theory of rational and irrational numbers
The following entry is some sort of index of articles in PlanetMath about the basic theory of rational and irrational numbers, and it should be studied together with its complement: the theory of algebraic and transcendental numbers (http://planetmath.org/TheoryOfAlgebraicNumbers). The reader should follow the links in each bullet-point to learn more about each topic. For a somewhat deeper approach to the subject, the reader should read about Algebraic Number Theory. In this entry we will concentrate on the properties of the complex numbers and the extension .
There is also a topic entry on rational numbers.
1 Basic Definitions
A real number is said to be irrational (http://planetmath.org/IrrationalNumber) if it is not rational, i.e. it cannot be expressed as a quotient of integers. The decimal expansion is non-periodic for any irrational, but periodic for any rational number.
For example, is irrational.
2 Small Results
is irrational (http://planetmath.org/SquareRootOf2IsIrrationalProof). Similarly is irrational as long as is not a perfect square.
The number e is irrational (this is not as difficult to prove as it is to show that e is transcendental). In fact, if then is also irrational (http://planetmath.org/ErIsIrrationalForRinmathbbQsetminus0). There is an easier way to show that e is not a quadratic irrational.
If is irrational then is irrational (see here (http://planetmath.org/IfAnIsIrrationalThenAIsIrrational)).
A surprising fact: an irrational to an irrational power can be rational.
and are irrational (http://planetmath.org/PiAndPi2AreIrrational).
3 BIG Results
The irrational numbers are, in general, “easily” understood. The BIG theorems appear in the theory of transcendental numbers. Still, there are some open problems: is Euler’s constant irrational? is rational?
|Title||theory of rational and irrational numbers|
|Date of creation||2013-03-22 15:14:10|
|Last modified on||2013-03-22 15:14:10|
|Last modified by||alozano (2414)|