical proof. In mathematics, insight and proof play different roles.
Practice Questions
- 1 List all partitions of 6 and find p(6).
- 2 Use Ramanujan's congruence p(5n + 4) is divisible by 5 to decide whether p(24) must be divisible by 5. Show the value of n if it applies.
- 3 Explain why Ramanujan's work is important even when some of his notebook formulas were not proved at the time he wrote them.