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Pre-algebra - Pre-algebra is a common name for a course in middle school mathematics. In the United States, it is generally taught between the seventh and ninth grades, although students have taken this course as early as fifth or sixth grade. The objective of pre-algebra is to prepare the student to the study of algebra. Pre-algebra includes several broad subjects: Review of natural- and whole-number arithmetic; introduction of new types of numbers such as integers, fractions, decimals and negative numbers; Factorization of natural numbers; Properties of operations (associative, distributive and so on); Simple roots and powers; Rules of evaluation of expressions, such as operator precedence and use of parentheses; Basics of equations, including rules for invariant manipulation of equations; Variables and exponentiation. Pre-algebra often includes some basic subjects from geometry, mostly the kinds that further understanding of algebra and show how it is used, such as area, volume, and perimeter. Wikipedia Pre-algebra.
Algebra I & II - Algebra is a branch of mathematics concerning the study of structure, relation, and quantity. Together with geometry, analysis, combinatory, and number theory, algebra is one of the main branches of mathematics. Elementary algebra is often part of the curriculum in secondary education and provides an introduction to the basic ideas of algebra, including effects of adding and multiplying numbers, the concept of variables, definition of polynomials, along with factorization and determining their roots. Algebra is much broader than elementary algebra and can be generalized. In addition to working directly with numbers, algebra covers working with symbols, variables, and set elements. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields. Wikipedia Algebra
Abstract Algebra - Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulas and algebraic expressions involving unknowns and real or complex numbers, often now called elementary algebra. The distinction is rarely made in more recent writings. Contemporary mathematics and mathematical physics make intensive use of abstract algebra; for example, theoretical physics draws on Lie algebras. Subject areas such as algebraic number theory, algebraic topology, and algebraic geometry apply algebraic methods to other areas of mathematics. Representation theory, roughly speaking, takes the 'abstract' out of 'abstract algebra', studying the concrete side of a given structure; see model theory. Wikipedia Abstract Algebra
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