Introducution to Algebraic Numbers and Integers

Date: 04/15/2008 at 23:57:56
From: Sujeet
Subject: A different way of classifying Complex Numbers

While researching the classification of Complex numbers on the 
internet, I came across the following classification -

[1]. Complex numbers contain -

  a. algebraic numbers 
  b. transcendental numbers 

[2]. Algebraic numbers contain -

  a. algebraic integers (entire algebraic numbers) 
  b. algebraic fractions (fractional algebraic numbers) 

[3]. Algebraic integers contain -

  a. rational integers 
  b. non-rational integers 

[4]. Algebraic fractions contain -

  a. rational fractions 
  b. non-rational fractions 

[5]. Transcendental numbers contain -

  a. real transcendental numbers 
  b. imaginary transcendental numbers 

If possible I would like to understand this classification in simple 
terms. 

How come we have two different classifications of Complex numbers, the
usual one we are taught in school and then this one?  Are the two 
classifications related to each other?  In what way?

I am able to follow the classification till Algebraic and 
Transcendental numbers but things become difficult and go beyond my 
present understanding of mathematics when we come to algebraic 
integers, rational integers and non-rational integers.  On 
researching the topic further on the internet, I find that this is a 
vast subject in Mathematics of which I may be seeing only the tip. 

It is beyond me at this time to learn this branch of mathematics but 
at the same time I would love to understand in simple terms this 
classification.  It will definitely enhance my understanding and love 
of mathematics. 

Date: 04/19/2008 at 04:54:16
From: Doctor Jacques
Subject: Re: A different way of classifying Complex Numbers

Hi Sujeet,

I will try to draw some kind of overall Venn diagram:


  +------------------------------------------------+
  |                                               C|
  |                                                |
  |                                                |
  |                                                |
  +----------+----------------------------+        |
  |         B|                           A|        |
  |          |                            |        |
  |          |                            |        |
  |          |                            |        |
  |          |                            |        |
  +----------+----------------------------+--------+
  |          |                            |       R|
  +----+-----+-------------+              |        |
  |   N|    Z|            Q|              |        |
  |    |     |             |              |        |
  |    |     |             |              |        |
  +----+-----+-------------+--------------+--------+
  O

In this diagram, I have identified some sets (but not all--see 
below) with a letter (N, Z, ...).  For any such a set, the letter 
identifies the top right corner of the rectangle that represents the 
set; the bottom left corner is the origin O.  For example, the whole 
rectangle corresponds to the set C.

The sets so identified are those that correspond to "positive" 
definitions:

  C : complex numbers, for example "pi" + 3.5*i
  R : real numbers, for example "e"
  Q : rational numbers, for example, 3/5
  Z : integers (also called "rational integers", see below), for
      example, -2
  N : Natural numbers, for example 3
  A : Algebraic numbers, for example sqrt(2) / sqrt(3) or i/2
  B : Algebraic integers, for example sqrt(2) or i

(there are no standard abbreviations for these latter two sets).

Those sets have "nice" properties (for example, they are all closed 
at least with respect to addition and multiplication).  Besides those 
sets, other sets can be defined as the complements of some sets 
(those are "negative" definitions):

  The transcendental numbers are those that are not algebraic (the
  complement of A, C \ A, where \ is the set difference operator).

  The irrational numbers are those that are not rational (C \ Q).

Those sets are, in some sense, less interesting, as they usually do 
not possess any structure.  For example, the irrational numbers are 
not closed under addition, since pi + (-pi) = 0 : the sum of two 
irrational numbers may be rational.

Let us look in more detail at the definitions.  I assume that you know 
the definitions of N, Z, Q, R, and C, so I will focus on the 
algebraic numbers.

An algebraic number is a root of a polynomial with integer 
coefficients.  Specifically, "a" is an algebraic number if and only if 
there exists a polynomial f(x) with integer coefficients such that f
(a) = 0.  The polynomial with the lowest degree whose a is a root is 
called the minimal polynomial of a; it is uniquely defined up to a 
constant coefficient (and this ambiguity can also be resolved).

For example, the roots of the equation:

  3x^2 - 5x + 6 = 0   [1]

are algebraic numbers.

Algebraic numbers can be real, like sqrt(2/3), which is a root of 
3x^2 - 2 = 0, or complex (non-real), like the roots of the equation 
[1] above.

It can be shown that algebraic numbers are closed with respect to 
addition, subtraction, multiplication, and division (except by 0, of 
course).

Any rational number is an algebraic number, since it is the root of a 
first degree polynomial with integer coefficients: for example, 3/5 
is a root of the equation 5x - 3 = 0.  In fact, an algebraic number is 
rational iff its minimal polynomial has degree 1.

An algebraic integer is a root of a polynomial with integer 
coefficient, and whose leading coefficient (the coefficient of the 
highest power of the indeterminate) is equal to 1.  For example:

  sqrt(2), which is a root of x^2 - 2 = 0
  i, which is a root of x^2 + 1 = 0
  (sqrt(5)-1)/2 (the golden ratio), which is a root of x^2 + x - 1 = 0

As you see, algebraic integers (like all algebraic numbers) can be 
real or complex).

Any algebraic integer is an algebraic number, since we merely added 
the condition on the first coefficient.

Any normal integer (element of Z) is an algebraic integer, since the 
integer n is a root of the equation x - n = 0, which has integer 
coefficients (1, n), and whose leading coefficient is 1.  In fact, an 
algebraic integer is a normal integer if and only if it is a rational 
number : the intersection of the sets B and Q is equal to Z (this 
corresponds to the fact that the minimal polynomial has degree 1).

It can be shown (this is more difficult) that algebraic integers are 
closed with respect to addition, subtraction, and multiplication (but 
not division).

In many books on algebraic number theory, the main objects of interest 
are the algebraic integers.  In such books, it is quite common to use 
the word "integer" to denote an algebraic integer, and "rational 
integer" to denote a plain integer (an element of Z). (You may have 
wondered why the set Z is sometimes called the set of "rational 
integers"; after all, any normal integer is a rational number--this is 
where the expression comes from.)

A few additional notes on some ambiguities related to terminology:

The expression "imaginary number" is often used in the sense of "pure 
imaginary", i.e., a complex number whose real part is 0.  The 
rigorously exact term should be "non-real complex number".  I used 
simply complex number in that sense, but this is not strictly correct, 
since any real number is also a complex number (whose imaginary part 
is 0).

Transcendental and irrational numbers include complex numbers; 
however, in many cases, those expressions are used only to refer to 
real transcendental and irrational numbers--this should be clear 
from the context.

The term "algebraic fractions" is not commonly used; I would recommend 
"fractional algebraic numbers".  Those are the set A \ B.  As the 
diagram shows, the intersection of that set with Q (which you call 
"rational fractions") is Q \ Z, the non-integer rationals.

Please feel free to write back if you want to discuss this further.

- Doctor Jacques, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 04/22/2008 at 09:53:34
From: Sujeet
Subject: Thank you (A different way of classifying Complex Numbers)

Dear Dr. Jacques,

Thanks a lot for your reply.  I have learned something new today :-).

With best regards,

Sujeet