## History

The ideas in this course span over two millennia, from ancient Greek mathematics to nineteenth-century foundational revolutions.

### Number Theory

The study of primes and divisibility begins with Euclid's *Elements* (c. 300 BC), which contains the proof that there are infinitely many primes and the algorithm for computing greatest common divisors that still bears his name. Euclid's approach — building a chain of propositions from axioms — is the prototype for the axiomatic method that pervades this course.

Number theory lay relatively dormant through the medieval period, with notable exceptions in Chinese mathematics (the Chinese Remainder Theorem appears in Sun Tzu's *Sunzi Suanjing*, c. 3rd century AD) and Indian mathematics (Brahmagupta's work on linear congruences, 7th century). The subject was revived in Europe by Pierre de Fermat in the seventeenth century, whose correspondence introduced many of the results in the [Elementary Number Theory](/page/Cambridge%20Part%20IA%20Numbers%20and%20Sets) chapter — including the "Little Theorem" ($a^p \equiv a \pmod{p}$, stated in a 1640 letter to Frénicle de Bessy) and observations about quadratic residues. Fermat rarely published proofs; most of his number-theoretic claims were established rigorously only later, by Leonhard Euler.

Euler (1707–1783) transformed number theory from a collection of isolated results into a coherent theory. He proved [Fermat's Little Theorem](/theorems/731) (1736), introduced the totient function $\varphi(n)$ and proved its generalisation [Fermat-Euler](/theorems/732) ($a^{\varphi(n)} \equiv 1 \pmod{n}$, 1763), and gave the first proof of the [Infinitude of Primes](/theorems/724) using analytic methods (the divergence of $\sum 1/p$). [Wilson's Theorem](/theorems/733) — that $(p-1)! \equiv -1 \pmod{p}$ — was stated by Edward Waring in 1770 and first proved by Joseph-Louis Lagrange in 1771.

The pinnacle of elementary number theory is the *Disquisitiones Arithmeticae* of Carl Friedrich Gauss (1801), which systematised modular arithmetic, introduced the congruence notation $a \equiv b \pmod{n}$, and proved quadratic reciprocity. The [Fundamental Theorem of Arithmetic](/theorems/730) (unique prime factorisation) receives its first complete proof in the *Disquisitiones*, though the existence half was implicit in Euclid. Gauss's work made number theory a rigorous deductive science and established the standard of proof that this course aims to teach.

The application of number theory to cryptography is much more recent. The RSA encryption scheme was published by Ron Rivest, Adi Shamir, and Leonard Adleman in 1977, though an equivalent system had been developed independently by Clifford Cocks at GCHQ in 1973 (classified until 1997). RSA demonstrated that ancient number-theoretic results — the [Fermat-Euler Theorem](/theorems/732), the [Chinese Remainder Theorem](/theorems/734) — have profound practical consequences in the digital age.

### The Real Numbers

The ancient Greeks knew that $\sqrt{2}$ is irrational (traditionally attributed to the Pythagorean school, c. 5th century BC), but a rigorous construction of the real number system did not appear until the nineteenth century. For over two thousand years, mathematicians worked with "continuous magnitudes" on an intuitive basis, relying on geometric reasoning rather than arithmetic axioms.

The crisis came in the early nineteenth century, when analysis became increasingly dependent on limiting processes — convergent sequences, infinite series, continuity, differentiability — without a firm logical foundation. Karl Weierstrass, in his Berlin lectures from the 1860s onwards, insisted on arithmetising analysis: reducing all concepts to properties of numbers rather than geometric intuition. The culmination was the independent constructions of $\mathbb{R}$ by Richard Dedekind (via "cuts" in $\mathbb{Q}$, published 1872) and Georg Cantor (via equivalence classes of Cauchy sequences, also 1872). Both constructions establish the completeness of $\mathbb{R}$ — the least upper bound property — as a theorem rather than an assumption.

The [irrationality of $e$](/theorems/746) was proved by Euler in 1737 using a continued fraction argument; the proof via the series $\sum 1/n!$ presented in this course is essentially due to Joseph Fourier (c. 1815). The transcendence of $e$ was established by Charles Hermite in 1873, and that of $\pi$ by Ferdinand von Lindemann in 1882, settling the ancient problem of squaring the circle. Joseph Liouville had constructed the first explicit [transcendental number](/theorems/747) in 1851, using the rapid-convergence argument presented in this course.

### Set Theory and Foundations

The language of sets and functions, now ubiquitous in mathematics, was largely created by Georg Cantor in the 1870s and 1880s. Cantor's original motivation was the study of trigonometric series (uniqueness of Fourier representations), which led him to consider the structure of point sets on the real line. This investigation produced the theory of infinite cardinalities: the proof that [ℝ is uncountable](/theorems/758) (1874, via a nested-intervals argument; the diagonal argument appeared in 1891), [Cantor's Theorem](/theorems/759) that no set bijects with its power set, and the introduction of transfinite cardinal and ordinal numbers.

Cantor's work was controversial in his lifetime — Leopold Kronecker opposed the use of completed infinities, and the paradoxes discovered by Cesare Burali-Forti (1897) and Bertrand Russell (1901) showed that naive set theory leads to contradictions. Russell's paradox, in particular, demonstrated that the collection $\{x : x \notin x\}$ cannot be a set, motivating the axiomatic approach to set theory developed by Ernst Zermelo (1908) and Abraham Fraenkel (1922). The Zermelo-Fraenkel axioms (ZF), supplemented by the Axiom of Choice (ZFC), remain the standard foundation for mathematics.

The [Schröder-Bernstein Theorem](/theorems/760) — that mutual injections imply a bijection — was stated by Cantor in 1887 and proved independently by Ernst Schröder (1896) and Felix Bernstein (1898, as a nineteen-year-old student). The proof by ancestor sequences presented in this course is essentially Bernstein's original argument.

### The Axiomatic Method

The Peano axioms for the natural numbers were published by Giuseppe Peano in 1889, building on earlier work by Dedekind (*Was sind und was sollen die Zahlen?*, 1888). Peano's contribution was to isolate precisely which properties of $\mathbb{N}$ are needed as axioms — the existence of a first element, the injectivity of the successor function, and the principle of induction — and to show that all of arithmetic can be derived from them. The [equivalence of induction and the well-ordering principle](/theorems/722), proved in this course, demonstrates that these axioms are tightly interlocked: no single one can be removed without losing essential content.