√2 - THE IRRATIONAL NUMBERS NUMBER INTRODUCTION: Most of the numbers we use in our daily life is rational. However, number of non-rational or irrational numbers exists in the number system such as √2, √3, √5, ∠etc. In particular the union of rational and irrational numbers is called real numbers. REAL NUMBERS RATIONAL NUMBERS IRRATIONAL NUMBERS As we are discussing the rational and irrational numbers let us first see the definitions. Rational: A rational number can always be expressed as quotient of two integers, say b & a where b=0, it is written as a/b. Irrational: The numbers which cannot be expressed in the above form is called incommensurable or Non-rational or Irrational numbers. √2 is the number which gave the idea of existence of irrational numbers in the number system. Let us have a look on the origin of irrational numbers and Euclid’s proof, and Dedekind’s cut on real line. Origin of Irrationals: We all know Pythagoras by his famous theorem on hypotenuse of right angle triangle called by his name. He together with his disciples called Pythagorean brotherhood. The philosophy of Pythagoras is that, Every thing in the universe is connected some way or other with a number, which has some thing in common with every other number. Thus he believed that any two lengths must have some definite length common to each. One of his disciples ran up against this, something very mysterious and baffling to them, and something that many people still find difficult to understand, the first time they encounter it. What he found is this: “There is no common measure for the length of a side of a square an the length of it’s diagonalâ€. He came to this conclusion when he working on squares whose sides are 1cm. He applied the Pythagoras theorem to find the length of the diagonal. Let us draw a square ABCD whose side is 1cm or I inch. Let us draw a line AC which is diagonal of this square ABCD. Since the sides of the square make an right angle we can apply Pythagoras Theorem and find the length AC. So AC=√AB+BC Here AB=1cm and BC=1cm Therefore AC=√2 which cannot be expressed as a rational number. Pythagoras was so horrified by this number √2, since it was a great setback for his philosophy. History says that he killed his disciple who raised this problem. Actually √2 is an non-terminating decimal number. The numbers after the decimal point does not have any pattern as terminating decimals. √2=1.414213562…….. NASA has calculated up to 100 million numbers after the decimal point using our modern day computers. After Pythagoras lot of people started calculating the value of √2, some of them thought that they can find a pattern after some decimal place and some believed there is no such pattern exists. EUCLID’S PROOF: After several years the author of “ELEMENTS†, one of the great mathematician Euclid of Alexandria has given a beautiful proof by the method “REDUCTIO AD ABSURUM†. i.e., proof by contradiction. PROOF: Before getting into the proof let us understand the following facts. (1). When a rational number is reduced to lower terms the GCD of the numerator and denominator is1. (2). If an integer is even, then it has 2 as its factor and may be written as ‘2k’. (3). If a perfect square is even, then its square root is even. Let us know get into the proof: Let us assume that √2 is rational. Then by definition it can be written as √2=p/q, Such that p and q are integers and their GCD is 1. Now squaring both sides we have 2=p/q q2=p This equation indicates that 2 is a factor of p and so p is even and p is also even. Since p is even, we can write in the form ‘2k’ where ‘k’ is an integer. 2q= (2k) =4k q=2k Since 2 is factor of q so q is even. But this is a contradiction to our assumption. Because, if p and q are even, then they have GCD as ‘2m’ for some integer ‘m’ contradicting ourassumption. Hence √2 is irrational. We would like to present another theorem or lemma whatever u may call, because it penned by our team. PROVE THAT THE SET OF ALL IRRATIONAL NUMBERS IS INFINITE: We know that √2 is an irrational number. We can generalize it as follows: √2=√1+1 Now put 1=n so we get √n+n=√2n=n√2. Let us prove that n√2 is irrational number for any nЄN by mathematical induction. Now put n=1 we get 1√2= √2 This is clearly irrational. Therefore n√2 is true for n=1, for any nЄN. So it’s true for n=2. Let us prove for n=k, for kЄN. n√2=k√2 is also irrational for any kЄN. And for n=k+1 n√2=(k+1)√2 is also irrational for any nЄN. We know that Natural numbers are infinite, so we can construct infinite number of irrationals using n√2. Remarks: (1).This n√2 can also extend for set of all positive and negative integers except for zero. (2). Even though we can find infinite irrational numbers using n√2, we cannot find all of the irrational numbers. Ex: ∠etc. We would like to present another theorem and the statement goes like this. Theorem: Between any two rational number there exists an irrational number. Proof: let us select two distinct rational, say m|n and r|s where m|n < r|s and m, n, r, s are all integers and n ≠0 and s≠0 Now m|n < r|s m|n-m|n < r|s-m|n 0<r|s-m|n<r|s say r|s-m|n=a|b , So 0<a|b<r|s We know that √2 irrational greater than 1. 1|√2 is an irrational less than 1. Therefore 0<1|√2 a|b < r|s Since1|√2 is an irrational so 1|√2 a|b is also irrational. Since1|√2 <1 so m|n > 1|√2 a|b So m|n <m|n+1|√2 a|b< r|s . Rational < irrational < rational. Hence there is an irrational number in between every distinct rational number. Irrational number on real line: Now let us try to locate the point where the √2 is on the real line. Let us draw a right angle isosceles triangle of side’s 1cm and hypotenuse √2. Let us place it on the real line. Now by using compass we can draw an arc of radius (hypotenuse) it cuts the real line at some points and we can take it as √2. Dedekind’s Cut: We already discussed that √2 is a non-terminating decimal, so we take√2 as 1.41 then we can point it on the on the real line if we take it as 1.414 then the points lies nearer to the previous point but not exactly on the same point when we see accurately. Since √2 is a non-terminating decimal number, we cannot find the exact or accurate point where the √2 lies on the real line. Richard Dedekind (1831-1916) has defined √2 has an cut not an exact point. His idea was to define it as cut of the rational numbers corresponding to each irrational number. He define this cut as the set of rational numbers which are less than 2, but which contains an infinite set of irrationals which progress to the right on the number line or real line, but which are always less than 2. irrationals as follows. Definition: A set k of rational numbers is said to be a cut if, (1) k contains at least one rational number, but not every rational number. (2) If p is in k and q is a rational number less than p, then q is in k , and (3) K contains no largest rational number. All the familiar properties of arithmetic in particular the field properties hold for Dedekind’s cut. It is also called as “ARITHMETIZATION OF IRRATIONAL NUMBERSâ€. Dedekind’s characterization of real number as cut of rational numbers is today the standard approach taken in most textbooks on REAL ANALYSIS and NUMBER THEORY.
