Irrational number

The mathematical constant π π {displaystyle pi }expressed in its decimal form.

Ten thousand first decimal figures of the number e{displaystyle {text{e}}}.

In mathematics, a irrational number is a value that cannot be expressed as a fraction mn{displaystyle {frac {m}{n}}}}Where m,n한 한 Z{displaystyle m,nin mathbb {Z} } and nI was. I was. 0{displaystyle nneq 0}. It is any real number that is not rational, and its decimal expression is neither accurate nor periodic.

A infinite decimal (i.e., with infinite numbers) aperiodicLike, √7 = 2,64575131106459059050161... cannot represent a rational number. Such numbers are named "irrational numbers". This denomination means the impossibility of representing such a number as reason of two whole numbers. Number pi (π π {displaystyle pi }), number and golden number (φ φ {displaystyle phi }) are other examples of irrational numbers.

History

Since in practice measuring the length of a line segment can only produce a fractional number, the Greeks initially identified numbers with the lengths of line segments. When identifying in the aforementioned way, the need arises to consider a class of numbers broader than that of fractional numbers. It is attributed to Hipaso de Metaponto, belonging to a group of Pythagorean mathematicians, the existence of line segments incommensurable with respect to a segment that is taken as a unit in a measurement system. Well, there are line segments whose measured length in this system is not a fractional number.

For example, in a square, its diagonal is immeasurable with respect to its sides. This fact caused a convulsion in the ancient scientific world. It caused a break between the geometry and arithmetic of that time, since the latter, at that time, was based on the theory of proportionality, which only applies to commensurable magnitudes.

They tried to overcome the obstacle by distinguishing between the concept of number and that of the length of a line segment, and took the latter as basic elements for their calculations. In this way, the incommensurable segments with respect to the unit taken as a standard of measurement were assigned a new type of magnitude: irrational numbers, which for a long time were not recognized as true numbers.

Notation

There is no universal notation to indicate them, as I{displaystyle mathbb {I} }that is generally accepted. The reasons are that the set of Irrational Numbers does not constitute an algebraic structure, as are the natural ones (N{displaystyle mathbb {N} }), the integers (Z{displaystyle mathbb {Z} }), rational (Q{displaystyle mathbb {Q} }), the real (R{displaystyle mathbb {R} }) and complexes (C{displaystyle mathbb {C} }on the one hand, and I{displaystyle mathbb {I} } is as appropriate to designate the set of irrational numbers as the set of imaginary numbers, which can create confusion. Get out of it,

I:=R Q={x한 한 R日本語x Q!{displaystyle mathbb {I}:=mathbb {R} backslash mathbb {Q} ={xin mathbb {R} Δxnotin mathbb {Q} }}{cH}}}

Classification

The irrational numbers are the elements of the real line that cover the gaps left by the rational numbers, since many sequences of rationals have as limit a number that is not a rational number.

Irrational numbers are the elements of the real line that cannot be expressed by the quotient of two integers and are characterized by having infinite non-recurring decimal figures. The irrational number can be defined as an infinite non-periodic decimal fraction. In general, any expression in decimal numbers is only an approximation in rational numbers to the referred irrational number, and it is properly said that the number √2 is approximately equal to 1.4142135 to 7 decimal places, or ok is equals to 1.4142135… where the three dots refer to the missing decimals. Because of this, the best known irrational numbers are identified by symbols:

1. π π {displaystyle pi } (Number "pi" 3,14159...): reason between the length of a circumference and its diameter.
2. e (Number "e" 2,7182...): limn→ → +∞ ∞ (1+1n)n{displaystyle lim _{nto +infty }left(1+{frac {1}{n}{n}right)^{n}}
3. ≈ ≈ {displaystyle Phi } (Number "aurus" 1,6180...): 1+52{displaystyle {frac {1+{sqrt {5}}}{2}}}}
4. real x solutions² - 3 = 0; x⁵ -7 = 0; x³ = 11; 3^x = 5; 7o, etc.

Irrational numbers are classified into two types:

1. Algebraic Number: They are the solution of some algebraic equation and can sometimes be represented by a finite number of free radicals or anitious in some cases. There are also algebraic numbers that cannot be expressed with sums of products or radicals, such is the case of the roots of polynomial x5− − 6x+3{displaystyle x^{5}-6x+3}, since your group of Galois turns out not to be soluble; if x{displaystyle x} represents this number, by eliminating radicals from the second member through reverse operations, there is an algebraic equation of a certain degree. All non-exact roots of any order are irrational algebraic. For example, the golden number is one of the roots of algebraic equation x2− − x− − 1=0{displaystyle x^{2}-x-1=0}So it's an irrational algebraic number.
3. Significant number: They are not a solution of any polynomial with rational coefficients; they come from the so-called transcendent functions (trigonometric, logarithmic and exponential, etc.) They also arise when writing non-recurrent decimal numbers randomly or with a pattern that has no definite period, respectively, as the following two:

0,193650278443757{displaystyle 0,193650278443757}...

0,101001000100001{displaystyle 0.101001000100001}...

The so-called transcendent numbers have special relevance since they cannot be a solution to any algebraic equation. The pi and e numbers are transcendent irrational, since they cannot be expressed by radicals.

Irrational numbers are not countable, that is, they cannot be bijected with the set of natural numbers. By extension, the real numbers are not countable either since they include the set of irrational numbers.

Properties

• Be the expressions k+lκ κ =m+nκ κ {displaystyle k+lkappa =m+nkappa } where k,l,m,n한 한 Q;κ κ 한 한 R− − Q=Qc{displaystyle k,l,m,nin Q;kappa in R-Q=Q^{c}}implies that k=m,l=n{displaystyle k=m,l=n}
• The sum and difference of a rational number and an irrational number is an irrational number:a한 한 Q,b한 한 Qc a± ± b한 한 Qc{displaystyle ain Q,bin Q^{c}implies apm bin Q^{c}}
• The inverse additive of an irrational number is an irrational number: a한 한 Qc − − a한 한 Qc{displaystyle ain Q^{c}implies - ain Q^{c}}
• The product of a rational different from zero by an irrational is an irrational number:a한 한 Q,b한 한 Qc a⋅ ⋅ b한 한 Qc{displaystyle ain Q,bin Q^{c}implies acdot bin Q^{c}
• The quotient between a non-null rational and an irrational number is an irrational number:a한 한 Q;b한 한 Qc a⋅ ⋅ b− − 1=ab한 한 Qc{displaystyle ain Q;bin Q^{c}implies acdot b^{-1}={frac {a}{b}}in Q^{c}}}
• The reverse of an irrational number is irrational number:a한 한 Qc a− − 1한 한 Qc{displaystyle ain Q^{c}implies a^{-1}in Q^{c}
• Be a binomial, formed by a rational plus a second order radical, or the sum of two second order radicals, which is irrational. Then your conjugate is irrational.
• The values of vulgar or natural logarithms and the values of trigonometric reasons, the vast majority not numberable, are irrational.
• The number of Gelfond (2^√2) is a transcendent irrational number
• The square root of a perfect non-square natural number is an irrational number; it is also the ensima root of a natural p that is not perfect ensima power.
• Between two different rationales, there is at least one irrational number
• The trigonometric reasons of an angle are irrational, exceptionally one of them in the case that two of the sides of the rectangle triangle are rational.
• Lebesgue measurement of any closed interval type [chuckles]a,b] I R{displaystyle scriptstyle [a,b]cap mathbb {I} subset mathbb {R} } equals the measure b-a. That means that if there was a procedure to randomly select a number of such interval, probably 1 the number obtained would be irrational.
• Any irrational number that is in an open interval of real numbers is accumulation point of the actual numbers of such interval, as of the irrational numbers thereof. For example: √5 is point of accumulation of the actual numbers of the interval K = ≤1;4as well as irrational numbers K.
• The set of irrational numbers is equivalent (the same cardinal) to the set of actual numbers.