The algebraic numbers are closed under addition and multiplication: equipped with these operations, they form a field (see "L'esprit de corps quadratique"). Consequently, the sum and product of two algebraic numbers are algebraic. The same does not hold for the sum or product of two transcendental numbers, which may well be algebraic—or even an integer—as with the following numbers:
(π+4)−πetπ×1π.(\pi+4)-\pi\quad {\rm{et}\quad \pi\times\frac{1}{\pi }.}
This immediately raises the question of the sum and product of the two most famous transcendental numbers: s = π + e and p = πe. Nothing is known about either number individually, but paradoxically, as we have seen (see the end of the article "Histoire d'e"), we know that at least one of them is transcendental.
What about powers? ------------------
For powers—that is, for *ab, where a is a strictly positive real number other than 1 and b* is real—we know a great deal, but not everything!