<P> Every nonnegative real number has a square root in R, although no negative number does . This shows that the order on R is determined by its algebraic structure . Also, every polynomial of odd degree admits at least one real root: these two properties make R the premier example of a real closed field . Proving this is the first half of one proof of the fundamental theorem of algebra . </P> <P> The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalized such that the unit interval (0; 1) has measure 1 . There exist sets of real numbers that are not Lebesgue measurable, e.g. Vitali sets . </P> <P> The supremum axiom of the reals refers to subsets of the reals and is therefore a second - order logical statement . It is not possible to characterize the reals with first - order logic alone: the Löwenheim--Skolem theorem implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first - order logic as the real numbers themselves . The set of hyperreal numbers satisfies the same first order sentences as R. Ordered fields that satisfy the same first - order sentences as R are called nonstandard models of R . This is what makes nonstandard analysis work; by proving a first - order statement in some nonstandard model (which may be easier than proving it in R), we know that the same statement must also be true of R . </P> <P> The field R of real numbers is an extension field of the field Q of rational numbers, and R can therefore be seen as a vector space over Q. Zermelo--Fraenkel set theory with the axiom of choice guarantees the existence of a basis of this vector space: there exists a set B of real numbers such that every real number can be written uniquely as a finite linear combination of elements of this set, using rational coefficients only, and such that no element of B is a rational linear combination of the others . However, this existence theorem is purely theoretical, as such a base has never been explicitly described . </P>

For which of the following values of x is not defined as a real number