What is algebra What are the uses

What is algebra - Serlo "Math for non-freaks"

Algebra is one of the most fundamental branches of mathematics, dealing with solving equations, arithmetic operations, and in general with the mathematical structures behind these operations. In this article we will briefly introduce this sub-area.

History [edit]

Period of the Greeks

Even the ancient Greeks studied algebraic equations[1]. However, unlike the Babylonians and Egyptians, they were not only interested in solving practical problems. They saw geometrical issues in particular as a central part of their philosophy. This was the beginning of algebra, geometry, and mathematics as a science as a whole.

For the Greeks, sides (mostly lines) of geometric objects represented the terms of algebraic equations. Using construction methods with compasses and rulers, they determined the corresponding solutions. Since the ancient Greek algebra was founded by geometry, it is called geometric algebra.

One of the most important Greek mathematicians was Euclid of Alexandria. The second volume of those he wrote elements contains a number of algebraic statements formulated in the language of geometry. Among other things, Euclid discussed the theory of surface application in the elements, which goes back to the ancient Pythagoreans. With this method one can solve certain linear and quadratic equations with an indeterminate from the point of view of modern algebra.

In the tenth book of the elements Euclid provided evidence of the irrationality of . Irrational proportions were already known to the Pythagoreans. These had also proven Euclid's above theorem in a more general form.

Diophantos of Alexandria, who probably lived around the year 250 AD, is considered to be the most important algebraist of antiquity. His first and most important work, the Arithmetica, originally consisted of thirteen individual books, of which only six have survived. With this work he completely replaced the then known arithmetic and algebra from geometry.

Geometry and Algebra [edit]

The Cartesian coordinate system with four points drawn in

The connection of geometry and algebra to what is known as analytical geometry was one of the greatest advances in mathematics. It enables the computational solution of many geometric problems[2]. The starting point is the Cartesian coordinate system, which is named after René Descartes[3]. René Descartes (1596 - 1650) was one of the founders of analytical geometry alongside Pierre de Fermat (1607 - 1665)[2].

In the Cartesian coordinate system, points in the plane or in space are expressed by numbers, which enables an algebraic description of geometric figures. So becomes by the equation a circle around the origin with the radius described. the equation represents a straight line through the zero point with a slope represent.

This algebraic description enables geometrical problems to be solved through certain calculations. For example, if you look at the intersections of the circle with the straight line want to calculate, you have to find the solutions of the following system of equations:

Summary: The analytical geometry records and examines geometrical structures with algebraic tools, whereby the introduction of the coordinate system was the starting point of this new theory. To date, the use of the computer in geometry is only possible thanks to analytical geometry.

The development of the concept of vector space

The treatment of coordinates led to the concept of vector space. This was only introduced in the 19th century by Hermann Graßmann (1809 - 1877) in his mathematically largest work "Expansion Theory"[4]. Here he was the first to describe one -dimensional Euclidean geometry using vectors. He is therefore considered to be the founder of vector calculus.

However, the importance of his scientific work was recognized late[4] and to this day he is hardly known as a mathematician. Graßmann's theory described many terms of today's linear algebra such as dimension, generating system and bases (only under different names)[4]. Giuseppe Peano (1858 - 1932) was the first to formulate the modern definition of a vector space based on Graßmann's theory[5].

In addition to its geometric meaning, which is particularly important in graphic data processing and robotics, linear algebra as the theory of vector spaces is an extremely important area in all of mathematics. Vector spaces are also needed in many areas of mathematics, from economics to physics.

Sections of algebra [edit]

  • Elementary algebra: Elementary algebra is algebra in the sense of school mathematics. It includes the calculation rules for natural, whole, fractional and real numbers. She also explores the use of expressions that contain variables and finds ways of solving simple algebraic problems such as solving quadratic equations.
  • Classical algebra: Classical algebra is concerned with solving general algebraic equations over real or complex numbers. Its central result is the fundamental theorem of algebra. This says that every non-constant polynomial -th degree in Linear factors can be decomposed with complex coefficients.
  • Linear Algebra: Linear algebra deals with vector spaces and linear mappings between them. Vector spaces are generalizations of the vector spaces known from school and . This particularly includes the consideration of linear systems of equations and matrices. The vector spaces With is called Euclidean vector spaces or Coordinate spaces. Linear algebra is also the basis for analytic geometry. This is a branch of geometry that provides algebraic tools (especially from linear algebra) for solving geometric problems. In many cases it makes it possible to solve geometrical tasks purely arithmetically without using visualization as an aid[6].
  • Abstract algebra: Abstract algebra is the basic discipline of modern mathematics. She deals with algebraic structures such as groups, rings, bodies, etc. In doing so, she examines their properties and the mappings that exist between these structures. The structures examined in abstract algebra appear in many branches of mathematics. Abstract algebra connects many mathematical theories and has many applications.

What is algebra? [Edit]

The definition of the mathematical sub-area algebra is ambiguous and the respective areas are often very different.

Originally, algebra was understood to mean solving and transforming equations by calculating with symbols. This is also evident in the origin of the name "algebra". The Arabic means "straightening broken bones". In mathematics, algebra means to a certain extent the "setting of mathematical terms".[7] So it's about finding solutions for unknown terms. In contrast to analysis, in algebra there is no limit value formation - equations should always be solved exactly in a finite number of calculation steps and not, for example, by series expansion or approximation.

If the equations are linear, it is called linear algebra; in principle, any number of linear equations with any number of variables can be solved by elimination. If the equations are not linear, this is no longer so easy. Linear and quadratic equations in one variable were studied in ancient times (first by the Babylonians). Today you learn to solve these equations in school. Corresponding general formulas for third and fourth degree equations were found in Italy in the 16th century. The search for solution formulas for equations of degree 5 was unsuccessful for a long time. In 1824 Abel finally proved, building on an incomplete proof by Ruffini, that closed formulas do not exist[8]. The proof of this amazing statement is a milestone in the history of mathematics. In the decades that followed, the theory was further developed considerably, including by Galois, which also made the principles behind Abel-Ruffini's theorem clearer. Out of this, under the influence of Arthur Cayley and others, the theory of groups emerged and with it gradually abstract algebra as a structure theory.

The systematic study of structures is an essential feature of modern mathematics. Today you learn in the first semester what bodies, vector spaces and groups are and get a feel for the abstract approach to the solution theory of equations and other questions. On the other hand, the pure structural theory seems a bit dry and unmotivated if it is not presented together with some of the concrete questions from which it arose. In addition to questions of algebra itself, such as the theory of solving algebraic equations, above all questions of geometry and number theory.

Algebra developed in particular out of the need to develop a theory for the solution of equations and systems of equations. Let's take the simple equation . We immediately realize that this is for is satisfied. Is This the Right Solution?

The solution for only applies if we are working in a range of numbers that is “large” enough. The rational numbers or the real numbers include, for example, the solution . However, let's work in the number range of natural numbers or in the number range of whole numbers , then the above equation has no solution. Thus, in algebra, we also deal with number ranges and their structure. The typical number ranges are given by the following chain of subsets:

Algebra examines the structures of these number ranges. It is typical of numbers that you can count on them. An arithmetic operation is a combination that takes two numbers and returns a new number as the result. One such link is addition, which is defined in all sets of the chain of subsets above. Let's take the range of numbers as an example of whole numbers. There it holds that the equation For has a solution. However, there are in natural numbers not this solution, because there are no negative numbers there.

Linear algebra deals with the mathematical structure of vector space. This was created by abstracting the descriptive vector calculation as it is taught in school. The so-called linear mappings between two identical vector spaces play a prominent role in the investigation of vector spaces. These linear mappings can be described in a simple and clear way using matrices. The matrix calculation is therefore the formalism for "calculating" with linear mappings. The development of linear algebra was, among other things, motivated by the need for a solution theory for systems of linear equations. The solving of linear systems of equations is traced back to the solving of matrix equations. Methods and objects from linear algebra can also be found in analytical geometry!

Variables in algebra

With the use of variables and terms, we approach “pure” mathematics. Where previously it was about concrete arithmetic with reference to everyday life (be it with the basic arithmetic operations, with geometric objects, with the rule of three or with the percentage calculation), a first step is now being made into abstraction. To some, this new mathematics may seem “aloof and empty” and no longer relevant to practice. Variables and terms, however, enable a clear, precise and very brief symbolic representation for (mathematical) situations in which many sentences would be necessary on the linguistic level.

Basic ideas about a variable [edit]

The different possibilities offered by dealing with variables represent the first difficulty. Unlike in everyday life, variables are not abbreviations for anything, but ultimately a placeholder for one or more numbers, for all numbers or for a term.

So a variable is a name for a space in a logical or mathematical expression. The term is derived from the Latin adjective variabilis (changeable). The already mentioned term “placeholder” or “changeable” are used equally. When several variables with functional connections between them come together, a distinction is made between dependent and independent variables. All independent variables belong to the domain of definition. The dependent variables belong to the range of values.

Variables are usually referred to with letters. You can insert values ​​(placement aspect) or calculate with the variables (calculus aspect). Variables are also used to describe something (object aspect).

Types of variables [edit]

There are different types of variables[9]:

  1. The variable as an unknown value of a searched number: This type of variable is probably the best known. Here the variable stands for a number to be determined. The value for the variable is to be determined in such a way that a given equation is fulfilled and a true statement is made. Example: , so is the number we were looking for .
  2. Variable in terms as a proof principle: These variables appear in a general description of a calculation law or a geometric formula. Any numbers that make sense for the context can be used for the variables. The variables are an example and in the binomial formula or the variables (Surface), (Length of the baseline) and (Height) in the formula for calculating the area of ​​a triangle.
  3. Independent variable: We speak of an independent variable if its value can be freely chosen within its definition range. If you look at the air pressure in height looks so is an independent variable.
  4. Dependent variable: The value of a variable depends on the values ​​of other variables. So is the air pressure a dependent variable, depending on the height, represented by the letter .
  5. parameter: A parameter is an inherently independent variable, which, however, at least in a given situation is more likely to be understood as a fixed quantity. For example is the braking distance of a vehicle mainly on its speed dependent, because , is there a parameter whose value, on closer inspection, depends on other parameters such as the grip of the road surface and the tread depth of the tires.
  6. Constants: Often also concrete immutable Numbers, fixed Sizes or due to measurement deviations unsafe or. incorrect Provide measured values ​​with a formula symbol that can now be used instead of the numerical specification. The symbol stands for the usually unknown true value. Examples are the circle number (depending on how many decimal places I allow, the value of the circle number changes ). Also the elementary charge