“Irrational numbers are those real numbers which are not rational numbers!”
Def.1: Rational Number
A rational number is a real number which can be expressed in the form of where and are both integers relatively prime to each other and being non-zero.
Following two statements are equivalent to the definition 1.
1. is rational if and only if and are integers relatively prime to each other and does not equal to zero.
Def. 2: Relatively Prime Numbers
Two integers and are said to be relatively prime to each other if the greatest common divisor of and is .
For example: The pairs (2, 9); (4, 7) etc. are such that each element is relatively prime to other.
Def. 3: Irrational Number
A real number, which does not hold atleast ‘one’ property of rational numbers is termed as an irrational number.
A silly question:
Let, in the definition of a rational numbers, and , then, as we know is a rational number, however can divide both integers and , i.e., . (Why?)
Primary ways to prove the irrationality of a real number
It is all clear that any real, if not rational, is irrational. So, in order to prove a number (real number, off-course!) irrational, we need to show that it is not a rational number (i.e., not satisying definition 1). Most popular method to prove irrationality in numbers, is the Proof by Contradiction, in which we first assume the given (irrational) number to be ‘almost’ rational and later we show that our assumption was untrue. There are many more ways to prove the irrational behaviour of numbers but those are more or less derived from the proof by contradiction.
Some methods, those I’ll discuss here briefly are:
1. Pythagorean Approach
2. Using Euclidean Algorithm
3. Power series expansion of special numbers
4. Continued Fraction representation of irrational numbers.
(1) Pythagorean Approach
This proof is due to Pythagoras and thus called Pythagorean Approach to irrationality. In this approach, we assume a number to be first. Later using the fundamental rules of arithmetic, we make sure whether or not our assumption was true. If our assumption was true, the number we took was rational, otherwise irrational.
Prove that the number is irrational.
Proof: Suppose, to the contrary, that is a rational number. Then as according to the defintion 1, we can write
where and are both integers with and .
Squaring equation (1),
From the equation (2), we can break our thoughts into two ways:
is a positive number, therefore we can assume and both to be positive.
or, (read as b divides a squared).
Since, is positive integer, . However, is impossible since corresponding is not an integer. Thus, and then according to fundamental theorem of arithmetic, there exists atleast one prime which divides .
Mathematically, but as . It is clear that . This implies that .
Since and , therefore . So for given number,the greatest common divisor of and is not , but another prime larger than . Thus, it fails to satisfy the definition 1. Thus our claim that is rational, is untrue. Therefore, is an irrational number.
As a deviation, we can proceed our proof from equation (2) by taking the fact into mind that is positive. The number (natural number) can either be odd or even.
Let be odd, i.e., where . Therefore would also be odd. Which contradicts (2), since is always even and that equals to . Therefore, must be an even number.
Let . Putting this into (2) we get,
Which is contradiction to our claim.
Thus is an irrational number.
In similar ways, one can prove , , etc. to be irrationals.
(2) Using Euclidean Algorithm
This is an interesting variation of Pythagorean proof.
Let with , then according to Euclidean Algorithm, there must exist integers and , satisfying .
or, . (From we put .)
This representation of leads us to conclude that is an integer, which is completely false. Hence our claim that can be written in form of is untrue. Thus, is irrational.
Similarly, we can use other numbers to prove so.
(3) Power Series Expansion
Some irrational numbers, like , can be proved to be irrational by expanding them and arranging the terms. Over all, it is another form of proof by contradiction but different from the Pythagorean Approach.
can be defined by the following infinite series:
Suppose, to the contrary, that is rational, and (say) where and are positive integers. Then for any and also ,
is positive, since is positive.
It is clear that is less than .
So, being positive integer is less than ? It is impossible for any integer. Thus our claim is not true and hence is irrational.
(4) Continued Fractions
Any number, that can be expressed in form of an infinite continued fraction is always irrational.
1) can be represented in form of infinite continued fractions, thus is irrational.
2) Similarly is irrational.
3) is also irrational.