Throughout history, numbers have been used more and more and with greater utility. However, with its use it was not enough to be able to create general calculations that were based on possible values that could correspond to quantities that responded to real-world problems, which were mostly geometry and, on many occasions, also related to astronomy.
For this reason, the Arabs chose to introduce a system that used letters and other symbols to create mathematical expressions. This gave rise to what we now know as Algebra. The first mathematician to write a treatise using algebraic language was Al-Khuwarizmi.
In this way it can be determined that algebra is the part of mathematics in charge of studying the relationship between different letters, numbers and symbols. For this reason, algebraic language can be defined as that which makes use of symbols and letters to represent numbers. The main function of this language is to structure a language that helps when dealing with the different arithmetic operations.
Algebraic language features
There are different aspects that characterize algebraic language and that you should know to better understand what it is about:
- It has great precision. That is, they have a level of specificity greater than that of numerical language. This makes it possible to express long statements in a more reduced way.
- It is possible to express numbers that are unknown and carry out mathematical operations with them in the same way.
- It allows to express different numerical properties and relationships of a general nature, that is, the various formulas that allow us to perform different calculations.
- Writing with this language can alter unknown quantities with symbols that are easy to write. This means that theorems can be simplified by formulating equations and also help the study to solve them.
In this way, the algebraic language seeks to create a language that aims to generalize operations that have to do with arithmetic. In this way it complements this one, in which its basic operations are addition, subtraction, division and multiplication.
On the other hand, it must be taken into account that an algebraic represents a set of numbers and letters that are combined through operation signs and that it is composed of exponents, coefficients and base. For example, in the operation 3×4, 3 is the coefficient, «x» the base and 4 the numerical exponent.
In this case, the coefficient indicates the amount of numeric character that is located to the left of the base, thus showing the number of times it must be added or subtracted, depending on the sign that accompanies it. In this case it would be 3×4= x4+x4+x4.
The numerical exponent is the quantity that is placed to the right of the base and that indicates the number of times in which the base is taken as a product. Ex: 3×2= 3 (x) (x).
In this way it can be known that the numerical value within an algebraic expression is the number that is the result of replacing the letters with numbers.
Elementary algebra
Algebra is the branch of mathematics that is focused on relationships, quantities and structures, a discipline known as elementary algebra, in which various arithmetic operations can be carried out, but unlike arithmetic, you can make use of different symbols that are responsible for replacing the numbers. This allows you to create, as we have already mentioned, different laws and formulas, being able to make references to numbers or values that are unknown and, therefore, without unknowns. This makes it possible to develop equations and solve them.
In what is known as elementary algebra there are different laws that allow people to have knowledge of the properties with which to carry out different arithmetic operations, such as addition (a+b) which is commutative (a+b = b +a), associative, and has an inverse operation and a neutral element. Sometimes these properties are shared by different operations.
As indicated by the so-called Fundamental Theorem of Algebra, in the case of a non-constant variable in which there are complex coefficients, a polynomial has as many roots as its degree indicates. This is because the roots are considered with their multiplicity.
Boolean algebra
Boolean algebra, also known as Boolean algebra, is a special branch of algebra that is used mainly in the field of digital electronics and was invented by George Boole, an English mathematician in the year 1854.
With this method, the simplification of logic circuits or logic commutation circuits is sought, within digital electronics. This algebra was created for control systems such as connectors and relays, which have two states, open (or conductive) or closed (non-conductive).
Both states are represented with a 1 (open) or a 0 (closed), which simplifies and studies these logical components more systematically. At the same time, different laws and properties are applied that are not directly related to the element in question. That is, it does not matter if it is a transistor, a logic gate or a relay.
Thus, Boole established that any component that is of the “all or nothing” type can be presented with a value of 1 or 0. Boole’s algebra has established different rules that must be taken into account in order to perform this type of operation on those variables.
Boole’s laws of algebra
When proceeding to formulate mathematical expressions for logic circuits, it is necessary to know the rules that are established within Boolean algebra. In this way, binary logical statements can be simplified. Before indicating the different laws, you should know that a bar over a symbol indicates that it is an inversion of the signal.
Fundamental laws
OR
A + 0 = A
A + 1 = 1
And also:
A + A = A
A + A* = 1
The * refers to a negation bar.
AND
A + 0 = 0
A + 1 = A
And also:
A + A = A
A + A = 0
NOT
A* = A
The * is equivalent in this case to two negative bars.
Commutative laws
A + B = B + A
A ∙ B = B ∙ A
Associative laws
(A + B) + C = A + (B + C)
(A ∙ B) ∙ C = A ∙ (B ∙ C)
Associative laws
A ∙ (B + C) = (A ∙ B) + (A ∙ C)
A + (B ∙ C) = (A + B) ∙ (A + C)
There are also other useful identities that must be known in order to master Boolean algebra. In any case, it must also be taken into account that it is possible to simplify the Boolean functions, for which both identities and the so-called Karnaugh Map can be used.
With the Karnaugh Map, algebraic functions can be simplified graphically through different patterns and without having to carry out complex and extensive calculations. However, in order to apply it, it is also necessary to have knowledge of its rules and the different groupings of values that can be obtained. This is important to know for those who work or are with the study of logical processes.