âˆš2 - 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 âˆš2, âˆš3, âˆš5, âˆ 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. âˆš2 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 Euclidâ€™s proof, and Dedekindâ€™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: â€œThere is no common measure for the length of a side of a square an the length of itâ€™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=âˆšABï€‡+BCï€‡ Here AB=1cm and BC=1cm Therefore AC=âˆš2 which cannot be expressed as a rational number. Pythagoras was so horrified by this number âˆš2, since it was a great setback for his philosophy. History says that he killed his disciple who raised this problem. Actually âˆš2 is an non-terminating decimal number. The numbers after the decimal point does not have any pattern as terminating decimals. âˆš2=1.414213562â€¦â€¦.. 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 âˆš2, some of them thought that they can find a pattern after some decimal place and some believed there is no such pattern exists. EUCLIDâ€™S PROOF: After several years the author of â€œELEMENTSâ€ , one of the great mathematician Euclid of Alexandria has given a beautiful proof by the method â€œREDUCTIO AD ABSURUMâ€ . 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 â€˜2kâ€™. (3). If a perfect square is even, then its square root is even. Let us know get into the proof: Let us assume that âˆš2 is rational. Then by definition it can be written as âˆš2=p/q, Such that p and q are integers and their GCD is 1. Now squaring both sides we have 2=pï€‡/qï€‡ qï€‡2=pï€‡ This equation indicates that 2 is a factor of pï€‡ and so pï€‡ is even and p is also even. Since p is even, we can write in the form â€˜2kâ€™ where â€˜kâ€™ is an integer. 2qï€‡= (2k) ï€‡=4kï€‡ qï€‡=2kï€‡ Since 2 is factor of qï€‡ so q is even. But this is a contradiction to our assumption. Because, if p and q are even, then they have GCD as â€˜2mâ€™ for some integer â€˜mâ€™ contradicting ourassumption. Hence âˆš2 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 âˆš2 is an irrational number. We can generalize it as follows: âˆš2=âˆš1ï€‡+1ï€‡ Now put 1=n so we get âˆšnï€‡+nï€‡=âˆš2nï€‡=nâˆš2. Let us prove that nâˆš2 is irrational number for any nÐ„N by mathematical induction. Now put n=1 we get 1âˆš2= âˆš2 This is clearly irrational. Therefore nâˆš2 is true for n=1, for any nÐ„N. So itâ€™s true for n=2. Let us prove for n=k, for kÐ„N. nâˆš2=kâˆš2 is also irrational for any kÐ„N. And for n=k+1 nâˆš2=(k+1)âˆš2 is also irrational for any nÐ„N. We know that Natural numbers are infinite, so we can construct infinite number of irrationals using nâˆš2. Remarks: (1).This nâˆš2 can also extend for set of all positive and negative integers except for zero. (2). Even though we can find infinite irrational numbers using nâˆš2, we cannot find all of the irrational numbers. Ex: âˆ 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 â‰ 0 and sâ‰ 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 âˆš2 irrational greater than 1. 1|âˆš2 is an irrational less than 1. Therefore 0<1|âˆš2 a|b < r|s Since1|âˆš2 is an irrational so 1|âˆš2 a|b is also irrational. Since1|âˆš2 <1 so m|n > 1|âˆš2 a|b So m|n <m|n+1|âˆš2 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 âˆš2 is on the real line. Let us draw a right angle isosceles triangle of sideâ€™s 1cm and hypotenuse âˆš2. 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 âˆš2. Dedekindâ€™s Cut: We already discussed that âˆš2 is a non-terminating decimal, so we takeâˆš2 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 âˆš2 is a non-terminating decimal number, we cannot find the exact or accurate point where the âˆš2 lies on the real line. Richard Dedekind (1831-1916) has defined âˆš2 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 Dedekindâ€™s cut. It is also called as â€œARITHMETIZATION OF IRRATIONAL NUMBERSâ€. Dedekindâ€™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.