Elementary algebra in the context of "Distribution (logic)"

Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.

Elementary algebra is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called systems of linear equations. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions.

In mathematics, a basic algebraic operation is a mathematical operation similar to any one of the common operations of elementary algebra, which include addition, subtraction, multiplication, division, raising to a whole number power, and taking roots (fractional power). The operations of elementary algebra may be performed on numbers, in which case they are often called arithmetic operations. They may also be performed, in a similar way, on variables, algebraic expressions, and more generally, on elements of algebraic structures, such as groups and fields.

An algebraic operation on a set ⁠${\displaystyle S}$⁠ may be defined more formally as a function that maps to ⁠${\displaystyle S}$⁠ the tuples of a given length of elements of ⁠${\displaystyle S}$⁠. The length of the tuples is called the arity of the operation, and each member of the tuple is called an operand. The most common case is the case of arity two, where the operation is called a binary operation and the operands form an ordered pair. A unary operation is an operation of arity one that has only one operand; for example, the square root. An example of a ternary operation (arity three) is the triple product.

The Concise Book of Calculation by Restoration and Balancing (Arabic: الكتاب المختصر في حساب الجبر والمقابلة, al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah; or Latin: Liber Algebræ et Almucabola), commonly abbreviated Al-Jabr or Algebra (Arabic: الجبر), is an Arabic-language mathematical treatise on algebra written in Baghdad around 820 by the Persian polymath Al-Khwarizmi. It was a landmark work in the history of mathematics, with its title being the ultimate etymology of the word "algebra" itself, later borrowed into Medieval Latin as algebrāica.

Al-Jabr provided an exhaustive account of solving for the positive roots of polynomial equations up to the second degree. It was the first text to teach elementary algebra, and the first to teach algebra for its own sake. It also introduced the fundamental concept of "reduction" and "balancing" (which the term al-jabr originally referred to), the transposition of subtracted terms to the other side of an equation, i.e. the cancellation of like terms on opposite sides of the equation. The mathematics historian Victor J. Katz regards Al-Jabr as the first true algebra text that is still extant. Translated into Latin by Robert of Chester in 1145, it was used until the sixteenth century as the principal mathematical textbook of European universities.

In mathematics, the additive inverse of an element x, denoted −x, is the element that when added to x, yields the additive identity. This additive identity is often the number 0 (zero), but it can also refer to a more generalized zero element.

In elementary mathematics, the additive inverse is often referred to as the opposite number, or the negative of a number. The unary operation of arithmetic negation is closely related to subtraction and is important in solving algebraic equations. Not all sets where addition is defined have an additive inverse, such as the natural numbers.

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form ⁠${\displaystyle \textstyle ax^{2}+bx+c}$⁠ to the form ⁠${\displaystyle \textstyle a(x-h)^{2}+k}$⁠ for some values of ⁠${\displaystyle h}$⁠ and ⁠${\displaystyle k}$⁠. In terms of a new quantity ⁠${\displaystyle x-h}$⁠, this expression is a quadratic polynomial with no linear term. By subsequently isolating ⁠${\displaystyle \textstyle (x-h)^{2}}$⁠ and taking the square root, a quadratic problem can be reduced to a linear problem.

The name completing the square comes from a geometrical picture in which ⁠${\displaystyle x}$⁠ represents an unknown length. Then the quantity ⁠${\displaystyle \textstyle x^{2}}$⁠ represents the area of a square of side ⁠${\displaystyle x}$⁠ and the quantity ⁠${\displaystyle {\tfrac {b}{a}}x}$⁠ represents the area of a pair of congruent rectangles with sides ⁠${\displaystyle x}$⁠ and ⁠${\displaystyle {\tfrac {b}{2a}}}$⁠. To this square and pair of rectangles, one more square is added, of side length ⁠${\displaystyle {\tfrac {b}{2a}}}$⁠. This crucial step completes a larger square of side length ⁠${\displaystyle x+{\tfrac {b}{2a}}}$⁠.

In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in pedagogy.

Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory gives a unified framework to study properties and constructions that are similar for various structures.

In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values true and false, usually denoted by 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as conjunction (and) denoted as ∧, disjunction (or) denoted as ∨, and negation (not) denoted as ¬. Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction, and division. Boolean algebra is therefore a formal way of describing logical operations in the same way that elementary algebra describes numerical operations.

Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic (1847), and set forth more fully in his An Investigation of the Laws of Thought (1854). According to Huntington, the term Boolean algebra was first suggested by Henry M. Sheffer in 1913, although Charles Sanders Peirce gave the title "A Boolian [sic] Algebra with One Constant" to the first chapter of his "The Simplest Mathematics" in 1880. Boolean algebra has been fundamental in the development of digital electronics, and is provided for in all modern programming languages. It is also used in set theory and statistics.