A mathematician is a person

Person

A person is a human being, or an entity that has certain capacities or attributes strongly associated with being human , for example in a particular moral or legal context...

 whose primary area of study is the field of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

. Mathematicians are concerned with quantity

Quantity

Quantity is a property that can exist as a magnitude or multitude. Quantities can be compared in terms of "more" or "less" or "equal", or by assigning a numerical value in terms of a unit of measurement. Quantity is among the basic classes of things along with quality, substance, change, and relation...

, structure

Structure

Structure is a fundamental, tangible or intangible notion referring to the recognition, observation, nature, and permanence of patterns and relationships of entities. This notion may itself be an object, such as a built structure, or an attribute, such as the structure of society...

, space

Space

Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

, and change

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

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Some scientists who research other fields, such as theoretical physics, are also considered mathematicians if their research provides insights into mathematics. Conversely, some mathematicians may provide insights into other fields of research—these people are known as applied mathematicians

Applied mathematics

Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...

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Mathematicians usually cover breadth of topics within mathematics in their undergraduate education, and then proceed to specialize in topics of their own choice at the graduate level.