Numerical domains . Understanding the need for the expansion of numerical domains and their structure is of extraordinary importance for the acquisition of knowledge and power of all people in general.
In the historical development of mathematics , the order of appearance or application of numerical sets was never the same for all civilizations, fractional numbers were used by man before negative numbers .
Summary
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- 1 Construction of numerical domains
- 2 Numerical sets
- 3 Related Links
- 4 Source
Construction of numerical domains
The construction of numerical domains begins from the first grade of primary education with the set of natural numbers and the first expansion of the numerical domain is done at the level itself, thus introducing fractional numbers in sixth grade. In elementary school , rational numbers and integers are worked on , and in pre-university, reals and complex numbers are studied .
In carrying out this guideline, it is important that the knowledge acquired in each of the numerical domains, as well as the skills, be applied consciously in all other subject complexes. This includes, above all, the performance of the calculation operations, the constant exercise of the corresponding procedures and the observance of the corresponding calculation laws.
Making an assessment of the mathematics programs from the first grade to the 12th grade, it is recognized that the order followed in the school in the construction of the numerical domains, differs from the order followed in the science of Mathematics .
Throughout the course of numerical domain extensions, it should be recognized that the motivation for expansion occurs in a unified way by the condition of performing an unrestricted calculation operation.
The construction principle for numerical sets is the formation of classes based on equivalence relations.
The old number domain is isomorphic to a subdomain of the new number domain. It is important that the knowledge acquired in each of the numerical domains, as well as the skills, are applied consciously in all other subject complexes. This includes, above all, the performance of the calculation operations, the constant exercise of the corresponding procedures and the observance of the corresponding calculation laws.
Numerical sets
- N: Set of natural numbers:
They arose in the learning process that man had when he discovered how to count. They are the simplest numbers that we use, they are formed by the numbers 1,2,3,4,5 …
- Z: Set of integers:
They arise as the need that man saw to reunite in a single set the positive integers (natural) with the negative integers and with the element zero . The set of integers includes naturals, (naturals are a subset of integers).
- Nature of positive integers:
The integers can be defined as follows: Let any two numbers whose result in the division is integer (do not give any remainder, or the remainder is zero) the result obtained is an integer. Another definition of an integer is the periodic number of the form z’000000 ….
- Negativeintegers :
They arise from the need that man had to express situations such as: Temperatures below zero, debts, positions below sea level. They are denoted by – and are formed by the additive inverse numbers of the natural ones. – = {……, – 4, – 3, – 2, – 1}
- Q: Set of rational numbers:
They arise from the man’s need to take some parts of the unit. They are denoted by and are all those fractions that can be expressed in the form where p and q are integers and, for example: 3/5, – 2/3. etc.
In general:
Integers are also rational because unit (1) can be placed as a denominator .
The following decimals are also considered rational numbers: a. Finite decimals: those that have a finite number of decimal places, such as: 0.23, 2.3, – 0.324
- IrrationalNumbers :
They arise from the need to find the exact measure of the hypotenuse of a right triangle ; likewise of the need to express the real inexact roots. They are denoted by ‘and are all real inexact roots and non-periodic infinite decimals, such as: 0.32456891…, π = 3.14157…, = 1.414213562…
- A: Set of real numbers:
They arise from the need to bring together the rational and the irrational in one set. They are denoted by R. Therefore we have that: R = U ‘.
- Imaginary numbers:
They arise from the need to obtain the even index roots of negative quantities. They are denoted by I . The unit of the imaginary numbers is the square root of – 1 and is denoted by i, so: i =.
You should keep in mind that: i =
I ^ 2 = -1, I ^ 3 = – i, i ^ 4 = 1.
The union of real numbers with imaginary numbers gives rise to the noted complex numbers C , so: C = RUI .
- A: Set of complex algebraic numbers:
An algebraic number is any real or complex number that is the solution of a polynomial equation.
- C: Set of complex numbers:
An expression of the form a + bi in which a and b are any two real numbers and i is the imaginary unit is called a complex number. a + bi is the binomial form of the complex number; a is the real part and b is the imaginary part.
