Numbers are understood as the abstract entities with which magnitudes are represented that may well be quantities, distances, indicate an order, as well as the signs that represent them, which are also called figures.
They are abstract tools that we use to represent various sets of objects and to count, as well as to represent commensurable and incommensurable magnitudes, in various sciences (such as physics, chemistry, astronomy, etc.), as well as to perform everyday calculations. They are the set of abstract concepts with which the science of mathematics works, in the various branches into which it is divided.
In the various branches of mathematics, numbers are used that are little known outside of those who are expressly dedicated to mathematics, to carry out various operations, as is the case, for example, of the Hyperreals that are currently used in non-standard analysis). in operations with infinite and infinitesimal numbers (these hyperreal numbers have been used empirically since the time of the Greek mathematicians), and the Transititos that are used as an extension for the realization of operations with real numbers . Likewise, there are the so-called , Transcendent ,? , and , and Quaternions, among others that are used for the expression of various quantities, during operations.
They are studied by the so-called theory of numbers, which is the branch of mathematical sciences that is dedicated to the study of numbers and their types, classifying them into:
Main types of numbers:
Arabic or Hindu Numbers.- They are the signs that we use to designate numbers habitually: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, in a decimal system to count and perform operations. Its origin is in India from where its knowledge was acquired by the Arabs who spread its use in the world, reaching Europe and from there to us. It is the most widespread numbering system (and symbols) in the world today.
Ordinal Numbers.- They are those that are used to express a position or order of an element or of one or more sets of elements, within an ordered sequence. Being able to be expressed by means of words (first, second, third, fourth, fifth, etc.), or by means of numbers followed by a suffix after the number (for example 1st, 5th, 9th), adding a letter (a), lowercase called feminine ordinal (1st, 2nd, 3rd), pronounced, respectively, in feminine first, second, third and in this way with other figures, or the degree sign (°)or masculine ordinal, as follows, 1st, 2nd, 3rd, pronounced first, second, third, and so on. Roman numerals are also often used as ordinal numbers, for example I, II, III, IV, V, which are read respectively as first, second, third, fourth and fifth, being used, for example, in the numbering of chapters, subsections, and others. .
Cardinal Numbers.- These are used to express amounts, they are expressed by writing the numbers in words, for example twenty, twenty-one, twenty-two, twenty-three, etc., to express an amount, let’s say that Juan received thirty-three pesos in coins, after having given a fifty-peso bill to pay for bread and milk.
Roman numerals.- These emerged using independent signs for the designation of quantities, but later they evolved using letters for their designation. Currently they are still used to make enumerations and in various uses, (for example in dates), but their use has declined in favor of “Arabic” numbering. It is a system that was basically based on the use of the letters I, V, X, L, C, D and M, making combinations of the order of these, to configure the various figures. For example, if X is 10, and the letter I is one, to add 13 they added X and three III as follows XIII to form the number 13, and depending on the position of the letter the number changed, for example in XXI and in XIX, which are read respectively 21 and 19, with the “Arabic” numbers.
Natural numbers.- It is called as natural, the set of numbers with which the elements of a set can be counted, these numbers are 0,1, 2, 3, 4, 5, 6, 7, 8, 9. These are the first numbers that were used by humans to count the various objects in their environment.
Negative numbers.- Negatives are understood as any number whose value is less than zero, that is, they are numbers that are counted from zero, negatively. This type of numbers can be easily explained by means of a number line, where zero is in the middle of the line and from zero the numbers to the right are positive and the numbers to the left of zero are negative. An example of its use is in a thermometer, where the positives (above zero) represent temperatures above zero degrees and the negatives (below zero) represent the graduation of degrees below zero.
Integer Numbers.- Positive natural numbers (other than zero), as well as negative ones, are called integers, that is, fractional numbers or fractions are excluded.
Prime Numbers.- Are those integers and positive numbers, which are different from zero and one, being that they can only be divided by the number 1 or by themselves, so that an exact result can be obtained, being that if they are divided by any number other than one or itself, the result would be inexact.
Irrational Numbers.- These are those that cannot be expressed by a fraction exactly, but rather by means of approximations, an example is the number pi, which is equivalent to approximately 3.14159.
Rational numbers.- They are called rational numbers to the set that is formed by all the fractional numbers and all the integers (which are designated with the letter Q), they can be expressed as a fraction, that is, as a quotient of two integers . It stands out that unlike integers or negatives that are followed by another number as follows: 1 follows 2, follows 3, follows 4 and so on and negatives -1, followed by -2, followed by -3, then -4, and so on. In the set of irrational numbers, this is not the case, since between each of these rational numbers, there are infinite numbers that separate them.
Complex Numbers.- They are an extension of real numbers that are used to be able to perform operations, representing the roots of polynomials, it is a tool created for ordinary algebra or algebra of complex numbers, being used to solve for example roots squares of negative numbers. They are used both in pure mathematics and in algebra, analysis and physics.
Negative Numbers.- Those that have a value less than zero are called negative, and the rest of the positive numbers. They are represented in the same way as the positive ones, but adding a minus or subtraction (-) sign , before the number in question, so that they can be easily identified. It is common for them to be used on number lines for grading temperatures (in thermometers), marking temperatures below zero, or expressed by putting the decimal point before them, to separate them from positive figures or from zero.
Odd Numbers.- They are called those that cannot be divided by the number 2 and obtain an exact result, these invariably have termination in any of the numbers 1, 3, 5, 7, or 9. For example the number 13, is not exactly divisible by 2, so it belongs to the group of odd numbers.
Even Numbers.- They are called like that, those that can be divisible by the number 2, or in their case that are multiples of this number (2), being that the pairs can only have the endings in digits: 0, 2, 4, 6, or 8, being that they are even numbers such as 18, 30, 42, or 88.
Perfect Numbers.- They are called perfect, those whose divisors less than it, if added, result in the same number, (including the number 1 among the minor divisors). An example of a perfect number is six, since its divisors (1, 2 and 3), added together give 6: 1+2+3 = 6. For a number to be counted in this category, it must be comply with the rule that it can be divisible by the numbers that, added together, result in said perfect number. That is, in the previous case, the rule is fulfilled, since 6 is divisible, both by 3, and between 2 and one.
Binary Numbers.- Binaries are understood as the numerical system used to perform calculations and operations by computers. It is a binary or diatic system, which consists in the complete representation of any figure, made through the exclusive use of zeros (0) and ones (1), that is, it is a system, based only on two figures. , for designation of numbers and performing operations. This system is used internally in computers, since they use two voltage levels, representing with a 0 and a 1 respectively, the absence or presence of voltage.