Development of Numbers 3

Georg Cantor proved that the infinity of irrational numbers was greater than the infinity of rational numbers. We will see different kinds of numbers which were subsets of Irrational Numbers.

Algebraic Numbers

With the development of algebra, an irrational number like could be seen as the solution of an algebraic equation = 2. Square roots of all Natural numbers, except the perfect squares, are irrational.

Irrationals of this type are called Algebraic Numbers.

The Golden Ratio “Phi” is also an Algebraic Irrational since it is a solution of the equation x2 - x - 1 = 0.

Transcendental Numbers

There are irrational numbers which are not solutions of algebraic equations. Prominent among them are numbers like π and e. These numbers are expressible only as a sum of an infinite series.

Irrationals of this type are called Transcendental Numbers.

Computable Numbers

With the rapid development of the science of computing, a new set of numbers called Computable Numbers has been defined. This includes any number which can be computed to any required accuracy, using a computer algorithm.

All rational numbers are computable. Algebraic Irrationals are also computable. Some of the transcendental numbers which can be expressed as an infinite series are also computable. As of now there are many irrationals which are not computable.