âˆš2 - THE IRRATIONAL NUMBERS\n\nNUMBER\n\nINTRODUCTION:\n\nMost of the numbers we use in our daily life is rational. However,\nnumber of non-rational or irrational numbers exists in the number\nsystem such as âˆš2, âˆš3, âˆš5, âˆ etc.\n\nIn particular the union of rational and irrational numbers is called\nreal numbers.\n\nREAL NUMBERS\nRATIONAL NUMBERS IRRATIONAL NUMBERS\n\nAs we are discussing the rational and irrational numbers let us first\nsee the definitions.\n\nRational:\nA rational number can always be expressed as quotient of two integers,\nsay b & a where b=0, it is written as a/b.\n\nIrrational:\nThe numbers which cannot be expressed in the above form is called\nincommensurable or Non-rational or Irrational numbers.\n\nâˆš2 is the number which gave the idea of existence of irrational\nnumbers in the number system. Let us have a look on the origin of\nirrational numbers and Euclidâ€™s proof, and Dedekindâ€™s cut on real\nline.\n\nOrigin of Irrationals:\nWe all know Pythagoras by his famous theorem on hypotenuse of right\nangle triangle called by his name. He together with his disciples\ncalled Pythagorean brotherhood.\n\nThe philosophy of Pythagoras is that,\nEvery thing in the universe is connected some way or other with a\nnumber, which has some thing in common with every other number.\n\nThus he believed that any two lengths must have some definite length\ncommon to each.\n\nOne of his disciples ran up against this, something very mysterious and\nbaffling to them, and something that many people still find difficult\nto understand, the first time they encounter it.\n\nWhat he found is this:\nâ€œThere is no common measure for the length of a side of a square an\nthe length of itâ€™s diagonalâ€.\n\nHe came to this conclusion when he working on squares whose sides are\n1cm. He applied the Pythagoras theorem to find the length of the\ndiagonal.\n\nLet us draw a square ABCD whose side is 1cm or I inch. Let us draw a\nline AC which is diagonal of this square ABCD.\n\nSince the sides of the square make an right angle we can apply\nPythagoras Theorem and find the length AC.\n\nSo AC=âˆšABï€‡+BCï€‡\nHere AB=1cm and BC=1cm\nTherefore AC=âˆš2 which cannot be expressed as a rational number.\n\nPythagoras was so horrified by this number âˆš2, since it was a great\nsetback for his philosophy. History says that he killed his disciple\nwho raised this problem.\n\nActually âˆš2 is an non-terminating decimal number. The numbers after\nthe decimal point does not have any pattern as terminating decimals.\nâˆš2=1.414213562â€¦â€¦..\nNASA has calculated up to 100 million numbers after the decimal point\nusing our modern day computers.\n\nAfter Pythagoras lot of people started calculating the value of âˆš2,\nsome of them thought that they can find a pattern after some decimal\nplace and some believed there is no such pattern exists.\n\nEUCLIDâ€™S PROOF:\n\nAfter several years the author of â€œELEMENTSâ€ , one of the great\nmathematician Euclid of Alexandria has given a beautiful proof by the\nmethod â€œREDUCTIO AD ABSURUMâ€ . i.e., proof by contradiction.\n\nPROOF:\n\nBefore getting into the proof let us understand the following facts.\n\n(1). When a rational number is reduced to lower terms the GCD of the\nnumerator and denominator is1.\n(2). If an integer is even, then it has 2 as its factor and may be\nwritten as â€˜2kâ€™.\n(3). If a perfect square is even, then its square root is even.\nLet us know get into the proof:\n\nLet us assume that âˆš2 is rational.\nThen by definition it can be written as\nâˆš2=p/q,\nSuch that p and q are integers and their GCD is 1. Now squaring both\nsides we have\n2=pï€‡/qï€‡\nqï€‡2=pï€‡\n\nThis equation indicates that 2 is a factor of pï€‡ and so pï€‡ is even\nand p is also even.\nSince p is even, we can write in the form â€˜2kâ€™ where â€˜kâ€™ is an\ninteger.\n\n2qï€‡= (2k) ï€‡=4kï€‡\nqï€‡=2kï€‡\nSince 2 is factor of qï€‡ so q is even. But this is a contradiction to\nour assumption. Because, if p and q are even, then they have GCD as\nâ€˜2mâ€™ for some integer â€˜mâ€™ contradicting ourassumption.\n\nHence âˆš2 is irrational.\n\nWe would like to present another theorem or lemma whatever u may call,\nbecause it penned by our team.\n\nPROVE THAT THE SET OF ALL IRRATIONAL NUMBERS IS INFINITE:\n\nWe know that âˆš2 is an irrational number. We can generalize it as\nfollows:\nâˆš2=âˆš1ï€‡+1ï€‡\nNow put 1=n so we get\nâˆšnï€‡+nï€‡=âˆš2nï€‡=nâˆš2.\n\nLet us prove that nâˆš2 is irrational number for any nÐ„N by\nmathematical induction.\nNow put n=1 we get\n1âˆš2= âˆš2\nThis is clearly irrational.\nTherefore nâˆš2 is true for n=1, for any nÐ„N.\nSo itâ€™s true for n=2.\nLet us prove for n=k, for kÐ„N.\nnâˆš2=kâˆš2 is also irrational for any kÐ„N.\nAnd for n=k+1 nâˆš2=(k+1)âˆš2 is also irrational for any nÐ„N.\nWe know that Natural numbers are infinite, so we can construct infinite\nnumber of irrationals using nâˆš2.\n\nRemarks:\n\n(1).This nâˆš2 can also extend for set of all positive and negative\nintegers except for zero.\n(2). Even though we can find infinite irrational numbers using nâˆš2,\nwe cannot find all of the irrational numbers. Ex: âˆ etc.\n\nWe would like to present another theorem and the statement goes like\nthis.\n\nTheorem:\n\nBetween any two rational number there exists an irrational number.\n\nProof: let us select two distinct rational, say m|n and r|s where m|n\n< r|s and m, n, r, s are all integers and n â‰ 0 and sâ‰ 0\n\nNow m|n < r|s\n\nm|n-m|n < r|s-m|n\n\n0<r|s-m|n<r|s\n\nsay r|s-m|n=a|b ,\n\nSo 0<a|b<r|s\n\nWe know that âˆš2 irrational greater than 1.\n\n1|âˆš2 is an irrational less than 1.\n\nTherefore 0<1|âˆš2 a|b < r|s\n\nSince1|âˆš2 is an irrational so 1|âˆš2 a|b is also irrational.\n\nSince1|âˆš2 <1 so m|n > 1|âˆš2 a|b\n\nSo m|n <m|n+1|âˆš2 a|b< r|s .\n\nRational < irrational < rational.\n\nHence there is an irrational number in between every distinct rational\nnumber.\n\nIrrational number on real line:\n\nNow let us try to locate the point where the âˆš2 is on the real line.\n\nLet us draw a right angle isosceles triangle of sideâ€™s 1cm and\nhypotenuse âˆš2. Let us place it on the real line. Now by using compass\nwe can draw an arc of radius (hypotenuse) it cuts the real line at some\npoints and we can take it as âˆš2.\n\nDedekindâ€™s Cut:\n\nWe already discussed that âˆš2 is a non-terminating decimal, so we\ntakeâˆš2 as 1.41 then we can point it on the on the real line if we\ntake it as 1.414 then the points lies nearer to the previous point but\nnot exactly on the same point when we see accurately.\n\nSince âˆš2 is a non-terminating decimal number, we cannot find the\nexact or accurate point where the âˆš2 lies on the real line.\n\nRichard Dedekind (1831-1916) has defined âˆš2 has an cut not an exact\npoint. His idea was to define it as cut of the rational numbers\ncorresponding to each irrational number. He define this cut as the set\nof rational numbers which are less than 2, but which contains an\ninfinite set of irrationals which progress to the right on the number\nline or real line, but which are always less than 2.\n[QUOTE]\nFrom this he has generalized his definition for â€œCUTâ€ for all[/QUOTE]\nirrationals as follows.\n\nDefinition: A set k of rational numbers is said to be a cut if,\n\n(1) k contains at least one rational number, but not every rational\nnumber.\n(2) If p is in k and q is a rational number less than p, then q is in k\n, and\n(3) K contains no largest rational number.\nAll the familiar properties of arithmetic in particular the field\nproperties hold for Dedekindâ€™s cut. It is also called as\nâ€œARITHMETIZATION OF IRRATIONAL NUMBERSâ€.\n\nDedekindâ€™s characterization of real number as cut of rational numbers\nis today the standard approach taken in most textbooks on REAL ANALYSIS\nand NUMBER THEORY.