In the cases of the rational numbers ( Q ) and the real numbers ( R ) there are no nontrivial field automorphisms.

In the case of the complex numbers, C, there is a unique nontrivial automorphism that sends R into R: complex conjugation, but there are infinitely ( uncountably ) many " wild " automorphisms ( assuming the axiom of choice ).

For example, the field extension R / Q, that is the field of real numbers as an extension of the field of rational numbers, is transcendental, while the field extensions C / R and Q (√ 2 )/ Q are algebraic, where C is the field of complex numbers.

* The set R of all real numbers is not compact as there is a cover of open intervals that does not have a finite subcover.

is a function defined on a subset I of the set R of real numbers.

This subset I is referred to as the domain of f. Possible choices include I = R, the whole set of real numbers, an open interval

The convex subsets of R ( the set of real numbers ) are simply the intervals of R.

* The cofinality of the real numbers with their usual ordering is ℵ < sub > 0 </ sub >, since N is cofinal in R. The usual ordering of R is not order isomorphic to c, the cardinality of the real numbers, which has cofinality strictly greater than ℵ < sub > 0 </ sub >.

In 1994 it was shown by W. R. ( Red ) Alford, Andrew Granville and Carl Pomerance that there really do exist infinitely many Carmichael numbers.

The space R of real numbers and the space C of complex numbers ( with the metric given by the absolute value ) are complete, and so is Euclidean space R < sup > n </ sup >, with the usual distance metric.

The space Q < sub > p </ sub > of p-adic numbers is complete for any prime number p. This space completes Q with the p-adic metric in the same way that R completes Q with the usual metric.

where n > 1 is an integer and p, q, and r are prime numbers, then 2 < sup > n </ sup >× p × q and 2 < sup > n </ sup >× r are a pair of amicable numbers.

where n > m > 0 are integers and p, q, and r are prime numbers, then 2 < sup > n </ sup >× p × q and 2 < sup > n </ sup >× r are a pair of amicable numbers.

If the exponent r is even, then the inequality is valid for all real numbers x.

For any real numbers x, r > 0, one has

0 ), the remainders r < sub >− 2 </ sub > and r < sub >− 1 </ sub > equal a and b, the numbers for which the GCD is sought.

Where a unique representation is needed for any point, it is usual to limit r to non-negative numbers () and θ to the interval or (− 180 °, 180 ° ( in radians, or (− π, π ).

A vector may also be multiplied, or re-scaled, by a real number r. In the context of conventional vector algebra, these real numbers are often called scalars ( from scale ) to distinguish them from vectors.

where p < sub > 1 </ sub > < p < sub > 2 </ sub > < ... < p < sub > r </ sub > are prime numbers and each k < sub > i </ sub > ≥ 1.

In other words, if ( r < sub > n </ sub >) is a sequence of distinct numbers such that ƒ ( r < sub > n </ sub >) = 0 for all n and this sequence converges to a point r in the domain of D, then ƒ is identically zero on the connected component of D containing r.

This yields an infinite list of the corresponding real numbers: r < sub > 1 </ sub >, r < sub > 2 </ sub >, ....

The preceding two paragraphs are an expression in English which unambiguously defines a real number r. Thus r must be one of the numbers r < sub > n </ sub >.

Richard claimed that the flaw in the paradoxical construction was that the expression for the construction of the real number r does not actually unambiguously define a real number, because the statement refers to the construction of an infinite set of real numbers, of which r itself is a part.

For Euclid ’ s method to succeed, the starting lengths must satisfy two requirements: ( i ) the lengths must not be 0, AND ( ii ) the subtraction must be “ proper ”, a test must guarantee that the smaller of the two numbers is subtracted from the larger ( alternately, the two can be equal so their subtraction yields 0 ).

On the Infinite was Hilbert ’ s most important paper on the foundations of mathematics, serving as the heart of Hilbert's program to secure the foundation of transfinite numbers by basing them on finite methods.

Eritrea ’ s poverty, the presence of large numbers of land mines, and the continued tensions that flare up between Eritrea and its neighbors have deterred the development of a tourist industry in Eritrea.

Assume that s > 1 is the product of prime numbers in two different ways:

Both orthometric and normal heights are heights in metres above sea level, whereas geopotential numbers are measures of potential energy ( unit: m² s < sup >− 2 </ sup >) and not metric.

As their numbers increased, Rapp ’ s group officially split with the Lutheran Church in 1785 and was banned from meeting.

The increasing numbers, which included followers outside of Rapp ’ s village, continued to concern the government, who feared they might become rebellious and dangerous to the state.

The most common is two ’ s complement, which allows a signed integral type with n bits to represent numbers from − 2 < sup >( n − 1 )</ sup > through 2 < sup >( n − 1 )</ sup >− 1.

The numbers at Jinnah ’ s meetings, once counted in thousands soon numbered only a few hundreds.

" On a method of investigating periodicities in disturbed series, with special reference to wolfer ’ s sunspot numbers ".

The Laplace transform of a function f ( t ), defined for all real numbers t ≥ 0, is the function F ( s ), defined by:

Gottfried Wilhelm Leibniz argued that idealized numbers containing infinitesimal s be introduced.

The numbers 1-7 are there denoted by the consonants l, s, n, m, t, f, u ( v ) and zero by the vowel o.

With Earth held stationary at the center of a nonrotating frame, the successive inferior conjunction s of Venus over eight Earth years trace a pentagram mic pattern ( reflecting the difference between the numbers in the ratio ).

After Fightback !, Keating ‘ did practically nothing ’ as Hawke ’ s support dwindled and the numbers moved in Keating ’ s favour.

For example, no two electrons in a single atom can have the same four quantum numbers ; if n, l, and m < sub > l </ sub > are the same, m < sub > s </ sub > must be different such that the electrons have opposite spins, and so on.

The following infinite series converges for all complex numbers s with real part greater than 1, and defines ζ ( s ) in this case:

where, by definition, the left hand side is ζ ( s ) and the infinite product on the right hand side extends over all prime numbers p ( such expressions are called Euler products ):

Hence the probability that s numbers are all divisible by this prime is 1 / p < sup > s </ sup >, and the probability that at least one of them is not is.

) Thus the asymptotic probability that s numbers are coprime is given by a product over all primes,

The family ’ s vast numbers allow it to control most of the kingdom ’ s important posts and to have an involvement and presence at all levels of government.