In school, the word “algebra” is used to denote manipulation of symbols and solving equations. However, mathematicians use the word “algebra” to denote a branch of mathematics dealing with number systems, polynomials, and even more abstract structures such as groups, fields, rings, vector spaces, fields, categories and many more.
Originally, algebra developed to solve concrete problems, for instance in conjunction with solving of first- and second-degree polynomial equations. In modern algebra, which developed with the axiomatic approach in 20th century, algebraists concentrated on algebraic structures and their general properties. This approach revealed that algebraic structures appear, sometimes unexpectedly, in different branches of mathematics. Surprisingly, the general understanding of their abstract properties helps to understand deeper results from other branches of mathematics, or even physics. This is the reason, why algebra became one of the most important branches of mathematics.
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