==================== Algebra ==================== .. note:: Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_ .. list-table:: Closure of binary operators on given sets of numbers * - Operation - :latex:`$+$` - :latex:`$\times$` - :latex:`$-$` - :latex:`$\divide$` - :latex:`$^$` - :latex:`$\sqrt{\text{ }}$` * - Positive Integers - Y - Y - N - N - Y - N * - Positive rationals - Y - Y - N - Y - Y - N * - Rationals (and zero) - Y - Y - Y - Y - Y - N * - Reals wrt positive integers - Y - Y - Y - Y - Y - Y * - Complex numbers - Y - Y - Y - Y - Y - Y Definitions ============= *involution* to raise a number to a given power *evolution* to take a given root of a number *associative* :latex:`$(a+b)+c=a+(b+c)$` *comutative* :latex:`$a+b=b+c$` *distributive* :latex:`$(a+b)c=ac+bc$` .. [rudin] `Principles of Mathematical Analysis (3rd ed)`:title:, by Walter Rudin. McGraw-Hill, 1976 .. [meserve] `Fundamental Concepts of Algebra`:title:, by Bruce Meserve.