A rational number is a real number that can be expressed in the form p/q where p and q are integers. A real number that is not rational is called irrational.
In his book Introductio in analysin infinitorum (1748), Euler proved that every rational number can be expressed as a finite continued fraction (Maor, 1994). For example,
13/8 = 1 + 1/1+ 3/5
Thus, every irrational number can be expressed as an infinite continued fraction. Euler derived several infinite continued fractions involving the number e, including the ones seen below (Maor, 1994).
Figures: Infinite Continued Fractions of e
The patterns seen in the continued fractions above suggest that the continued fractions are infinite. Euler did not prove that these patterns continue, but he knew that showing this would prove e is irrational (O'Connor & Robertson, 2001).
Proof by Contradiction
The following proof by contradiction that e is irrational has been adapted from Eli Maor's book e: The story of a number (1994).
Assume that e is rational. By definition, this implies e= p/q where p and q are integers. We know that 2 < e < 3 (shown by Bernoulli), so e cannot be an integer. This implies that the denominator q must be at least 2. Using the series expansion of e (to learn about this click on the series link in the menu above), we also know that:
The left hand side is clearly an integer because the set of all integers is closed. The expression inside the brackets on the right hand side is also an integer for the same reason. However, the remaining terms on the right hand side are not integers. We now need to show that their sum is also not an integer. Recall that q ≥ 2, this implies that:
The series 1/3 + 1/32 + ... is an infinite geometric series whose sum is 1/2. Thus,
1/q+1 + 1/(q+1)(q+2) +... < 1/2
This means that the remaining terms on the right hand side sum to a non-integer. The sum of an integer and a non-integer is not an integer, so the right hand side is a non-integer. Since the left hand side is an integer and the right hand side is not an integer, we have a contradiction. Since we logically deduced this contradiction from the assumption that e is rational, we conclude that this assumption must be false. Therefore, e is irrational. QED.
Interesting Fact:
To remember the first nine digits after the decimal point in the number e just think of Andrew Jackson.
Figure: Andrew Jackson, 1767-1845
Andrew Jackson was the 7th president of the United States. He was elected in 1828 and he served as president for two terms. So,
