The Algebraic and Transcendental Numbers

A real or complex number z is called algebraic if it is the root of a polynomial equation zn + an − 1zn − 1 + … + a1z + a0 = 0, where the coefficients a0, a1, … an − 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 z2 + 2 = 0; similarly, i, a root of z2 + 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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