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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  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 + is transcendental.
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