er is irrational for r{0}

We here present a proof of the following theorem:


er is irrational for all rQ{0}

To begin with, note that it is sufficient to show that eu is irrational for any positive integer ( this entry, :={1,2,3,} and 0:={0}. u (for if er=euv were rational, so would (euv)v=eu). Next, we look at some simple properties of polynomial fn(x):=xn(1-x)nn!:

  • fn(x)=1n!i=n2ncixi, with ci for all i.

  • fn(k)(0) and fn(k)(1) are integers for all k0: as 0 is a root ( of order n, fn(k)(0)=0 unless nk2n, in which case fn(k)(0)=k!n!ck, an integer. Since fn(k)(x)=(-1)kfn(k)(1-x), the same is true for fn(k)(1).

  • For all 0<x<1 we have 0<fn(x)<1n!.

Now we can readily prove the theorem:


Assume that eu=ab for some (a,b)2 and let


which is actually a finite sum since fn(k)(x)=0 for all k>2n. Differentiating Fn(x) yields Fn(x)=u2n+1fn(x)-uFn(x) and thus:


Now consider the sequenceMathworldPlanetmath


Given the remarks on fn(x), wn should be an integer for all n, yet it is clear that wn<bu2n+11n!=an!u2n+1 and so limnwn=0, a contradictionMathworldPlanetmathPlanetmath. ∎

The result could also easily have been obtained by starting with wn and integrating by parts 2n times. Note also that much stronger statements are known, such as “ea is transcendental for all a𝔸{0}22𝔸 denotes the set of algebraic numbersMathworldPlanetmath.. We conclude this entry with the following evident corollary:


For all rQ+,logr is irrational.


  • 1 M. Aigner & G. M. Ziegler: Proofs from THE BOOK, 3rd edition (2004), Springer-Verlag, 30–31.
  • 2 G. H. Hardy & E. M. Wright: An Introduction to the Theory of Numbers, 5th edition (1979), Oxford University Press, 46–47.
Title er is irrational for r{0}
Canonical name ErIsIrrationalForRinmathbbQsetminus0
Date of creation 2013-03-22 15:07:46
Last modified on 2013-03-22 15:07:46
Owner Cosmin (8605)
Last modified by Cosmin (8605)
Numerical id 12
Author Cosmin (8605)
Entry type Theorem
Classification msc 11J72
Synonym er is irrational for non-zero rational r
Synonym irrationality of the exponential functionDlmfDlmfMathworld on
Related topic Irrational
Related topic EIsIrrationalProof
Related topic EIsIrrational
Related topic EIsTranscendental