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 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)

    “Ah! I have penetrated to those meadows on the morning of many a first spring day, jumping from hummock to hummock, from willow root to willow root, when the wild river valley and the woods were bathed in so pure and bright a light as would have waked the dead, if they had been slumbering in their graves, as some suppose. There needs no stronger proof of immortality. All things must live in such a light. O Death, where was thy sting? O Grave, where was thy victory, then?”
    —Henry David Thoreau (1817–1862)

    “If any doubt has arisen as to me, my country [Virginia] will have my political creed in the form of a “Declaration &c.” which I was lately directed to draw. This will give decisive proof that my own sentiment concurred with the vote they instructed us to give.”
    —Thomas Jefferson (1743–1826)