Construction of The Real Numbers
On December 12, 2010,
in Analysis,
by ufukkaya
having at least two distinct elements and satisfying the following five axioms is called the set of real numbers and each element of 
defined as 
for each 
in 
satisfies the following properties:
,
,
Tagged with: absolute value • absolute value of a real number • addition • additive inverse • additive inverses • associative • associativity • associativity of addition • associativity of multiplication • associativity of the addition • associativity of the multiplication • axioms of addition • axioms of multiplication • bounded interval • bounded set • bounded subset • bounded subset of real numbers • bounded subset of the real numbers • bounded subsets of real numbers • bounded subsets of the real numbers • closed interval • commutativity • commutativity of addition • commutativity of multiplication • commutativity of the addition • commutativity of the multiplication • completeness of the real numbers • construction of real numbers • construction of the real numbers • criteria of supremum and infimum • criterion of supremum and infimum • distance between two real numbers • distributivity of multiplication over addition • entending of the real numbers • existence of additive inverse • existence of infimum • existence of multiplicative inverse • existence of supremum • existence of supremum and infimum • existence of supremum and infimum in reals • existence of supremum and infimum in the real numbers • extended real number line • extended real number system • extended real numbers • greatest lower bound • greatest lower bound property • identity element • infimum • infimum in reals • infimum of a set • infinity • interval • interval lied real numbers • interval lied reals • interval with the endpoint a and b • least upper bound • least upper bound property • left closed left open interval • left open right closed interval • length of an interval • lower bound • lower bounds • maximum • maximum element • maximum element of a set • maximum element of a subset • measure of an interval • minimum • minimum element • minimum element a set • minimum element a subset • modulus • modulus of a real number • multiplication • multiplication of negative and negative is positive • multiplication of negative number and negative number is positive number • multiplication of positive and negative is negative • multiplication of positive and positive is positive • multiplication of positive number and negative number is negative number • multiplication of positive number and positive number is positive number • multiplicative inverse • multiplicative inverses • negative • negative infinity • negative real numbers • open interval • operation addition • operation multiplication • order • order of real numbers • ordering of the real numbers • positive • positive infinity • positive real numbers • proof of infimum • proof of reverse triangle inequality • proof of supremum • proof of triangle inequality • properties of modulus • properties of the absolute value • properties of the extended real numbers • real number • real numbers • reverse triangle inequality • sets supremum infimum • size of an interval • supremum • supremum and infimum in reals • supremum in reals • supremum infimum • supremum of a set • the axiomatic structure of the real numbers • the axioms of the addition • the axioms of the multiplication • the identity of multiplication • the identity of the multiplication • the negative real numbers • the order • the positive real numbers • the real numbers • the set of lower bound of a set • the set of real numbers • the set of real numbers is associative • the set of real numbers is complete • the set of the real numbers • the set of upper bound of a set • the zero of addition • the zero of the addition • triangle inequality • unbounded interval • uniqueness of identity element • uniqueness of zero • upper bound • upper bounds • width of an interval • zero
