Elementary Algebraic Specifications of the Rational Complex Numbers
نویسندگان
چکیده
From the range of techniques available for algebraic specifications we select a core set of features which we define to be the elementary algebraic specifications. These include equational specifications with hidden functions and sorts and initial algebra semantics. We give an elementary equational specification of the field operations and conjugation operator on the rational complex numbers Q(i) and discuss some open problems.
منابع مشابه
Simple Equational Specifications of Rational Arithmetic
We exhibit an initial specification of the rational numbers equipped with addition, subtraction, multiplication, greatest integer function, and absolute value. Our specification uses only the sort of rational numbers. It uses one hidden function; that function is unary. But it does not use an error constant, or extra (hidden) sorts, or conditional equations. All of our work is elementary and se...
متن کاملThe Arithmetic of Algebraic Numbers: An Elementary Approach
Let Q and R be the fields of rational and real numbers respectively. Recall that a real number r is algebraic over the rationals if there is a polynomial p with coefficients in Q that has r as a root, i.e., that has p(r) = 0. Any college freshman can understand that idea, but things get more challenging when one asks about arithmetic with algebraic numbers. For example, being the roots of x2− 3...
متن کاملNon-commutative Rational Power Series and Algebraic Generating Functions
Sequences of numbers abound in combinatorics whose generating functions are algebraic over the rational functions. Examples include Catalan and related numbers, numbers of words expressing an element in a free group, and diagonal coe cients of 2-variable rational generating functions (Furstenberg's theorem). Algebraicity is of of practical as well as theoretical interest, since it guarantees an...
متن کاملDeveloping a model to enhance elementary teachers’ ability to foster functional thinking and algebraic reasoning in elementary students
Abstract: This study explores the process of change of grades 3 to 5 elementary teachers, who participated in a professional development program "Algebraic Thinking: Foundation of Elementary Mathematics". Algebraic thinking as a functional thinking was the centerpiece of the program. The “Concern Base Adaptation Model” (CBAM) was used as methodology. The results of the study showed the signific...
متن کاملAlgebraic and Transcendental Numbers
To begin with, recall that a complex number α is said to be a root of a polynomial P (X) if P (α) = 0. A complex number α is said to be algebraic if there is a nonzero polynomial P (X), with integer coefficients, of which α is a root. The set of algebraic numbers is denoted by Q̄. A complex number α which is not algebraic is said to be transcendental. The following numbers are obviously algebrai...
متن کامل