The Algebraic and Transcendental Numbers
A real or complex number z is called algebraic if it is the root of a polynomial equation z n + a n - 1 z n - 1 + … + a 1 z + a 0 = 0, where the coefficients a 0, a 1, … a n - 1 are all rational; if z cannot be a root of such an equation, it is said to be transcendental. The number 2 is algebraic because it is a root of the equation z 2 + 2 = 0; similarly, i, a root of z 2 + 1 = 0, is also algebraic. However, F. Lindemann showed (1882) that π is transcendental, and using this fact he proved the impossibility of "squaring the circle" by straight edge and compass alone (see geometric problems of antiquity). The number e has also been found to be transcendental, although it still remains unknown whether e + π is transcendental.
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