the cosine function (also referred to as the Dottie number ) – the unique real solution to the equation , where is in radians (by the Lindemann–Weierstrass theorem). if is algebraic and not equal to 0 or 1, for any branch of the logarithm function (by the Lindemann–Weierstrass theorem). if and are positive integers not both powers of the same integer (by the Gelfond–Schneider theorem). The Bessel function , its first derivative, and the quotient {{math|{{sfrac|''Jν(x)|Jν(x)}}}} are transcendental when ν is rational and x is algebraic and nonzero, and all nonzero roots of and are transcendental when ν is rational. if is algebraic and nonzero, for any branch of the Lambert W Function (by the Lindemann–Weierstrass theorem), in particular: the omega constant , the square super-root of any natural number is either an integer or transcendental (by the Gelfond–Schneider theorem) , , and . The numbers , and are also known to be transcendental. The numbers and are also transcendental. 0.64341054629..., Cahen's constant. The Champernowne constants, the irrational numbers formed by concatenating representations of all positive integers.. , Chaitin's constant (since it is a non-computable number). The so-called Fredholm constants, such as. The name 'Fredholm number' is misplaced: Kempner first proved this number is transcendental, and the note on page 403 states that Fredholm never studied this number. which also holds by replacing 10 with any algebraic . Gauss's constant and the lemniscate constant. The aforementioned Liouville constant for any algebraic . The Prouhet–Thue–Morse constant.. The Komornik–Loreti constant. Any number for which the digits with respect to some fixed base form a Sturmian word. For where is the floor function. 3.300330000000000330033... and its reciprocal 0.30300000303..., two numbers with only two different decimal digits whose nonzero digit positions are given by the Moser–de Bruijn sequence and its double. The number , where and are Bessel functions and is the Euler–Mascheroni constant. Nesterenko proved in 1996 that and are algebraically independent. Possible transcendental numbers Numbers which have yet to be proven to be either transcendental or algebraic: Most sums, products, powers, etc. of the number and the number , e.g. , , , , , , , , are not known to be rational, algebraic, irrational or transcendental. A notable exception is (for any positive integer ) which has been proven transcendental. The Euler–Mascheroni constant : In 2010 M. Ram Murty and N. Saradha found an infinite list of numbers containing such