\documentstyle{article} \begin{document} \begin{center} \huge{DETECTING IRRATIONALITY\\From \(\sqrt{2}\) to $\zeta (3)$} \end{center} {\bf Introduction}. In this article we give a survey of some methods used to show irrationality of numbers.Our style is expository. We do not entend to give a proof of every theorem we state; rather we refer the reader to the suitable reference. {\bf Sec1. The Fundamental Theorem of Arithmetic.} This theorem simply states that ``Every natural number can be written as a product of primes".This marvelous theorem was the first tool used to detect irrationality. One can go one step farther and say it is the one that brought irratoinal numbers into the realm of mathematics.The story is very short.Using our modern terminology, the Phythagoreans thought that all numbers are rational.They were shocked when they found that the length of the diagonal of the unit square is not rational. That is to say, $\sqrt{2}$ is not rational.For if $\sqrt{2} = \frac{a}{b}$ for some relatively prime integers a and b, then we would have $a^{2}=2b^{2}$, this says a square is divisible by an odd power of 2,which contradicts the fundamental theorem of arithmatic, since a square (if divisible by 2) is divisible by an even power of 2. Thus \(\sqrt{2}\) is irrational. The same argument can be used to show that \(\sqrt[n]{m}\), where m is not a power of n,is also irrational. An almost similar argument can show that $\log[m] n $ is irrational if (i) n is not a power of m and (ii) there is a prime that divides n or m, but not both. {\bf sec2. The Rational zeros Theorem.} In the theory of equations, we have the following {\bf Theorem:}Let $a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x +a_{0}=0$, where $a_{i}$ are integers, $a_{n}\neq0$.If $\frac {p}{q}$, where (p,q)=1, is a zero of the given equation, then p divides $a_{0}$ and q divides $a_{n}$. So this theorem determines all rational zeros of equations with integer coefficients. Now let $\alpha$ be a real number.If we can show that $\alpha$ is a root of some polynomial equation with integer coefficients and with no rational roots,then $\alpha$ is necessarily irrational. For example,let $\alpha$=\(\sqrt{3}\)+\(\sqrt{5}\). Then $\alpha$ is a root of $x^{4}-16x^{2}+4=0$.The possible rational zeros are: +1, -1, +2, -2, +4, -4. None of which is a zero. So $\alpha$ is irrational. {\bf sec3. Decimal Expansions.} In the theory of decimal expansions, there is a fact which states that ``A real number is rational if and only if its decimal expansion terminates or recurs." So this gives a way to detect irrationality. But, How practical is this method? One, in fact, can use decimal expansions to construct irrational numbers. For example, the numbers $\alpha =.a_{1}a_{2}a_{3}...$, where $a_{i}=1$ if i is prime and 0 otherwise, and $\beta = .p_{1}p_{2}p_{3}...$, where $p_{i}$ is the sequence of primes given in increasing order, are irrational. But the proofs are not trivial, particularly for $\beta$.See[1,sec 9.4]. The main point to observe here is that there is a rule defining the digits in $\alpha$ and $\beta$. Then using the theory of primes, one can show their irrationality. But suppose one constructed a decimal number by giving the digits in a random way. This number is most likely to be irrational; but, how to prove it? No clue. {\bf Sec4. Lindemann Theorem.} Equally difficult is the question of determining the transcendence of numbers. The transcndence of $\pi$ was proved by Hermite in 1873 and that of e was proved by Lindemann im 1882. Lidemann gave a general theorem that proves the transcendence of a class of numbers including $\pi$ and e. Before we state the theorem and its corollary, we remind the reader that every transcendental number is irrational as can be trivially shown. {\bf Lindemann Theorem:} Given any distinct algebraic numbers $\alpha_{1},\ldots,\alpha_{n}$, the values $e^{\alpha_{1}},\ldots, e^{\alpha_{n}}$ are linearly independent over the field of algebraic numbers.[2,ch.9] {\bf Corollary:}The following numbers are transcendental: e, $\pi$, $e^{\alpha}$, $sin\alpha$, $cos\alpha$, $tan\alpha$, $sinh\alpha$, $cosh\alpha$, $tanh\alpha$, ($\alpha\neq$0, algebraic) $log\beta$, $arcsin\beta$, $arccos\beta$, $arctan\beta$, $arcsinh\beta$, $arccosh\beta$, $arctanh\beta$, ($\beta\neq$0,1, algebraic) Before we leave this section, we point out that the irrationality of e and $\pi$ can be proved independetly of being transcendental. See[1]. {\bf Sec5. Gelfand-Schneider Theorem.} This theorem, proved independently by Gelfand and Schneider, gives another class of transcendental and hence irrational numbers. {\bf Theorem:} If $\alpha$ and $\beta$ are algebraic numbers with $\alpha\neq$0, 1 and $\beta$ is not a real rational number, then every value of $\alpha^{\beta}$ is transcendental. See [2] for a proof. For example, the numbers $2^{i}, 3^{\sqrt{5}}, \sqrt{2}^{\sqrt{5}},i^{-2i}$ are all transcendental. Since $e^{\pi}$ is one value of $i^{-2i}$, then $e^{\pi}$ is also transcendental. {\bf Sec6. Irrationality of $\zeta(3)$.} The series $\zeta(k)=\sum_{n=1}^{\infty} \frac{1}{n^{k}}$ converges for $k>1$. In fact we know more; $\zeta(2k)=\frac{(-1)^{k}(2\pi)^{2k} B_{2k}}{2(2k)!}$ and so $\zeta(2k)$ is irrational for $k\geq1$. But What about $\zeta(2k+1)$ for $k\geq1$?. Recently, Apery showed the irrationality of $\zeta(3)$. The other values, $\zeta(2k+1)$, $k>1$, are still unknown. How did Apery prove it?. For practical purposes, irrational numbers are always approximated by rationals. In this direction, a whole theory was developed. One of its beautiful theorems is the following characterization of irrationality: {\bf Theorem:} If there is a $\delta>0$ and a sequence $\frac{a_{n}}{b_{n}}$ of rational numbers such that $\frac{a_{n}}{b_{n}} \neq \alpha$ and $|\alpha- \frac{a_{n}}{b_{n}}|<\frac{1}{q_{n}^{1+\delta}}, n=1, 2, 3,...$, then $\alpha$ is irrational. For $\zeta(3)$, Apery was able find a $\delta$ and a sequence $\frac{a_{n}}{b_{n}}$ of rational numbers satisfying the inequality of the above theorem. Namely, given the recursion \[ n^{3}u_{n}+(n-1)^{3}u_{n-2}=(34n^{3}-51n^{2}+27n-5)u_{n-1}\] define $ u_{0}=a_{0}=0, u_{1}=a_{1}=6, a_{n}=u_{n}$ for $n \geq 2$, and $u_{0}=b_{0}=1, u_{1}=b_{1}=5, b_{n}=u_{n}$ for $n \geq 2$. Further, take $\delta =0.080 529 ... >0$. For more details, the reader is encouraged to read [4]. {\bf Conclusion.} As the reader observed, there is no one method that works for all. Rather, different methods take care of different classes of numbers. But, still, the story is not complete. There are some numbers whose irrationality or transcendency are not yet known. Examples include Euler's constant $\gamma, \pi^{\sqrt{2}}, \pi^{e}, e+\pi, 2^{e}, 2^{\pi}$. {\bf References:} 1. G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers. Fifth edition, 1979, Oxford Science Publications. 2. Ivan Niven. Irrational Numbers. 1956. The Mathematical Association of America. 3. H. E. Rose. A Course in Number Theory. Second edition,1994, Oxford Science Publications. 4. Alfred van der Poorten. A Proof that Euler Missed...The Mathematical Intelligencer,Vol 1,pp.195-203,1979.