Algebra
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Algebra is a branch of mathematics in which numbers are represented by symbols (usually letters such as x, y, a, b) to express relationships between quantities. Instead of calculating specific numbers one by one, algebra develops general rules that work in many cases.
The most fundamental part is solving equations, which are statements asserting that two expressions are equal. Solving means finding the value or values of the unknown letter that make the equation true. Algebra also studies functions, matrices, and structures such as groups (kooxaha) and fields (goobaha).
Etymology and history
The word 'algebra' comes from the Arabic word 'al-jabr', which was part of the title of a book written by Muhammad ibn Musa al-Khwarizmi in the early centuries of Baghdad's era. The book systematically explained methods for solving first- and second-degree equations. The name al-Khwarizmi is also the origin of the word 'algorithm'. Before and after, mathematicians from Babylon, Greece, India, and Europe developed new methods.
The quadratic equation
The quadratic equation is an equation in which the highest power of the unknown is two, with the form ax² + bx + c = 0, where a is non-zero. It can be solved three ways: factoring, completing the square, and the quadratic formula. The number of solutions depends on the value of b² − 4ac: if it is greater than zero there are two distinct solutions, if it equals zero there is one solution, and if it is less than zero the solutions are complex numbers.
Major branches
Elementary algebra focuses on expressions, equations, and inequalities. Abstract algebra studies general structures based on operations (such as addition and multiplication) and the laws they follow, including groups, rings (giraanaha) and fields. Linear algebra deals with vectors, matrices, and systems of linear equations; it is widely used in engineering, computer science, and statistics.
Algebra is foundational to other areas of mathematics such as analytic geometry, in which geometric objects are represented by equations, and calculus. It is also present in financial modeling, computer programming, and data encryption.