History of Mathematics (Subtest III)
Ancient and Classical Foundations
The History of Mathematics domain tests how mathematical ideas developed, not just who is credited. Babylonian mathematics used a sexagesimal (base-60) system, whose influence survives in 60-minute hours and 360-degree circles. Egyptian arithmetic relied on unit fractions, as in the Rhind Papyrus. Greek mathematics shifted toward deductive proof: Thales is associated with early geometric reasoning, Pythagoras with the famous theorem and a number-centered school, and Euclid's Elements organized definitions, postulates, and proofs into an axiomatic model. Archimedes used exhaustion to approximate π and areas; Apollonius studied conics.
Non-Western Contributions
Indian mathematicians developed positional decimal notation and a symbol for zero; Brahmagupta gave rules for zero and negative numbers, and Aryabhata worked with trigonometry. Chinese texts such as the Nine Chapters on the Mathematical Art include elimination methods resembling Gaussian elimination and remainder problems. Al-Khwarizmi's al-Kitāb al-mukhtaṣar systematized algebra, and his name yields the word algorithm; Islamic scholars also preserved and extended Greek works.
Early Modern to Modern
Cardano and Tartaglia solved cubics and quartics in the sixteenth century; Viète introduced symbolic notation and Descartes connected algebra and geometry. Newton and Leibniz independently developed calculus. Euler standardized notation (e, i, π, f(x)) and proved many foundational results. Gauss, Lobachevsky, Bolyai, and Riemann reshaped geometry; Cantor developed set theory and different sizes of infinity; Hilbert's problems and Gödel's incompleteness theorems shaped twentieth-century foundations. Noether connected symmetry and conservation laws.
Worked Example
Euclid's proof that there are infinitely many primes is a model of contradiction reasoning. Assume finitely many primes p₁, p₂, …, pₙ. Form N = p₁p₂···pₙ + 1. Dividing N by any pᵢ leaves remainder 1, so no listed prime divides N. But N > 1 must have a prime divisor, which is therefore not in the list-contradiction. Hence infinitely many primes exist. The argument's power lies in constructing an object that escapes any proposed finite list.
Traps That Cost Marks
- Crediting calculus solely to Newton or Leibniz rather than both independently.
- Confusing al-Khwarizmi with later developers of algorithms.
- Attributing non-Euclidean geometry to a single mathematician when several developed it.
- Assuming zero was invented in only one culture.
- Mixing up Euler's and Bernoulli's contributions to notation and series.