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v. t. e. Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." [1]
Noncommutative algebra. v. t. e. Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with rational coefficients), in both qualitative and quantitative ways.
Algebraic number theory. Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory. Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations.
In mathematical logic, a non-standard model of arithmetic is a model of first-order Peano arithmetic that contains non-standard numbers. The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, …. The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the ...
Pisot–Vijayaraghavan number. Salem number. Transcendental number. e (mathematical constant) pi, list of topics related to pi. Squaring the circle. Proof that e is irrational. Lindemann–Weierstrass theorem. Hilbert's seventh problem.
The developers of calculus used real numbers and limits without defining them rigorously. In his Cours d'Analyse (1821), Cauchy made calculus rigorous, but he used the real numbers without defining them, and assumed without proof that every Cauchy sequence has a limit and that this limit is a real number.
Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in and the study of these lattices provides fundamental information on algebraic numbers. [1] The geometry of numbers was initiated by Hermann Minkowski (1910). The geometry of ...
Arithmetic dynamics[1] is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Part of the inspiration comes from complex dynamics, the study of the iteration of self-maps of the complex plane or other complex algebraic varieties. Arithmetic dynamics is the study of the number-theoretic properties of integer ...