--- format: markdown categories: math toc: no ... # Algebra *Note: Most of the definitions and notation in the section are based on [^rudin] or [^meserve].* ----------- ----------------- -------------- --------------- ---------- --------------------- -------- Name Symbol Pos. Integers? Pos. Rationals? Rationals? Reals (wrt Pos Int.)? Complex? ----------- ----------------- -------------- --------------- ---------- --------------------- -------- addition $a + b$ Y Y Y Y Y product $a\times b$ Y Y Y Y Y subtraction $a-b$ N N Y Y Y division $\frac{a}{b}$ N Y Y Y Y power $a^b$ Y Y Y Y Y root $\sqrt{\text{a}}$ N N N Y Y ----------- ----------------- -------------- --------------- ---------- --------------------- -------- Table: Closure of binary operators on given sets of numbers ## Definitions involution : to raise a number to a given power evolution : to take a given root of a number associative : $(a+b)+c=a+(b+c)$ commutative : $a+b=b+c$ distributive : $(a+b)c=ac+bc$ [^rudin]: **Principles of Mathematical Analysis (3rd ed)**, by Walter Rudin. McGraw-Hill, 1976 [^meserve]: **Fundamental Concepts of Algebra**, by Bruce Meserve.