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 - . Complex numbers contain - a. algebraic numbers b. transcendental numbers . Algebraic numbers contain - a. algebraic integers (entire algebraic numbers) b. algebraic fractions (fractional algebraic numbers) . Algebraic integers contain - a. rational integers b. non-rational integers . Algebraic fractions contain - a. rational fractions b. non-rational fractions . 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  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  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
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