A number in mathematics is the word or symbol used to designate quantities or entities that behave as quantities.
A number in mathematics is the word or symbol used to designate quantities or entities that behave as quantities.
Numbers are grouped into diverse sets or structures; each contains the one before it and is more complete than it and with greater possibilities in its operations. They are listed below.
This is the mathematical notion of fundamental importance, introduced more or less consciously since antiquity, in order to be able to operate on quantities of elements constituting sets or on quantities expressing measures of material entities. Many numerical sets can be introduced axiomatically together with the corresponding operations, such as algebraic and topological particulars. Vice versa, one can proceed constructively, introducing successively larger numerical sets.
The natural numbers 1, 2, 3,... are introduced as cardinal or as ordinal, i.e. as entities in a position to represent the order of finite sets and the positions of sequences (Peano axioms); zero is introduced as the order of the empty set.
Zero and the natural numbers constitute the set of non-negative numbers. Negative numbers are introduced as the inverses of the positive numbers with respect to addition, and in order to be able to perform unrestricted subtraction.
Rational numbers are introduced in order to perform unrestricted division. The extension to algebraic numbers is done to guarantee the existence of zeros of polynomials with integer coefficients.
Real numbers are introduced in order to be able to perform with minimal restrictions operations passing to the limit.
Finally, the real field is extended to that of the complex numbers to guarantee the existence of n roots for each polynomial of degree n.
- Polynomials with integer coefficients are introduced so that operations can be performed with minimal restrictions on passing to the limit.
- Fermat's number: Any number of the form 22n+1, for each n=1,2,3, ... It has been shown that its author's first conjecture that these numbers were all prime is not true.
- Perfect number: positive integer equal to the sum of its positive divisors, excluding itself. It is not known whether perfect odd numbers exist.
- Polygonal number: natural number of the sequence n0 = 1, n1 ... nr ... where nr = nr-1 + (m-2)r +1, where m is a natural number greater than two. For m = 3,4,5..., we obtain the triangular, quadrangular, pentagonal numbers.... The number nr is the number of points marked in a geometrical scheme formed with triangles, squares, pentagons..., respectively.
- Transfinite number: cardinal number which is not integer.
- Transcendent number: number that is not the root of any algebraic equation with rational coefficients.
- Triangular number: natural number of the sequence n0 = 1, n1 ... nr ... in which nr = nr-1 + r +1, . The number nr is the number of marked points in a geometric scheme formed with triangles.
- Friendly numbers: pair of positive integers such that the sum of the positive divisors of each number less than itself is equal to the other number.
- Pythagorean numbers: triads of positive integers such that the square of one of them is equal to the sum of the squares of the other two. If the lengths of the two sides of a triangle are integers and Pythagorean, the triangle is right-angled.
These are the ones used to count the elements of sets:
N = {0, 1, 2, 2, 3,..., 9, 10, 11, 11, 12,...}
There are infinities. They can be added and multiplied and with both operations the result is, in all cases, a natural number. However, they can not always be subtracted or divided (neither 3 - 7 nor 7 : 4 are natural numbers).
These are the natural numbers and the corresponding negatives:
Z = {..., -11, -10, -9,..., -3, -2, -1, 0, 1, 2, 3,..., 9, 10, 11,...}
In addition to being added and multiplied in all cases, they can be subtracted, so this structure improves on that of the naturals. However, in general, two integers cannot be divided. This is why we move on to the following number structure.
These are the ones that can be expressed as a quotient of two integers. The set Q of rational numbers is composed of the integers and the fractional numbers. They can be added, subtracted, multiplied and divided (except by zero) and the result of all these operations between two rational numbers is always another rational number.
Unlike the natural numbers and the integers, the rational numbers are not arranged in such a way that they can be ordered one at a time. That is, there is no "next" rational number, because between any two rational numbers there are infinitely many others, so that if they are represented on a line, it is densely occupied by them: if we take a piece of line, a segment, however small, contains infinitely many rational numbers. However, in between these numbers densely located on the line there are also infinite other points that are not occupied by rationals. These are the irrational numbers.
The set formed by all rational and irrational numbers is the set of real numbers, so that all the numbers mentioned so far (natural, integer, rational, irrational) are real. These numbers occupy the number line point by point, so it is called real line.
The same operations are defined among the real numbers as among the rationals (addition, subtraction, multiplication and division, except for zero).
The product of a real number by itself is always 0 or positive, so the equation x2 = -1 has no solution in the real number system. If one wants to give a value to x, such that x = Á, this cannot be a real value, no longer in a mathematical sense but also not in a technical sense. A new set of numbers (different from that of the real numbers), that of the imaginary numbers, is used for this purpose. The symbol i represents the unit of the imaginary numbers and is equivalent to Á. These numbers allow one to find, for example, the solution of the equation , which can be written as
x = 3 × i or x = 3i
The numbers bi,b ≠ 0, are called pure imaginary numbers.
An imaginary number is obtained by adding together a real number and a pure imaginary number.
In its general form, a complex number is represented as a+ bi, where a and b are real numbers. The set of complex numbers consists of all the real numbers and all the imaginary numbers.
Complex numbers are usually represented in the so-called Argand diagram. The real and imaginary parts of a complex number are placed as points on two perpendicular lines or axes. In this way, a complex number is represented as a single point on a plane, known as the complex plane.
Complex numbers are of great use in the theory of alternating electric current as well as in other branches of physics, in engineering and in the natural sciences.
Complex numbers are also useful in the theory of alternating electric current as well as in other branches of physics, in engineering and in the natural sciences.