Editing The Road to Reality Study Notes (section)

===3.2 The real number system===

'''Natural Numbers:'''

‘Counting’ numbers from 1 to <math>\infty</math>.

'''Whole Numbers:'''

All counting numbers including 0, cannot be a fraction.

'''Integers:'''
 
All natural numbers and their negative counterparts and 0.  If  and  are integers, then their sum , their difference , and their product  are all integers (that is, the integers are closed under addition and multiplication), but their quotient  may or may not be an integer, depending on whether  can be divided by with no remainder.

'''Rational Numbers:'''

A number that can be expressed as the ratio a/b of two integers (or whole numbers) a and b, with b non-zero.  The decimal expansion is alyas ultimately periodic, at a certain point the infinite sequence of digits consists of some finite sequence repeated indefinitely.

'''Irrational Numbers:'''

A number that cannot be expressed as the ratio of two integers. When an irrational number is expressed in decimal notation it never terminates nor repeats. 

:'''Quadratic Irrational Numbers'''
:Arise in the solution of a general quadratic equation:

:<math>Ax^{2} + Bx + C = 0</math>

:With A non-zero, the solutions being (derived from the quadratic formula):

:<math>-\frac{B}{2A}\sqrt{\left(\frac{B}{2A}\right)^2}+\frac{C}{A}, \quad -\frac{B}{2A}\sqrt{\left(\frac{B}{2A}\right)^2}-\frac{C}{A}</math>

:where, to keep within the realm of real numbers, be must have B2 greater than 4AC.   When A, B, and C are integers or rational numbers, and where there is no rational solution to the equation, the solutions are quadratic irrationals. 

'''Real Numbers:'''

A number in the set of all numbers above that falls on the real number line. It can have any value.

:'''Algebraic Real Numbers:'''

:Any number that is the solution to a polynomial with rational coefficients.

:'''Transcendental Real Numbers'''

:Any number that is not the solution to a polynomial with rational coefficients.