**e**
** and **
** (irrationality). **

**The irrationality of **
**e**
** was first established by Euler, who concluded it from his derivation of the **
**infinite**
** **
**continued fraction**
** expansion of **
**. I will not dwell much on this since I cannot presume a familarity with continued fraction theory on behalf of a general reader. However I show some brief computations:**

`> `
**with(numtheory): # for use of 'cfrac'**

`Warning, the protected name order has been redefined and unprotected`

`> `
**cfrac((exp(1) - 1)/2, 5); # to the 5th 'partial quotient'**

`> `

**where the pattern from the partial quotient '6' (the **
** of the standard continued fraction notation) is continued **
**ad infinitum**
**:**

`> `
**cfrac ((exp(1) - 1)/2, 50, 'quotients'); **

`> `

**Everyone knows the standard classic irrationality proof of (Euler's number) **
**e**
**, the proof, given by Fourier in 1815, that exploits:**

**e**
** = **
** (i)**

**Suppose **
** for some (positive) whole numbers **
**a**
** and **
**b**
**. Then multiplying throughout (i) by **
**n**
**! gives:**

** ... (ii)**

**where **

** = **
**, and thus **
** **

**Thus (i) is impossible since (ii) - for sufficiently large **
**n**
** - leads to an integer (the LHS) being equal to (an integer, plus a positive error term that is less than 1). Thus **
**e**
** is irrational.**

**_________**

**Any obvious attempt at giving a similar irrationality proof for (say) **
**appears at first sight to be doomed (try it to see what I mean). In his 1949 Princeton book on Transcendental Numbers, C. L. Siegel gave an elementary proof - which I had always assumed was his own, since he didn't credit it to anyone - not only that **
** is irrational, but more generally that **
**e**
** is not a quadratic algebraic number (i.e. **
**e**
** does not satisfy any quadratic equation **
** with integers **
**a**
**, **
**b**
** and **
**c**
**, and **
**). While preparing my Manchester 1972-73 course I made a minor improvement on Siegel's proof, by formulating another elementary proof that **
**is irrational. That alternative proof had the advantage that incorporating the idea in Siegel's proof led to an elementary proof that **
** is irrational. **

** Although I gave a 'splinter group' talk (attended by Masser and Serre, who clearly had nothing better to do) on my proof at the BMC in 1973(?) I didn't bother to write up the proof for possible publication until 2002. I submitted it to the MAA **
**Monthly**
** in that year, and was just a little disappointed to learn from a scholarly referee that Siegel's proof had already been given by Liouville in 1840, and that 'my' proof was also given by Liouville in the same year (though I believe I give a better explanation that Liouville!). An interested reader may consult the submitted paper in the **
**esquared**
** corner of my web site.**

** Of course all of these elementary results are completely put in the shade with Hermite's (1873) result that **
**e**
** is a transcendental number.**

**__________**

**The irrationality of **
** was first demonstrated by Lambert in 1766. As with Euler, continued fractions were central to his proof.**

** There are many different proofs that **
** is irrational, but it should be said that none of them are elementary. Here my sole aim is to show you a very beautiful way of seeing that **
** isn't the oft-quoted **
**. I believe this could be understood, and how nice it would be to have it appreciated, by competent school pupils. I only came to know of this way in August 1996 while browsing in a bookshop in Blackwell's of Oxford: there I came upon a lovely integral, in a paper by van der Poorten and Bombieri, which I had never seen before: an integral with positive integrand, with value (**
**). What I saw fairly set my heart thumping, and I read no further as I wanted to have an opportunity to play. This is what I saw in their paper, and I hardly need comment on the obvious implications: **

**It's an easy school exercise to evaluate that integral, and I quickly show that Maple can cope with it, and more:**

`> `
**Int(x^4*(1 - x)^4/(1 + x^2), x = 0..1);**

`> `
**int(x^4*(1 - x)^4/(1 + x^2), x = 0..1);**

`> `
**evalf(22/7 - Pi);**

`> `
**plot(x^4*(1 - x)^4/(1 + x^2), x = 0..1); **

# Notice the scale on the y-axis

`> `

**But there is so much more to be investigated, discovered, and **
**proved**
**. Can one explain this, can one explain that...? I did a lot of related work in August 1996, but have never done anything about it. I drop some passing **
**hints**
**:**

`> `
**int(x^40*(1 - x)^40/(1 + x^2), x = 0..1);**

`> `
**ifactor(262144);**

`> `
**ifactor(216850257105757801880233554675); **

# the denominator of the approximating denominator

`> `

**I need hardly point out that all the odd primes between 3 and 79 occur there, except for 19.**

`> `
**int(x^68*(1 - x)^68/(1 + x^2), x = 0..1);**

`> `
**ifactor(4294967296);**

`> `
**ifactor(9825114844989333870090599983047666848673348322555875);**

# the denominator of the approximating denominator

`> `

**One will notice that the primes 13 and 107 are 'missing' between 3 and 131.**

**And here I have varied the powers in the integrand:**

`> `
**int(x^52*(1 - x)^76/(1 + x^2), x = 0..1);#**

`> `
**ifactor(68719476736);**

`> `
**ifactor(287400053611342363537442949475728231373512239326125);**

# the denominator of the approximating denominator

`> `

**An obvious question to ask is: can **
**some such integral**
** be used to give a proof that **
** is irrational?**