Irrational Numbers

We saw in a previous chapter that a decimal fraction can be written in a rational form i.e as "a/b (b>0)" if the decimal part has

1. A fixed number of numerals or

2. A certain numeral sequence repeating (recurring) infinitely

There are many fractions which do not follow the above rules. In these fractions the decimal part has an infinite sequence of numerals which do not form any pattern. Such fractions cannot be written in the form "a/b" and they are called "irrational numbers". The term "irrational" has no relation to the value loaded word "rational" which is used in daily language. A better phrase could have been non-rational.

How Irrationals Were Discovered

The discovery of irrational numbers has a very interesting history.

Greek Mathematicians came up with the idea of rational numbers. They were also convinced that any "part" of a "whole" can be represented as a fraction of in this manner. After all, they reasoned, you have the freedom to divide the whole into an immensely large number of small parts, out of which you again have the freedom to choose any number of these parts.

Hence ancient mathematicians were convinced that all fractions can be represented as rational numbers. They were in for a surprise!

Applying Pythagoras Theorem to a right triangle with both the smaller sides being 1, the hypotenuse of that triangle came out to be a number √2 which could not be represented in a rational form.

Euclid provided a very simple proof that could not be written in the form "a/b"! Pythagoras Theorem & the proof that √2 cannot be a rational number were possibly some of the earliest instances in the history of mathematics of a "proof". Euclid used a method which is now called "proof by contradiction".

They also discovered other similar fractions which contained fractional parts which were less than 1 but could not be written in the rational form. They called such fractions as Irrational Numbers in the sense that they were not rational or cannot be represented as ratios.

Secrecy about Irrational Numbers

There is an apocryphal story in the history of mathematics which reveals how difficult it was for Greek mathematicians to accept the existence of irrational numbers. Members of the Pythagorean school believed that the discovery of irrational numbers was something which had to be hidden from public knowledge. So, they swore themselves to secrecy that the existence of such numbers would never be revealed to outsiders. It is even said that a member, Hippasus of Metapontum, who accidentally revealed this secret to outsiders was drowned in the sea!

Study of Irrational Numbers

Georg Cantor, in the later 19thcentury, pioneered the study of infinities of numbers and came up with many counter-intuitive results!

He proved that the infinity of irrational numbers was far greater than the infinity of rational numbers! He showed that all rational numbers can be placed in an "ordered list" each of which can be numbered with a whole number. Hence by the principle of "one to one correspondence" the infinity of rational numbers was the same as that of whole numbers!

He also showed that it was not possible to list irrational numbers in the same way. Hence the infinity of irrational numbers was more than that of whole numbers!

This means that if you randomly "locate" any point on the Real Number Line, it is more likely to be an irrational than a rational!

The numerals in the decimal part of irrationals do not show any pattern. Though the infinity of irrationals is more than rational, we actually "know" or "are familiar with" only a few of them like π & e. The majority of them cannot even be written down!

Identifying Irrational Numbers is Difficult

A related problem is that if a number shows no patterns in the decimal part, until a certain number of digits, we cannot assume it is irrational. The patterns may occur after many more digits are discovered! Many of the universal physical constants discovered recently by scientists are of this kind. They have been calculated up to certain number of decimals. Even if they show no patterns, we cannot say with certainty that they are irrational. The speed of light is one such constant.

In one sense, we can say that up to rational numbers, the idea of numbers was based on the idea of counting. From irrational numbers onwards the idea of numbers is based on geometry.

Computation with Irrationals

Even though irrational numbers are ubiquitous in science & technology, we can never use them directly in mechanical or electronic computations. This is because computers can only work with certain precisions. We can compute with irrationals only by using rational approximations to the accuracy needed in the context. For example 22/7 and 355/113 are used as approximations of π.

But interestingly we can handle irrational numbers like √2 or √3 directly in paper & pencil computations, provided they cancel out and the end result is a rational number!

Degree of Irrationality

The idea of approximating an irrational number with a rational approximation gave rise to the idea of "degree of irrationality". It can be worked out depending on how many terms of the decimal part of the irrational need to be considered to get a good rational approximation.

From this point of view, π, and ϕ are said to be in an increasing degree of irrationality. The number ϕ is said to be the most irrational numbers.

Continued Fractions & Infinite Series

Many irrationals can be written as (infinitely) continued fractions. By truncating the continued fraction at any point, we can get a rational number approximation of the irrational.

Irrationals like π, ϕ and e can be written as an infinite series. By summing up more and more terms, we can get values which yield more decimal digits.

Types of Irrationals

There are two kinds of irrationals - algebraic & transcendental.

Algebraic irrationals like √2 can be a root of an algebraic equation like x2 - 2 =0.

Irrationals like π & e cannot be got as roots of an algebraic equation. They can be represented as an infinite series. They are called Transcendental irrationals. It is very difficult to identify irrationals which are also transcendental.

Uses of Irrational Numbers

Irrational numbers play a key role in the complex math that makes possible high-frequency stock trading, modelling, forecasting and most statistical analysis — all activities that keep our society humming.