Knowra Pure mathematics Pure mathematics Pure mathematics studies abstract structures, patterns, and theories for their intrinsic interest, without requiring an immediate practical application.
Abstract algebra : The study of algebraic structures such as groups, rings, and fields. It investigates symmetry and operations through structures defined by general rules.
Mathematical proof : A deductive argument establishing that a mathematical statement follows from accepted premises. Proof distinguishes established results from conjectures across pure mathematics.
Euclid's Elements : An ancient Greek mathematical treatise presenting geometry and number theory through definitions, propositions, and proofs. Its axiomatic presentation shaped mathematical exposition for more than two millennia.
Applied mathematics : The use of mathematical methods to formulate and solve problems in science, engineering, and other fields. It prioritizes problems arising outside mathematics, unlike the intrinsic aims emphasized here.
Cryptography : The use of mathematical and computational methods to secure information and communication. Number theory and abstract algebra underpin widely used public-key cryptographic systems.
Number theory : The study of integers and their properties, including divisibility and prime numbers. It develops deep results about whole numbers without requiring practical motivation.
Axiomatic method : A method of developing a theory from explicit assumptions and rules of inference. Axioms let mathematicians explore what follows from a chosen set of foundations.
Greek mathematics : The mathematical traditions of ancient Greek-speaking societies, known for systematic proof and geometry. Greek proof-based geometry became a durable model for mathematics pursued beyond immediate calculation.
Theoretical mathematics : Mathematical research focused on general principles and abstract theory rather than immediate computation or implementation. Its scope overlaps substantially with pure mathematics, though the labels are not always identical.
General relativity : Einstein's theory of gravity, describing gravitation through the geometry of spacetime. Differential geometry became essential to expressing the theory's physical ideas.
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