Infinity, Structure, and Computation
What pure mathematicians actually study, in one traverse: primes and modular arithmetic, the sizes of infinity, the foundations that nearly cracked, the algebra of structure, measure and the Lebesgue integral, and the limits of computation. Cross-book on purpose: the proof text supplies the rigor, the survey supplies the reach.
You can run the Euclidean algorithm and compute with congruences, prove a set countable or show that no listing of it can exist, use Cantor-Schroder-Bernstein in place of an explicit bijection, say what the axiom of choice buys and what Godel's theorems forbid, recognize a group, ring, or field when a structure presents one, explain why the Cantor set is uncountable yet has measure zero, and sort a problem into tractable or NP-hard before writing a line of code.
The arithmetic of the integers
The Euclidean algorithm and Bezout's identity are the working tools of the whole chapter, and unique factorization is the theorem that makes the integers legible.
Congruence turns infinite arithmetic into finite arithmetic; the care lives with inverses and cancellation, because division does not survive the trip to the clock face.
Euler's theorem collapses enormous powers into single residues, and RSA is the payoff: that one theorem, stitched to the hardness of factoring, gives working public-key encryption.
The sizes of infinity
Same size means a bijection exists, nothing more; the definition is the whole subject, and it already pairs the naturals with sets that look much larger.
Cantor's diagonal argument is the first proof that infinity comes in sizes, and the closure facts, subsets, countable unions, finite strings, are what make countability proofs routine.
Cantor-Schroder-Bernstein trades one impossible bijection for two easy injections; most cardinality equalities in practice are proved exactly this way.
Choice, Zorn's lemma, and well-ordering are one axiom in three costumes, and Banach-Tarski is the price of admission stated honestly.
Foundations
Cardinal arithmetic puts numbers on the sizes of infinity, and Russell's paradox is why set theory had to be rebuilt on axioms rather than intuition.
The survey's verdict on foundations: choice is independent and indispensable, and Godel showed that any consistent axiomatization of arithmetic leaves true statements unprovable.
Structure
A group is symmetry made algebraic; actions and representations are how the abstraction earns its keep, by moving concrete objects.
Rings and ideals generalize the integer arithmetic of the first stage, and Galois theory answers a question two thousand years old about which equations can be solved by radicals.
Measure
Measure assigns length to sets far wilder than intervals, the Cantor set shows uncountable and negligible can coincide, and the convergence theorems are why analysts left Riemann behind.
Computation
Big-O is the yardstick, sorting and graph search are the specimens, and P versus NP is the open question that decides what computation can reach. Euclid's algorithm from the first stage was the prototype all along.