Algebra

Algebra (from Arabic: الجبر al-ŷabr ' reintegration, recomposition' and data collection) is the branch of mathematics that studies the combination of elements of abstract structures according to certain rules. Originally these elements could be interpreted as numbers or quantities, so that algebra was in a sense originally a generalization and extension of arithmetic. outside, etc).

Elementary algebra differs from arithmetic in its use of abstractions, such as the use of letters to represent numbers that are unknown or can take on many values. For example in the letter is unknown, but by applying the opposite its value can be revealed: . In , the letters Y are variables, and the letter is a constant, the speed of light in a vacuum. Algebra provides methods for writing formulas and solving equations that are much clearer and easier than the old method of writing everything in words.

The word algebra is also used in certain specialized ways. A special type of mathematical object in abstract algebra is called an algebra, and the word is used, for example, in the phrases linear algebra and algebraic topology.

Etymology

The word algebra comes from the Arabic الجبر​ and data calculus from the title of the early 9th-century book Ilm al-jabr wa l-muqābala, The Science of Restoration and Equilibrium by the Persian mathematician and astronomer Muḥammad ibn Mūsā al-Khwārizmī. In his work, the term al-jabr referred to the operation of moving a term from one side of an equation to the other, المقابلة al-muqābala "balancing" referred to adding equal terms to both sides. Shortened to just algeber or algebrain Latin, the word ended up entering the English language during the fifteenth century, either from Spanish, Italian or medieval Latin. It originally referred to the surgical procedure of fixing broken or dislocated bones. The mathematical meaning was first recorded (in English) in the 16th century.

Introduction

Unlike elementary arithmetic, which deals with numbers and fundamental operations, in algebra -to achieve generalization- symbols (usually letters) are also introduced to represent parameters (variables or coefficients), or unknown quantities (unknowns); the expressions thus formed are called " algebraic formulas ", and express a general rule or principle. Algebra forms one of the great areas of mathematics, together with number theory, geometry, and analysis.

The word «algebra» comes from the Arabic word الجبر al-ŷabar (in dialectal Arabic by progressive assimilation it was pronounced [alŷɛbɾ] from which the terms of European languages ​​derive), which translates as 'restoration' or 'replenishment, reintegration'. It derives from the treaty written around the year 820 AD. C. by the Persian mathematician and astronomer Muhammad ibn Musa al-Jwarizmi (known as Al Juarismi), entitled Al-kitāb al-mukhtaṣar fī ḥisāb al-ŷarabi waˀl-muqābala (Compendium of calculation by reintegration and comparison), which provided symbolic operations for the systematic solution of linear and quadratic equations. Many of its methods derive from the development of mathematics in medieval Islam, highlighting the independence of algebra as a mathematical discipline independent of geometry and arithmetic. Algebra can be considered as the art of making calculations in the same way as in arithmetic., but with non-numeric mathematical objects.

The adjective " algebraic " usually denotes a relation to algebra, as for example in algebraic structure. For historical reasons, it can also indicate a relationship to the solutions of polynomial equations, algebraic numbers, algebraic extension, or algebraic expression. It is convenient to distinguish between:

• Elementary algebra is the part of algebra that is usually taught in math courses.
• Abstract algebra is the name given to the study of "algebraic structures" proper.

Algebra is usually based on studying the combinations of finite strings of signs and, while mathematical analysis requires studying limits and sequences of an infinite number of elements.

History of algebra

Algebra in ancient times

The roots of algebra can be traced back to ancient Babylonian mathematics, who had developed an advanced arithmetical system with which they were able to do calculations in an algorithmic way. With the use of this system they managed to find formulas and solutions to solve problems that today are usually solved by linear equations, second degree equations and indeterminate equations. In contrast, most Egyptians of this time, and most Greek and Chinese mathematicians of the first millennium BC, normally solved such equations by geometric methods, such as those described in the Rhind Papyrus, Euclid's Elements and The Nine Chapters on the Mathematical Art.

• Ahmes Papyrus; dated between 2000 to 1800 a. c.
• The Nine Lessons of the Mathematical Art; compiled during the 2nd and 1st centuries BC. c.
• Euclid's Elements, ca. 300 BC c.

Ancient Greek mathematicians introduced an important transformation by creating a geometric-type algebra, where "terms" were represented by the "sides of geometric objects", usually lines to which they associated letters. The Hellenic mathematicians Hero of Alexandria and Diophantus, as well as Indian mathematicians such as Brahmagupta, followed the traditions of Egypt and Babylon, although Diophantus' Arithmetica and Brahmagupta's Brahmasphutasiddhanta are at a much higher level of development. For example, the first arithmetical solution complete (including zero and negative solutions) for quadratic equations was described by Brahmagupta in his book Brahmasphutasiddhanta. Later, Arab and Muslim mathematicians would develop algebraic methods to a much higher degree of sophistication.

Diophantus (3rd century AD), sometimes called "the father of algebra", was an Alexandrian mathematician, author of a series of books entitled Arithmetica. These texts deal with solutions to algebraic equations.

Arab influence

The Babylonians and Diophantus mostly used special ad hoc methods to solve equations, Al-Khwarizmi's contribution was fundamental; he solves linear and quadratic equations without algebraic symbolism, negative numbers, or zero, so he must distinguish various types of >jab.

Persian mathematician Omar Khayyam developed algebraic geometry and found the geometric solution of the cubic equation. Another Persian mathematician, Sharaf Al-Din al-Tusi, found the numerical and algebraic solution to various cases of cubic equations; he also developed the concept of function. The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji, and the Chinese mathematician Zhu Shijie solved various cases of equations of degree three, four, and five, as well as higher order polynomial equations by numerical methods.

Modern age

During the European Modern Age numerous innovations take place, and results are achieved that clearly exceed the results obtained by Arab, Persian, Indian or Greek mathematicians. Part of this encouragement comes from the study of third and fourth degree polynomial equations. The solutions for polynomial equations of the second degree were already known by the Babylonian mathematicians whose results spread throughout the ancient world.

The discovery of the procedure to find algebraic solutions of third and fourth order occurred in Italy in the 16th century. It is also notable that the notion of determinant was discovered by the Japanese mathematician Kowa Seki in the 17th century, followed by Gottfried Leibniz ten years later, in order to solve systems of simultaneous linear equations using matrices. Between the 16th and 17th centuries, the notion of complex number was consolidated, with which the notion of algebra began to move away from measurable quantities. Gabriel Cramer also did work on matrices and determinants in the 18th century. Leonhard Euler, Joseph-Louis Lagrange, Adrien-Marie Legendre, and numerous eighteenth-century mathematicians also made notable advances in algebra.

19th century

Abstract algebra developed in the 19th century, initially focusing on what is now known as Galois theory and on issues of constructability. Gauss's works generalized numerous algebraic structures. The search for a rigorous mathematical foundation and a classification of the different types of mathematical constructions led to the creation of areas of abstract algebra during the 19th century that were completely independent of arithmetic or geometric notions (something that had not happened with the algebra of previous centuries).

Math areas with the word algebra in their name

Some areas of mathematics that fall under the abstract algebra classification have the word algebra in their name; linear algebra is an example. Others don't: Group theory, ring theory, and field theory are examples. In this section, we list some areas of mathematics with the word "algebra" in the name.

• Elementary algebra, the part of algebra that is usually taught in elementary math courses.
• Abstract algebra, in which algebraic structures such as groups, rings, and fields are defined and investigated.
• Linear algebra, in which the specific properties of linear equations, vector spaces, and matrices are studied.
• Boolean algebra, a branch of algebra that abstracts calculus with the truth values ​​false and true.
• Commutative algebra, the study of commutative rings.
• Computer algebra, the implementation of algebraic methods as algorithms and computer programs.
• Homological algebra, the study of fundamental algebraic structures for the study of topological spaces.
• Universal algebra, in which properties common to all algebraic structures are studied.
• Algebraic number theory, in which the properties of numbers are studied from an algebraic point of view.
• Algebraic geometry, a branch of geometry, which in its primitive form specifies curves and surfaces as solutions of polynomial equations.
• Algebraic combinatorics, in which algebraic methods are used to study combinatorial questions.
• Relational algebra: set of finite relations that are closed under certain operators.

Many mathematical structures are called algebras:

• Algebra over a field or more generally algebra over a ring.Many classes of algebras on a field or on a ring have a specific name:
  • associative algebra
  • Non-associative algebra
  • Lie algebra
  • Hopf algebra
  • C*-algebra
  • symmetric algebra
  • foreign algebra
  • tensor algebra
• In measure theory,
  • σ-algebra
  • Algebra over a set
• In category theory
  • Algebra F and co-algebra F
  • Algebra T
• In logic,
  • Relation algebra, an expanded boolean residual algebra with an involution called a converse.
  • Boolean algebra, a complement to the distributive lattice.
  • Heyting algebra

Algebraic notation

It is that numbers are used to represent known and determined quantities. Letters are used to represent all kinds of quantities, whether known or unknown. Known quantities are expressed by the first letters of the alphabet: a, b, c, d, … Unknown quantities are represented by the last letters of the alphabet: u, v, w, x, y, z.

The signs used in algebra are of three kinds: operation signs, relationship signs, and grouping signs.

Signs of operation

In algebra, the same operations are verified with quantities as in arithmetic: addition, subtraction, multiplication, elevation to powers and extraction of roots, which are indicated with the main signs of arithmetic except the multiplication sign. Instead of the × sign, a period is usually used between the factors and multiplication is also indicated by placing the factors in parentheses. So a ⋅ b and (a)(b) equals a × b.

Relationship signs

These signs are used to indicate the relationship between two quantities. The main ones are:

• =, which is read equal to. Thus, a = b is read “ a equals b ”.
• >, which is read greater than. Thus, x + y > m is read “ x + y greater than m ”.
• <, which is read less than. Thus, a < b + c is read “ a less than b + c ”.

Grouping signs

The grouping signs are: parentheses or ordinary parentheses (), brackets or angle brackets or rectangular brackets [ ], braces { }, and the bar or link | |. These signs indicate that the operation placed between them must be performed first. Thus, (a + b) c indicates that the result of the sum a and b must be multiplied by c; [ a – b ] m indicates that the difference between a and b should be multiplied by m, { a + b } ÷ { c – d} indicates that the sum of a and b should be divided by the difference of c and d. The order of these signs are as follows | { [ () ] } |, for example: | { [ (a + b) – c ] × d } ÷ e | indicates that the result of the addition of a + b must be subtracted c, then the result of the subtraction must be multiplied by d, and at the end the result of the multiplication must be divided by e.

Most common signs and symbols

Signs and symbols are used in algebra —and in general in set theory and set algebra— with which equations, matrices, series, etc. are constituted. Its letters are called variables, since that same letter is used in other problems and its value varies.

Here some examples:

signs and symbols
Expression	Use
+	In addition to expressing addition, it is also used to express binary operations.
c or k	They express constant terms
First letters of the alphabeta, b, c, …	They are used to express known quantities.
Last letters of the alphabet…, x, y, z	They are used to express unknowns.
n	Express any number (1, 2, 3, 4, …, n)
Exponents and subscripts	Express quantities of the same kind, of different magnitude.

Symbology of Sets
Symbol	Description
{}	Set
∈	It is an element of the set or belongs to the set.
∉	It is not an element of the set or does not belong to the set.
⎜	such that
n(C)	Cardinality of set C
OR	Universe Set
Φ	empty set
⊆	subset of
⊂	proper subset of
⊄	is not a proper subset of
>	Greater than
<	Smaller than
≥	Greater than or equal
≤	less than or equal to
∩	Intersection of sets
∪	Union of sets
A'	Complement of set A
=	equality symbol
≠	is not equal to
…	The set continues
⇔	If and only if
¬ (sometimes ∼)	No, logical denial (it is false that)
∧	Y
∨	EITHER

Algebraic language

algebraic language
Common language	Algebraic language
any number	m
Any number increased by seven	m + 7
The difference of any two numbers	f - q
Twice a number exceeded by five	2x + 5 _
The division of an integer by its predecessor	x /(x -1)
half of a number	d /2
The square of a number	and ^2
The mean of the sum of two numbers	(b + c)/2
Two-thirds of a number decreased by five equals 12.	2/3 (x -5) = 12
Three consecutive natural numbers.	x, x + 1, x + 2.
The largest part of 1200, if the smallest is w	1200- w
The square of a number increased by seven	b +7
Three-fifths of a number plus half of its consecutive equals three.	3/ 5p + 1/2 (p +1 ) = 3
The product of a number with its predecessor equals 30.	x (x -1) = 30
The cube of a number plus three times the square of that number	x + 3x

Algebraic structure

In mathematics, an algebraic structure is a set of elements with certain operational properties; that is, what defines the structure of the set are the operations that can be performed with the elements of said set and the mathematical properties that these operations possess. A mathematical object constituted by a non-empty set and some internal composition laws defined in it is an algebraic structure. The most important algebraic structures are:

Structure	internal law	Associativity	Neutral	Reverse	commutativity
Magma
semigroup
monoid
abelian monoid
Group
abelian group

Structure (A,+,)	(A,+)	(A,·)
half ring	abelian monoid	monoid
Ring	abelian group	semigroup
Body	abelian group	abelian group