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# 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.