====================
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 name
     - addition
     - product
     - subtraction
     - division
     - power
     - root
   * - Operation symbol
     - :latex:`$a + b$`
     - :latex:`$a\times b$`
     - :latex:`$a-b$`
     - :latex:`$\frac{a}{b}$`
     - :latex:`$a^b$`
     - :latex:`$\sqrt{\text{a}}$`
   * - 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.