In mathematics, the series expansion of the number e For other meanings of mathematics or math, see mathematics (disambiguation). ... e is the unique number such that the value of the derivative (slope of a tangent line) of f (x)=ex for any value of x is equal to the value of f (x). ...

can be used to prove that e is irrational. In mathematics, an irrational number is any real number that is not a rational number, i. ...

Summary of the proof:

This will be a proof by contradiction. Initially e will be assumed to be rational. The proof is constructed to show that this assumption leads to a logical impossibility. This logical impossibility, or contradiction, implies that the underlying assumption is false, meaning that e must not be rational. Since any number that is not rational is by definition irrational, the proof is complete. This article or section does not cite its references or sources. ...

Proof:

Suppose e = a/b, for some positive integers a and b. Construct the number

We will first show that x is an integer, then show that x is less than 1 and positive. The contradiction will establish the irrationality of e.

• To see that x is an integer, note that

The last term in the final sum is b! / b! = 1 (i.e. it can be interpreted as an empty product). Clearly, however, every term is an integer.

• To see that x is a positive number less than 1, note that

	and so 0 < x. But:
x

Here, the last sum is a geometric series.

So 0 < x < 1 and since there does not exist a positive integer less than 1, we have reached a contradiction, and so e must be irrational. Q.E.D. In mathematics, an empty product, or nullary product, is the result of multiplying no numbers. ... In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ... Q.E.D. is an abbreviation of the Latin phrase (literally, which was to be demonstrated). In simple terms, the use of this Latin phrase is to indicate that something has been definitively proven. ...

See Also

• Proof that 0.999... equals 1
• Proof that 22/7 exceeds π
• Proofs of trigonometric identities
• Proof that the sum of the reciprocals of the primes diverges