Types of Numbers Part 4 – Irrational Numbers

Famous constants such as \pi, the Golden Ratio, Euler’s Number are all irrational numbers. These numbers cannot be expressed as a fraction,as a result all rational, natural and integer numbers are not irrational. Thus, Pi, the Golden Ratio and Euler’s Number cannot be expressed as fractions.

Irrational Numbers
There are no symbol for irrational numbers, however it can be represented in set notation as the set of all real numbers without rational numbers.

Though constants such as \pi have been known by even the earliest cultures, the concept of the irrational number is not nearly as old. The Pythagorean school of mathematics, a school of mathematics founded by Pythagoras, believed that all numbers are expressed by fractions, according to them \pi is a a fraction somewhere between 22/7 and 223/71 interestingly enough the concept of irrational numbers was discovered by Pythagorean mathematicians as a result of the Pythagorean Theorem. It turns out that \sqrt{2} cannot be expressed as  a fraction.

Many irrational constants are deeply rooted in physics and engineering. Fortunately all irrational numbers can be approximated by a series of rational numbers. For example the series, 3/1, 31/10, 314/100, 3141/1000, 31415/10000, ... approximates \pi. This fact is essential in engineering because precision can mean the difference between success and failure. As a result entire fields of mathematics and computer science have been dedicated to finding efficient methods for approximating irrational numbers.

There are also more irrational numbers than rational numbers. This also means there are more irrational numbers than integer and natural numbers. The rational number is considered uncountably infinite since there are no bijective functions between it and the natural numbers.