linguistic:paraphrasing
Other branches of mathematics

The algebraization of mathematics is the process of applying algebraic methods and principles to other branches of mathematics. It happens by employing symbols in the form of variables to express mathematical insights on a more general level, allowing mathematicians to develop formal models describing how objects interact and relate to each other. This is possible because the abstract patterns studied by algebra have many concrete applications in fields like geometry, topology, number theory, and calculus.

Geometry is interested in geometric figures, which can be described with algebraic statements. For example, the equation y = 3 x − 7 {\displaystyle y=3x-7} describes a line in two-dimensional space while the equation x 2 + y 2 + z 2 = 1 {\displaystyle x^{2}+y^{2}+z^{2}=1} corresponds to a sphere in three-dimensional space. Of special interest to algebraic geometry are algebraic varieties, which are solutions to systems of polynomial equations that can be used to describe more complex geometric figures. Algebraic reasoning can also solve geometric problems. For example, one can determine whether and where the line described by y = x + 1 {\displaystyle y=x+1} intersects with the circle described by x 2 + y 2 = 25 {\displaystyle x^{2}+y^{2}=25} by solving the system of equations made up of these two equations. Topology studies the properties of geometric figures or topological spaces that are preserved under operations of continuous deformation. Algebraic topology relies on algebraic theories like group theory to classify topological spaces. For example, homotopy groups classify topological spaces based on the existence of loops or holes in them. Number theory is concerned with the properties of and relations between integers. Algebraic number theory applies algebraic methods and principles to this field of inquiry. Number theorists employ algebraic expressions to describe general laws, such as Fermat's Last Theorem, and analyze how numbers form algebraic structures, such as the ring of integers. The insights of algebra are also relevant to calculus, which utilizes mathematical expressions to examine rates of change and accumulation. It relies on algebra to understand how these expressions can be transformed and what role variables play in them.

Abstract algebra has various uses in applied mathematics, ranging from electronics and robotics to cryptography. More specific applications are the use of group theory to solve puzzles like Sudoku and Rubik's cubes and the reliance on algebraic tools to analyze origami.