There are thus two ways in which to select such special numbers . One way is to set the derivative of the exponential function equal to , and solve for . The other way is to set the derivative of the base logarithm to and solve for . In each case, one arrives at a convenient choice of base for doing calculus. It turns out that these two solutions for are actually "the same", the number . Other characterizations of are also possible: one is as the limit of a sequence, another is as the sum of an infinite series, and still others rely on integral calculus. So far, the following two (equivalent) properties have been introduced:   The following four characterizations can be proven equivalent:    where is the factorial of . (By convention formula_29.) As in the motivation, the exponential function is important in part because it is the unique nontrivial function (up to multiplication by a constant) which is its own derivative  and therefore its own antiderivative as well:  The number is the unique real number such that  for all positive . Also, we have the inequality  for all real , with equality if and only if . Furthermore, is the unique base of the exponential for which the inequality holds for all . This is a limiting case of Bernoulli's inequality. Steiner's problem asks to find the global maximum for the function  This maximum occurs precisely at . For proof, the inequality formula_38, from above, evaluated at formula_39 and simplifying gives formula_40. So formula_41 for all positive "x". Similarly, is where the global minimum occurs for the function  defined for positive . More generally, for the function  the global maximum for positive occurs at for any ; and the global minimum occurs at for any . The infinite tetration  converges if and only if (or approximately between 0.0660 and 1.4447), due to a theorem of Leonhard Euler. The real number is irrational. Euler proved this by showing that its simple continued fraction expansion is infinite. (See also Fourier's proof that is irrational.) Furthermore, by the Lindemann–Weierstrass theorem, is transcendental, meaning that it is not a solution of any non-constant polynomial equation with rational coefficients. It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare with Liouville number); the proof was given by Charles Hermite in 1873.