Irrational Numbers

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We saw in a previous chapter that a decimal fraction can be written in a rational form i.e as &ldquo;a/b&rdquo; if the decimal part has

There are many numbers which do not follow the above rules. In these numbers the decimal part has an infinite sequence of numerals which do not form any pattern. Such numbers cannot be written in the form &ldquo;a/b&rdquo; and they are called &ldquo;irrational numbers&rdquo;. The term &lsquo;irrational&rsquo; has no relation to the value loaded word &lsquo;rational&rsquo; which is used in daily language. A better phrase could have been non-rational.
 * 1) A fixed number of numerals or
 * 2) A certain numeral sequence repeating infinitely

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 &lsquo;part&rsquo; of a &lsquo;whole&rsquo; can be represented as a fraction of thetypein this manner. After all 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. Euclid provided a very simple proof that could not be written in the form &lsquo;a/b&rsquo;! Pythagoras Theorem &amp; the proof that cannot be a rational number were possibly some of the earliest instances in the history of mathematics of a &ldquo;proof&rdquo;. Euclid used a method which is now called &ldquo;proof by contradiction&rdquo;.

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

There is an apocryphal story in the history of mathematics which reveals how difficult it was for Greek mathematicians to accept 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. A member who accidentally revealed this secret to outsiders was drowned in the sea!

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 &ldquo;ordered list&rdquo; each of which can be numbered with a whole number. Hence by the principle of &ldquo;one to one correspondence&rdquo; 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!

The numerals in the decimal part of irrationals do not show any pattern. Though the infinity of irrationals is more than rational, we actually &ldquo;know&rdquo; only a few of them like, etc &amp; &pi; &amp; e. The majority of them cannot even be written down!

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.

Computation with Irrationals

We can compute with irrational only by using rational approximations. For example 22/7 and 355/113 are used as approximations of &pi;.

Continued Fractions

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.

Order of Irrationality

An order of &ldquo;irrationality&rdquo; of any irrational can be worked out depending on how many terms of the irrational need to be considered to get a good rational approximation. From this point of view, &pi;, and ϕ are said to be in an increasing order of irrationality. The number ϕ is said to be the most irrational numbers.

Types of Irrationals

There are two kinds of irrationals – algebraic &amp; transcendental.

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

Irrationals like &pi; &amp; e cannot be got as roots of an algebraic equation. They can be represented as an infinite series. They are called Transcendental irrationals.

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