Mathematics

Calculus and analysis

Main articles: Calculus and Mathematical analysis
Calculus, formerly called infinitesimal calculus, was introduced in the 17th century by Newton and Leibniz, independently and simultaneously. It is fundamentally the study of the relationship of two changing quantities, called variables, in the case that one depends on the other. Calculus was expanded in the 18th century by Euler, with the introduction of the concept of a function, and many other results. Presently "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts.
Analysis is further subdivided into real analysis, where variables represent real numbers and complex analysis where variables represent complex numbers. Analysis includes many subareas, sharing some with other areas of mathematics; they include:

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  Multivariable calculus
  
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  Functional analysis, where variables represent varying functions;
  
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  Integration, measure theory and potential theory, all strongly related with Probability theory;
  
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  Ordinary differential equations;
  
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  Partial differential equations;
  
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  Numerical analysis, mainly devoted to the computation on computers of solutions of ordinary and partial differential equations that arise in many applications.
  

Discrete mathematics

Main article: Discrete mathematics
Discrete mathematics is a recently-emerging wide area of mathematics that aggregates several existing areas that deal with finite mathematical structures and processes where continuous variations are not found. These areas have in common that, because of the discrete aspect, the standard methods of calculus and mathematical analysis do not apply directly.^[c] These areas have in common that algorithms, their implementation and their computational complexity play a major role. Despite the many different objects of study, they share often similar methods.
Discrete mathematics includes:

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  Combinatorics, the art of enumerating mathematical objects that satisfy some given constraints. Originally, these objects were elements or subsets of a given set; this has been extended to various objects, which establishes a strong link between combinatorics and other parts of discrete mathematics. For example, discrete geometry includes counting configurations of geometric shapes
  
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  Graph theory and hypergraphs
  
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  Coding theory, including error correcting codes and a part of cryptography
  
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  Matroid theory
  
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  Discrete geometry
  
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  Discrete probabilities
  
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  Game theory (although continuous games are also studied, most common games, such as chess and poker are discrete)
  
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  discrete optimization, including combinatorial optimization, integer programming, constraint programming
  

Four color theorem and optimal sphere packing are two major problems of discrete mathematics that have been solved since the second half of the 20th century. The open problem P=NP is important for discrete mathematics, since its solution would impact most parts of discrete mathematics, regardless of the solution.