Definitions

Algebra (from the Arabic “al-jabr”, part of the title of the treatise “Kitab al-jabr val-mukabala” , which uses systems of arbitrary nature in the corresponding algebraic operations, similar to the power of numbers.

History

In our time, algebra is one of the most important parts of mathematics, which is used not only in theory but also in practice.

Methods of expanding equals were known in the II human millennium BC. rewriting ancient Egypt (protested without the use of letter symbols). The mathematical papyri preserved to this day are not such problems that are reduced to equal first degrees with the same, but also a problem that is reduced to the level of the form ax2 = b (quadratic planning).

Read more: Essay on History and Importance of Algebra

Even more complex problems changed the solution at the beginning of the second working year BC. e. In ancient Babylon: in mathematical texts formed by cuneiform on clay tablets, square and biquadratic levels, systems are equal to two unknowns and are used by the simplest cubic equations.

At the same time, the Babylonians also did not use letter notations, but gave solutions to typical problems, reducing the divergence of similar problems to replace the number of values. In some cases, the rules of the rules of identical transformations were also cited. If the length is to be found by the square root of a number that does not have a tonic square, the value of the root x, which was the arithmetic mean of the numbers x and a / x, is approached.

The average mathematicians of ancient Greece (from the VI century BC) were fully accepted all the algebraic hardness in the geometric system. Property added numbers that spoke of adding segments, the resulting two numbers were explained as the plane of the rectangle, and the product used using a rectangular parallelepiped. Algebraic formulas looked at each other between other areas and objects.

For example, it was said that the area of ​​an area constructed on two two segments is equal to the area of ​​a square constructed on sections increased by the underwater areas of a rectangle constructed on these shades. Thus, they represented the terms “square digits” (derived values ​​on themselves), “cube numbers”, “geometric mean”. The geometric formula in the oars acquired and diverged to a square level – they looked for the sides of the rectangle on a given perimeter and area.

The geometric approach to algebraic problems limited the further use of science. Probably, it was impossible to add large different sizes (long, flat, volume), it was impossible to speak about extraction of more multipliers, etc.

The idea of ​​spectacles from the geometric interpretation appeared in Diophantus of Alexandria, who lived in the III century. His book “Arithmetic” presents letter symbols and special symbols for degrees up to the 6th degree. Any you understand for the tracked degrees, negative smokers, as well as the equal sign (there was no special sign to add yet), stylistic notation of multipliers added to negative numbers.

Having given a larger number of algebraically strong problems studied by Diophantus, which come to complex system algebraic levels, it has so far been shown that the system, where the number was smaller, the number is still unknown. For such levels, Diophantus sought additional rational solutions.

From the VI century. the center of mathematical research is moving to India, China, the Middle East and Central Asia.

Chinese scientists have developed a method of sequential exclusion of unknowns for solving systems of linear equations, gave new methods for the approximate solution of equations of higher degrees.

Indian mathematicians used negative numbers, improved letter symbolism.

However, only in the works of scientists from the Middle East and Central Asia did algebra take shape as an independent branch of mathematics that deals with the solution of equations. In the ninth century. Uzbek mathematician and astronomer Muhammad al-Khorezmi wrote a treatise “Kitab al-Jabr val-mukabala”, which gave general rules for solving first degree equations.

The word “al-jabr” (restoration), from which the new science got its name, meant the transfer of negative members of the equation from one part to another with a change of sign.

Oriental scientists have studied the solution of cubic equations, although they have not been able to obtain a general formula for their roots.

In Europe, the study of algebra began in the XIII century. One of the great mathematicians of this time was the Italian Leonardo of Pisa (Fibonacci) (c. 1170 – after 1228). His Book of Abacus (1202) was a treatise containing information on arithmetic and algebra up to and including quadratic equations.

The first great independent achievement of Western European scientists was the discovery in the XVI century. formulas for solving the cubic equation. This was the merit of the Italian algebraists S. del Ferro, N. Tartaglia, and J. Cardano. The student J. Cardano L. Ferrari solved the equation of the 4th degree. The study of some questions related to the roots of cubic equations led the Italian algebraist R. Bombelli to the discovery of complex numbers.

The lack of convenient and well-developed symbolism hampered the further development of algebra: the most complex formulas had to be taught in verbal form. At the end of the XVI century. the French mathematician Francois Viet introduced letter notation not only for unknowns but also for arbitrary constants. The symbolism of Viet was perfected by his followers. The final look was given to it in the XVII century. French philosopher and mathematician Descartes Rene, who introduced (used to this day) notation for exponents.

Gradually expanded the stock of numbers with which you could perform actions. Negative numbers won citizenship rights, then complex numbers, and scientists began to use irrational numbers freely. It turned out that, despite such an expansion of the stock of numbers, the previously established rules of algebraic transformations remain valid.

Finally, Descartes succeeded in freeing algebra from its uncharacteristic geometric form. All this allowed us to consider the solution of equations in the most general form, to apply the equations to the solution of geometric problems.

For example, the problem of finding the point of intersection of two lines was reduced to the solution of a system of equations that satisfied the points of these lines. This method of solving geometric problems is called analytical geometry.

The development of letter symbolism allowed to establish general statements about algebraic equations:

  • Bez’s theorem on the divisibility of a polynomial P (x) into a binomial (x − a), where a is the root of this polynomial;
  • Vieta’s formula for the ratio between the roots of a quadratic equation and its coefficients;
  • rules that allow you to estimate the number of real roots of the equation;
    general methods for excluding unknowns from systems of equations.

Especially far in the field of solving systems of linear equations managed to move in the XVIII century. – formulas were obtained for them, which allow to express the solution through coefficients and free terms. Further study of such systems of equations led to the theory of matrices and determinants.

At the end of the XVIII century. it has been proved that any algebraic equation with complex coefficients has at least one complex root. This statement is called the basic theorem of algebra.

K. Gauss For two and a half centuries, the attention of algebraists was focused on the problem of deriving a formula for solving the general equation of the 5th degree. It was necessary to express the solution of this equation through its coefficients using arithmetic operations and roots (to solve equations in radicals).

Only in the nineteenth century. the Italian P. Rufini and the Norwegian N. Abel independently proved that such a formula does not exist. These studies were completed by the French mathematician E. Galois, whose methods made it possible to determine for such an equation whether it is solved in radicals or not.

One of the most prominent mathematicians, K. Gauss, found out when it is possible to construct a regular n-gon with a compass and a ruler: this problem was directly related to the study of the roots of the equation xn = 1. It turned out that it is solvable only when the number n is a Fermat prime number or the product of several different Fermat primes. Thus, a young student (Gauss was then only 19 years old) solved a problem that scientists have been unsuccessfully engaged in for more than two millennia.

In the early nineteenth century, the main problems facing algebra in the first millennium of its development were solved. The rules of letter calculus for rational and irrational expressions were developed, the question of the possibility of solving equations in radicals was clarified, and a strict theory of complex numbers was constructed.

After the creation of the theory of complex numbers, the question arose about the existence of “hypercomplex numbers” – numbers with several “imaginary units”. Such a system of numbers, which had the form a + bi + cj + dk, where i2 = j2 = k2 = −1, was built in 1843 by the Irish mathematician William Hamilton, calling them “quaternions”.

With operations whose properties only partially resemble the properties of arithmetic operations, mathematics of the nineteenth century. faced in other matters. In 1858, the English mathematician A. Kelly introduced the general operation of multiplying matrices and studied its properties.

It turned out that the multiplication of matrices is reduced to many previously studied operations. English logician George Bull in the mid-nineteenth century. began to study operations on statements, which allowed two of these statements to build a third, and in the late nineteenth century. German mathematician G. Cantor introduced operations on sets: union, intersection, and so on.

Having studied the properties of operations of addition and multiplication over sets of rational, real, and complex numbers, mathematicians have created a general concept of the field – the set where these two operations are defined, and their usual properties are fulfilled. The study of the operation of multiplication of matrices led to the selection of the concept of group, which is now one of the most important not only in algebra but also in all mathematics.