Proof That e Is Irrational - Proof

Proof

Towards a contradiction, suppose that e is a rational number. Then there exist positive integers a and b such that e = a/b where clearly b > 1.

Define the number

x = b!\,\biggl(e - \sum_{n = 0}^{b} \frac{1}{n!}\biggr)\!

To see that if e is rational, then x is an integer, substitute e = a/b into this definition to obtain

x = b!\,\biggl(\frac{a}{b} - \sum_{n = 0}^{b} \frac{1}{n!}\biggr) = a(b - 1)! - \sum_{n = 0}^{b} \frac{b!}{n!}\,.

The first term is an integer, and every fraction in the sum is actually an integer because n ≤ b for each term. Therefore x is an integer.

We now prove that 0 < x < 1. First, to prove that x is strictly positive, we insert the above series representation of e into the definition of x and obtain

because all the terms with n ≤ b cancel and the remaining ones are strictly positive.

We now prove that x < 1. For all terms with n ≥ b + 1 we have the upper estimate

\frac{b!}{n!} =\frac1{(b+1)(b+2)\cdots(b+(n-b))} \le\frac1{(b+1)^{n-b}}\,.\!

This inequality is strict for every n ≥ b + 2. Changing the index of summation to k = n – b and using the formula for the infinite geometric series, we obtain

x =\sum_{n = b+1}^\infty \frac{b!}{n!} < \sum_{n=b+1}^\infty \frac1{(b+1)^{n-b}} =\sum_{k=1}^\infty \frac1{(b+1)^k} =\frac{1}{b+1} \biggl(\frac1{1-\frac1{b+1}}\biggr) = \frac{1}{b} < 1.

Since there is no integer strictly between 0 and 1, we have reached a contradiction, and so e must be irrational. Q.E.D.

The above proof can be found in Proofs from THE BOOK, where the stronger result that eq is irrational for any non-zero rational q is also proved.

Read more about this topic:  Proof That e Is Irrational

Famous quotes containing the word proof:

    “The fact that several men were able to become infatuated with that latrine is truly the proof of the decline of the men of this century.”
    —Charles Baudelaire (1821–1867)

    “The source of Pyrrhonism comes from failing to distinguish between a demonstration, a proof and a probability. A demonstration supposes that the contradictory idea is impossible; a proof of fact is where all the reasons lead to belief, without there being any pretext for doubt; a probability is where the reasons for belief are stronger than those for doubting.”
    —Andrew Michael Ramsay (1686–1743)

    “The chief contribution of Protestantism to human thought is its massive proof that God is a bore.”
    —H.L. (Henry Lewis)