The Real Numbers 2
Besides the ordinary properties of multiplication and addition, there are more properties fundamental to the set of real numbers.
The real numbers form an ordered field with addition and multiplication as well as division by nonzero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication.
The set of real numbers has the least upper bound property and the greatest lower bound property. If a nonempty set of real numbers has an upper bound, then it has a least upper bound and if it has a lower bound then it has a greatest lower bound.
The set of real numbers is unbounded. Given any positive real numberthere is another real numbersuch thatand given any negative real numberthere is a negative real numbersuch that
The set of real numbers has no gaps in it – the set forms a continuum.
The set of real numbers is made up of the rational and irrational numbers. Between any two rational numbers there is a rational number and between any two irrational numbers there is a rationals numbers. In fact we can go further. Between any two rational numbers there are infinitely many irrationa numbers and between any two irrational numbers there are infinitely many rational numbers.
The rational and irrational numbers are both dense inThat is, every open setcontains infinitely many rational and irrational numbers.
The irrational and irrational numbers form two disjoint sets. We can write the set of rational numbers in more than one way.
The real numbers form a complete set. That is, if we have a sequenceof real numbers that converges to a limit, that limit is a member of
The real numbers are uncountable. They cannot be put into a one to one correspondence with the set of natural numbers.
The set of real numbers forms a metric space.
The real numbers are locally compact, since every closed subset ofcontains it's limit points, but not compact, since the limit point of the sequenceis infinity but
Transcendental numbers are defined as numbers which are not the root of a polynomial equation with rational coefficients. All real transcendental numbers are irrational.