Search results
Results from the WOW.Com Content Network
In 1840, Liouville published a proof of the fact that e 2 is irrational [10] followed by a proof that e 2 is not a root of a second-degree polynomial with rational coefficients. [11] This last fact implies that e 4 is irrational. His proofs are similar to Fourier's proof of the irrationality of e.
This same constant can be used to prove that "an irrational elevated to an irrational power may be rational", even without first proving its transcendence. The proof proceeds as follows: either 2 2 {\displaystyle {\sqrt {2}}^{\sqrt {2}}} is a rational which proves the theorem, or it is irrational (as it turns out to be) and then
In 1948, Erdős showed that the constant E is an irrational number. [3] Later, Borwein provided an alternative proof. [4] Despite its irrationality, the binary representation of the Erdős–Borwein constant may be calculated efficiently. [5] [6]
Since e is an irrational number (see proof that e is irrational), it cannot be represented as the quotient of two integers, but it can be represented as a continued fraction. Using calculus, e may also be represented as an infinite series, infinite product, or other types of limit of a sequence.
In other words, the n th digit of this number is 1 only if n is one of 1! = 1, 2! = 2, 3! = 6, 4! = 24, etc. Liouville showed that this number belongs to a class of transcendental numbers that can be more closely approximated by rational numbers than can any irrational algebraic number, and this class of numbers is called the Liouville numbers ...
The values of a and b are not restricted to real numbers; complex numbers are allowed (here complex numbers are not regarded as rational when they have an imaginary part not equal to 0, even if both the real and imaginary parts are rational).
The number e is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function.It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted .
Bertrand's postulate and a proof; Estimation of covariance matrices; Fermat's little theorem and some proofs; Gödel's completeness theorem and its original proof; Mathematical induction and a proof; Proof that 0.999... equals 1; Proof that 22/7 exceeds π; Proof that e is irrational; Proof that π is irrational