Remarkable number

In Mathématiques some Nombre S are distinguished from the others, play a key part, or appear curiously in many formulas. These numbers, considered as important, are called remarkable numbers and bear a name, which is sometimes that of a mathematician, of a geometrical figure,… Some call them constant mathematics , although constant does not correspond in Mathématiques to a quantity or a number, but with a constant Fonction. It is thus necessary to interpret a constant mathematics like a particular number.

Many numbers in mathematics have a particular significance and appear in various contexts. For example, in the following theorem: there exists a single holomorphic function $\backslash \; exp\; (Z)$ such as

$\backslash \; exp\; \text{'}(Z)\; =\; \backslash \; exp\; (Z)\; \backslash ,\; \backslash !$

$\backslash \; exp\; (0)\; =\; 1\; \backslash ,$

The number $\backslash \; exp\; (1)$ is then E the exponential number of one. Moreover $\backslash \; exp\; (Z)$ is a periodic function, of period $2\; \backslash \; pi\; i$, another remarkable number.

The remarkable numbers are typically elements of the body of the complex real numbers or . In any case, these particular numbers are always definable, and those currently existing have one (or several) rigorous definition. In addition, they are almost always calculable. But there exist remarkable numbers for which only approximate values coarse are known. Certain remarkable real numbers, can be classified according to their representation in the form of continued Fraction.

We will give a list of all the numbers considered usually as remarkable, while starting with most important those which some describe as constant important and which are certainly most remarkable because they are often met in various fields of mathematics.

Notice that the rise in a number to the remarkable row of number is rather arbitrary. Why not consider $\backslash \; ln\; 3$ for which it is certainly possible to find properties mathematical interesting? Or quite simply any prime number?

We are constrained to make a choice among all the existing numbers.

Remarkable entireties

to Voir : Category: Inventory of numbers for the complete listing of the article devoteds to a given entirety. The concept of remarkable entireties is difficult to define because of the paradox of the entireties remarquables.

• 0: neutral element of the additive Z , remarkable group for its history;

• 1: neutral element of the multiplicative group Z , first identified quantity;
• 2: only the Prime number even;
• the prime numbers.
• In September 2006, the greatest prime number known was a prime number of Mersenne, 2^{: 32582657}-1.
• the largest couple of Prime numbers twins is:

$100314512544015\; \backslash \; times\; 2^\; \{2171960\}\; \backslash \; pm\; 1\; \backslash ,$

• the perfect numbers, which are equal to the sum of their dividers. One knows 44 of them, including eight lower than $10^\; \{21\}$.
• 6
• 28
• 496
• 8.128
• 33.550.336
• 8.589.869 056
• 137.438.691 328
• 2.305.843 008.139.952 128
• the gogol = $10^\; \{100\}$ is higher than the number of Atome S in the Univers.
• the number of Shannon , 10¹²⁰, is an estimate of the complexity Jeu of failures.
• Number of Graham, which is known to be the greatest entirety ever used in a demonstration. Its last ten figures are… 2464195387.

Remarkable rational numbers

• the decimal numbers have a decimal Développement limited.

Remarkable algebraic numbers

• the square Root of two, is a irrational Nombre, solution of the equation $x^2\; =\; 2$. It is perhaps the first irrational one to be highlighted by the Greeks; it is equal to the length of the diagonal of a square on side one; it intervenes in the formulas giving volumes of the Tétraèdre and the Octaèdre;

• $i$ called number '' I '', solution of the equation $x^2+1=0$; at the base of the right-hand side of imaginary and definition of the complex numbers;
• Golden section: φ.
• $\backslash \; sqrt\; \{11\}\; =\; =\; \backslash \; overline\; \{3,6\}$ belongs to the irrational numbers which have a development in Fraction continues periodic pure.

Nonconstructible algebraic numbers

• the Heptagone is not constructible with the rule and the compass because $x\; =\; \backslash \; cos\; (\backslash \; pi/7)$ is not a constructible Nombre. Reality X is however algebraic, since it is root of $8x^3\; -\; 4x^2\; -\; 4x\; +\; 1\; =\; 0$.

transcendent Numbers

• π

• '' E ''
• One knows that at least one of the numbers $\backslash \; pi~+~e\; \backslash ,$ and $\backslash \; pi\; E\; \backslash ,$ must be transcendent.
• the Constant of Prouhet-Thue-Morse $\backslash \; tau\; \backslash ,$.

Transcendent numbers not calculable

• the constant Oméga of Chaitin Ω is well defined but is not calculable.

normal Numbers

• the number of Champernowne

0,1234567891011121314151617… who contains in his decimal development the concatenation of all the natural numbers is normal bases 10 of them, but it is not in certain for it other bases.

• the Constant of Copeland-Erdős

0,2357111317192329313741… obtained by concatenant the prime numbers is known as being a normal number in 10 bases.

It is not known if √2, π, ln (2) or '' E '' is normal.

Remarkable complex numbers

• According to the Assumption of Riemann, zero the noncommonplace ones of the Fonction Zeta of Riemann have all to some extent real 1/2. This conjecture constitutes one of the unsolved problems most important of current mathematics.

Real numbers with the unspecified statute

• γ : Constant of Euler-Mascheroni.
• Constant of Khintchine $K$.
• Constant of Catalan

See too

Related articles

• Table of constant mathematics

External bonds

• List of remarkable numbers
• Of constant mathematics
• Constant in Wolfram

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