The algebra is a branch of mathematics that uses numbers, letters and signs to refer to the various arithmetic operations performed. Today algebra as a mathematical resource is used in relationships, structures and quantity. Elementary algebra is the most common since it is the one that uses arithmetic operations such as addition, subtraction, multiplication and division since, unlike arithmetic, it uses symbols such as xy being the most common instead of using numbers.
What is algebra
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It is the branch that belongs to mathematics, which allows developing and solving arithmetic problems through letters, symbols and numbers, which in turn symbolize objects, subjects or groups of elements. This allows to formulate operations that contain unknown numbers, called unknowns and that makes the development of equations possible.
Through algebra, man has been able to count in an abstract and generic way, but also more advanced, through more complex calculations, developed by mathematical and physical intellectuals such as Sir Isaac Newton (1643-1727), Leonhard Euler (1707- 1783), Pierre de Fermat (1607-1665) or Carl Friedrich Gauss (1777-1855), thanks to whose contributions we have the definition of algebra as it is known today.
However, according to the history of algebra, Diophantus of Alexandria (date of birth and death unknown, believed to have lived between the 3rd and 4th centuries), was actually the father of this branch, as he published a work called Arithmetica, which It consisted of thirteen books and in which he presented problems with equations that, although they did not correspond to a theoretical character, were adequate for general solutions. This helped define what algebra is, and among many of the contributions he made, it was the implementation of universal symbols for the representation of an unknown within the variables of the problem to be solved.
The origin of the word "algebra" comes from Arabic and means "restoration" or "recognition". In the same way it has its meaning in Latin, which corresponds to "reduction", and, although they are not identical terms, they mean the same thing.
As an additional tool for the study of this branch, you can have the algebraic calculator, which are calculators that can graph algebraic functions. Allowing in this way to integrate, derive, simplify expressions and graph functions, make matrices, solve equations, among other functions, although this tool is more appropriate for a higher level.
Within algebra is the algebraic term, which is the product of a numerical factor of at least one letter variable; in which each term can be differentiated by its numerical coefficient, its variables represented by letters and the degree of the term by adding the exponents of the literal elements. This means that for the algebraic term p5qr2, the coefficient will be 1, its literal part will be p5qr2, and its degree will be 5 + 1 + 2 = 8.
What is an algebraic expression
It is an expression made up of integer constants, variables and algebraic operations. An algebraic expression is made up of signs or symbols and is made up of other specific elements.
In elementary algebra, as well as in arithmetic, the algebraic operations that are used to solve problems are: addition or addition, subtraction or subtraction, multiplication, division, empowerment (multiplication of a multiple factor times) and radication (inverse operation of potentiation).
The signs used in these operations are the same as those used for arithmetic for addition (+) and subtraction (-), but for multiplication, the X (x) is replaced by a point (.) Or they can be represented with grouping signs (example: cd and (c) (d) are equal to element “c” multiplied by element “d” or cxd) and in the algebraic division two points (:) are used.
Grouping signs are also used, such as parentheses (), square brackets, braces {}, and horizontal stripes. Relationship signs are also used, which are those used to indicate that there is a correlation between two data, and among the most used are equal to (=), greater than (>) and less than (<).
Also, they are characterized by using real numbers (rational, which include positive, negative and zero; and irrational, which are those that cannot be represented as fractions) or complex, which are part of the real, forming an algebraically closed field.
These are the main algebraic expressions
There are expressions that are part of the concept of what algebra is, these expressions are classified into two types: monomials, which are those that have a single addend; and polynomials, which has two (binomials), three (trinomials) or more addends.
Some examples of monomials would be: 3x, π
While some polynomials can be: 4 × 2 + 2x (binomial); 7ab + 3a3 (trinomial)
It is important to mention that if the variable (in this case "x") is in the denominator or within a root, the expressions would not be monomials or polynomials.
What is linear algebra
This area of mathematics and algebra is the one that studies the concepts of vectors, matrices, systems of linear equations, vector spaces, linear transformations and matrices. As can be seen, linear algebra has various applications.
Its usefulness varies from the study of the space of functions, which are those that are defined by a set X (horizontal) to a set Y (vertical) and are applied to vector or topological spaces; differential equations, which relate a function (value that depends on the second value) with its derivatives (instantaneous rate of change that makes the value of a given function vary); operations research, which applies advanced analytical methods to make sound decisions; to engineering.
One of the main axes of the study of linear algebra is found in vector spaces, which are made up of a set of vectors (segments of a line) and a set of scalars (real, constant or complex numbers, which have magnitude but not the direction vector characteristic).
The main finite dimensional vector spaces are three:
- The vectors in Rn, which represent Cartesian coordinates (horizontal X axis and vertical Y axis).
- The matrices, which are rectangular systems expressions (represented by numbers or symbols), are characterized by a number of rows (usually represented by the letter "m") and a number of columns (denoted by the letter "n"), and they are used in science and engineering.
- The vector space of polynomials in the same variable, given by polynomials that do not exceed degree 2, have real coefficients and are found on the variable "x".
Algebraic functions
It refers to a function that corresponds to an algebraic expression, while it also satisfies a polynomial equation (its coefficients can be monomials or polynomials). They are classified as: rational, irrational and absolute value.
- The integer rational functions are those expressed in:, where "P" and "Q" represent two polynomials and "x" the variable, where "Q" is different from the null polynomial, and the variable "x" does not cancel the denominator.
- Irrational functions, in which the expression f (x) represents a radical, like this:. If the value of "n" is even, the radical will be defined so that g (x) is greater than and equal to 0, and the sign of the result must also be indicated, since without it, it would not be possible to speak of a function, since for each value of "x" there would be two results; while if the index of the radical is odd, the latter is not necessary, since the result would be unique.
- The absolute value functions, where the absolute value of a real number will be its numerical value leaving aside its sign. For example, 5 will be the absolute value of both 5 and -5.
There are explicit algebraic functions, in which its variable "y" will be the result of combining the variable "x" a limited number of times, using algebraic operations (for example, algebraic addition), which include elevation to potencies and the extraction of roots; this would translate to y = f (x). An example of this type of algebraic function could be the following: y = 3x + 2 or what would be the same: (x) = 3x + 2, since “y” is only expressed in terms of “x”.
On the other hand, there are the implicit ones, which are those in which the variable “y” is not expressed only as a function of the variable “x”, so y ≠ f (x). As an example of this type of function, we have: y = 5x3y-2
Examples of algebraic functions
There are at least 30 types of algebraic functions, but among the most prominent, there are the following examples:
1. Explicit function: ƒ () = sin
2. Implicit function: yx = 9 × 3 + x-5
3. Polynomial function:
a) Constant: ƒ () = 6
b) First degree or linear: ƒ () = 3 + 4
c) Second degree or quadratic: ƒ () = 2 + 2 + 1 or (+1) 2
d) Third degree or cubic: ƒ () = 2 3 + 4 2 + 3 +9
4. Rational function: ƒ
5. Potential function: ƒ () = - 1
6. Radical function: ƒ () =
7. Function by sections: ƒ () = if 0 ≤ ≤ 5
What is Baldor algebra
When talking about what Baldor algebra is, it refers to a work developed by the mathematician, professor, writer and lawyer Aurelio Baldor (1906-1978), which was published in 1941. In the professor's publication, who was born in Havana, Cuba, 5,790 exercises are reviewed, equivalent to an average of 19 exercises per test.
Baldor published other works, such as "Plane and Space Geometry", "Baldor Trigonometry" and "Baldor Arithmetic", but the one that has had the most impact in the field of this branch has been the "Baldor Algebra".
This material, however, is more recommended for the intermediate educational level (such as secondary school), since for higher levels (university) it would hardly serve as a complement to other more advanced texts according to that level.
The famous cover featuring the Persian Muslim mathematician, astronomer and geographer Al-Juarismi (780-846), has represented confusion among the students who have used this famous mathematical tool, since it is thought that this character is about its author Baldor.
The content of the work is divided into 39 chapters and an appendix, which contains tables of calculations, a table of basic forms of factor decomposition and tables of roots and powers; and at the end of the text are the answers to the exercises.
At the beginning of each chapter there is an illustration that reflects a historical review of the concept that will be developed and explained below, and mentions prominent historical figures in the field, according to the historical context in which the reference of the concept is located. These characters range from Pythagoras, Archimedes, Plato, Diophantus, Hypatia, and Euclid, to René Descartes, Isaac Newton, Leonardo Euler, Blas Pascal, Pierre-Simon Laplace, Johann Carl Friedrich Gauss, Max Planck, and Albert Einstein.
What was the fame of this book due to?
Its success lies in the fact that it is, in addition to a famous compulsory literary work in Latin American secondary schools, the most consulted and complete book on the subject, as it contains a clear explanation of the concepts and their algebraic equations, as well as historical data on the aspects to study, in which the algebraic language is handled.
This book is the initiation par excellence for students into the algebraic world, even though for some it represents a source of inspirational studies and for others it is feared, the truth is that it is a mandatory and ideal bibliography for a better understanding of the topics covered..
What is Boolean algebra
The English mathematician George Boole (1815-1864), created a group of laws and rules to perform algebraic operations, to the point that a part of it was given its name. For this reason, the English mathematician and logician is considered one of the forerunners of computer science.
In the logical and philosophical problems, the laws that Boole developed allowed to simplify them in two states, which are the true state or the false state, and these conclusions were reached through a mathematical way. Some implemented control systems, such as contactors and relays, use open and closed components, the open being the one that conducts and the closed one being the one that does not. This is known as all or nothing in Boolean algebra.
Such states have a numerical representation of 1 and 0, where 1 represents the true and 0 the false, which makes their study easier. According to all this, any component of any type or nothing can be represented by a logical variable, which means that it can present the value 1 or 0, these representations are known as binary code.
Boolean algebra makes it possible to simplify logic or logic switching circuits within digital electronics; also through it, calculations and logic operations of the circuits can be performed in a more express way.
In Boolean algebra there are three fundamental procedures, which are: the logical product, the AND gate or intersection function; the logical sum, OR gate, or union function; and logical negation, NOT gate or complement function. There are also several auxiliary functions: logical product negation, NAND gate; negation of logical sum, NOR gate; exclusive logic sum, XOR gate; and negation of exclusive logical sum, gate XNOR.
Within Boolean algebra, there are a number of laws, among which are:
- Cancellation law. Also called the cancellation law, it says that in some exercise after a process, the independent term will be canceled, so that (AB) + A = A and (A + B). A = A.
- Identity law. Or of identity of elements 0 and 1, it establishes that a variable to which the null element or 0 is added, will be equal to the same variable A + 0 = A in the same way as if the variable is multiplied by 1, the result is the same A.1 = a.
- Idempotent law. States that a particular action can be performed several times and the same result, so that, if you have a combination A + A = A and if it is a disjunction AA = A.
- Commutative law. This means that no matter the order in which the variables are, so A + B = B + A.
- Double negation law. O involution, states that if a denial is given another denial a positive result, so that (A ') = A.
- Morgan's theorem. These say that the sum of some amount of negated variables in general will be equal to the product of each negated variable independently, so (A + B) '= A'.B' and (AB) '= A' + B '.
- Distributive law. It establishes that when some variables are joined, which will be multiplied by another external variable, it will be the same as multiplying each variable grouped by the external variable, as follows: A (B + C) = AB + AC.
- Absorption law. It says that if a variable A implies a variable B, then the variable A will imply A and B, and A will be "absorbed" by B.
- Associative law. In the disjunction or when joining several variables, the result will be the same regardless of their grouping; so that in the addition A + (B + C) = (A + B) + C (the first element plus the association of the last two, is equal to the association of the first two plus the last).
