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Pure mathematics is a branch of mathematics which focuses on the theoretical aspects of mathematics and the manipulation of abstract objects without any direct application to the physical world It strives for logical rigor, generality and abstraction and is heavily dependent on deductive reasoning. Pure mathematics is often divided into two main subdivisions: algebra and analysis. Algebra is the branch of pure mathematics focused on the properties of numerical operations and their structures as well as the representation, analysis, and manipulation of systems of equations. Analysis is the branch of pure mathematics concerned with the study of the behavior of functions and their limits, convergence and series.
The five best examples of topics from pure mathematics are algebraic topology, linear algebra, calculus, Fourier analysis, and Boolean algebra. Algebraic topology is the branch of mathematics which studies the properties of the underlying structure of objects and can be used to study physical systems and the manipulation of abstract objects. Linear algebra is the branch of mathematics which is focused on the methods and techniques used to solve systems of linear equations as well as the properties and applications of matrices and vectors. Calculus is the branch of pure mathematics which is focused on the study of the behavior of continuous functions and the computation of areas and volumes under curves. Fourier analysis is the branch of mathematics which deals with the representation of a function in terms of the sums of its components of waves of different amplitudes and frequencies. Boolean algebra is the branch of mathematics which deals with the algebraic manipulation of logic operations and logical statements.