proof that e is not a natural number

Here, we are going to show that the natural log base e is not a natural numberMathworldPlanetmath by showing a sharper result: that e is between 2 and 3.

PropositionPlanetmathPlanetmathPlanetmath. 2<e<3.


There are several infinite series representations of e. In this proof, we will use the most common one, the Taylor expansionMathworldPlanetmath of e:

i=01i!=10!+11!+12!++1n!+. (1)

We chop up the Taylor expansion of e into two parts: the first part a consists of the sum of the first two terms, and the second part b consists of the sum of the rest, or e-a. The proof of the proposition now lies in the estimation of a and b.

Step 1: e>2. First, a=10!+11!=1+1=2. Next, b>0, being a sum of the terms in (1), all of which are positive (note also that b must be bounded because (1) is a convergent seriesMathworldPlanetmath). Therefore, e=a+b=2+b>2+0=2.

Step 2: e<3. This step is the same as showing that b=e-a=e-2<3-2=1. With this in mind, let us compare term by term of the series (2) representing b and another series (3):

12!+13!++1n!+ (2)


122-1+123-1++12n-1+. (3)

It is well-known that the second series (a geometric seriesMathworldPlanetmath) sums to 1. Because both series are convergent, the term-by-term comparisons make sense. Except for the first term, where 12!=12=122-1, we have 1n!<12n-1 for all other terms. The inequalityMathworldPlanetmath 1n!<12n-1, for n a positive number can be translated into the basic inequality n!>2n-1, the proof of which, based on mathematical induction, can be found here (

Because the term comparisons show

  • that the terms from (2) the corresponding terms from (3), and

  • that at least one term from (2) < than the corresponding term from (3),

we conclude that (2) < (3), or that b<1. This concludes the proof. ∎

Title proof that e is not a natural number
Canonical name ProofThatEIsNotANaturalNumber
Date of creation 2013-03-22 15:39:52
Last modified on 2013-03-22 15:39:52
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Proof
Classification msc 40A25
Classification msc 40A05
Classification msc 11J72
Related topic EIsTranscendental