Transendenttiset

Transendenttiset refers to a mathematical concept concerning numbers or functions that cannot be expressed as the root of a polynomial equation with integer coefficients. This means that a number like pi or *e* is transcendental, as it cannot be a solution to an equation like ax^n + bx^(n-1) + ... + z = 0, where a, b, ..., z are integers and n is a non-negative integer.

The study of transcendental numbers began in the 19th century. Ferdinand von Lindemann proved that pi is transcendental in 1882, which in turn proved the impossibility of "squaring the circle," a problem that had challenged mathematicians for centuries. This proof confirmed that a square with the same area as a given circle cannot be constructed using only a compass and straightedge.

Other examples of transcendental numbers include *e*, the base of the natural logarithm, proven transcendental by Charles Hermite in 1873. Liouville numbers are also a class of transcendental numbers, constructed to be "very well approximable" by rational numbers, which implies their transcendence. The existence of transcendental numbers was first established by Joseph Liouville in 1844. Most real numbers are, in fact, transcendental, although proving the transcendence of specific numbers can be very difficult. The set of transcendental numbers is infinite and is considered to be of the same "size" (cardinality) as the set of all real numbers.