Search results for author:"David E Dobbs"
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Detecting Prime Numbers via Roots of Polynomials
David E. Dobbs
International Journal of Mathematical Education in Science and Technology Vol. 43, No. 3 (2012) pp. 381–387
It is proved that an integer n [greater than or equal] 2 is a prime (resp., composite) number if and only if there exists exactly one (resp., more than one) nth-degree monic polynomial f with coefficients in Z[subscript n], the ring of integers...
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Generalizing a Limit Description of the Natural Logarithm
David E. Dobbs
International Journal of Mathematical Education in Science and Technology Vol. 41, No. 5 (2010) pp. 687–691
If f is a continuous positive-valued function defined on the closed interval from a to x and if k[subscript 0] is greater than 0, then lim[subscript k[right arrow]0[superscript +] [integral][superscript x] [subscript a] f (t)[superscript k-k...
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On Rank and Nullity
David E. Dobbs
International Journal of Mathematical Education in Science and Technology Vol. 43, No. 2 (2012) pp. 271–283
This note explains how Emil Artin's proof that row rank equals column rank for a matrix with entries in a field leads naturally to the formula for the nullity of a matrix and also to an algorithm for solving any system of linear equations in any...
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A Single Instance of the Pythagorean Theorem Implies the Parallel Postulate
David E. Dobbs
International Journal of Mathematical Education in Science and Technology Vol. 33, No. 4 (2002) pp. 596–600
This note could find use as enrichment material in a course on the classical geometries; its preliminary results could also be used in an advanced calculus course. It is proved that if a , b and c are positive real numbers such that a[squared] + b...
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Polynomial Asymptotes of the Second Kind
David E. Dobbs
International Journal of Mathematical Education in Science and Technology Vol. 42, No. 2 (March 2011) pp. 276–282
This note uses the analytic notion of asymptotic functions to study when a function is asymptotic to a polynomial function. Along with associated existence and uniqueness results, this kind of asymptotic behaviour is related to the type of asymptote ...
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Why the Square Root Function Is Not Linear
David E. Dobbs
International Journal of Mathematical Education in Science and Technology Vol. 33, No. 5 (September 2002) pp. 742–747
Six proofs are given for the fact that for each integer n [greater than or equal to] 2, the nth root function, viewed as a function from the set of non-negative real numbers to itself, is not linear. If p is a prime number, then [Zeta]/p[Zeta] is...
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Iteration, Not Induction
David E. Dobbs
International Journal of Mathematical Education in Science and Technology Vol. 40, No. 4 (2009) pp. 517–523
The main purpose of this note is to present and justify proof via iteration as an intuitive, creative and empowering method that is often available and preferable as an alternative to proofs via either mathematical induction or the well-ordering...
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On the Definition of the Ordinary Points and the Regular Singular Points of a Homogeneous Linear Ordinary Differential Equation
David E. Dobbs
Mathematics and Computer Education Vol. 39, No. 2 (2005) pp. 125–130
The author discusses the definition of the ordinary points and the regular singular points of a homogeneous linear ordinary differential equation (ODE). The material of this note can find classroom use as enrichment material in courses on ODEs, in...
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Recognizing Exponential Growth. Classroom Notes
David E. Dobbs
International Journal of Mathematical Education in Science and Technology Vol. 35, No. 1 (2004) pp. 153–158
Two heuristic and three rigorous arguments are given for the fact that functions of the form Ce[kx], with C an arbitrary constant, are the only solutions of the equation dy/dx=ky where k is constant. Various of the proofs in this self-contained note ...
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On the Well-Definedness of the Order of an Ordinary Differential Equation
David E. Dobbs
International Journal of Mathematical Education in Science and Technology Vol. 37, No. 3 (Apr 15, 2006) pp. 358–362
It is proved that if the differential equations "y[(n)] = f(x,y,y[prime],...,y[(n-1)])" and "y[(m)] = g(x,y,y[prime],...,y[(m-1)])" have the same particular solutions in a suitable region where "f" and "g" are continuous real-valued functions with...
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On Solving Linear Recurrences
David E. Dobbs
International Journal of Mathematical Education in Science and Technology Vol. 44, No. 2 (2013) pp. 310–315
A direct method is given for solving first-order linear recurrences with constant coefficients. The limiting value of that solution is studied as "n to infinity." This classroom note could serve as enrichment material for the typical...
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An Elementary Proof of a Criterion for Linear Disjointness
David E. Dobbs
International Journal of Mathematical Education in Science and Technology Vol. 44, No. 4 (2013) pp. 614–617
An elementary proof using matrix theory is given for the following criterion: if "F"/"K" and "L"/"K" are field extensions, with "F" and "L" both contained in a common extension field, then &...
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Polynomial Asymptotes
David E. Dobbs
International Journal of Mathematical Education in Science and Technology Vol. 41, No. 7 (2010) pp. 943–950
This note develops and implements the theory of polynomial asymptotes to (graphs of) rational functions, as a generalization of the classical topics of horizontal asymptotes and oblique/slant asymptotes. Applications are given to hyperbolic...
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On Positive Functions with Positive Derivatives
David E. Dobbs
International Journal of Mathematical Education in Science and Technology Vol. 33, No. 6 (2002) pp. 895–898
Three proofs are given for the fact that the derivative of an everywhere-positive non-constant real polynomial function must change sign. This self-contained note could find classroom use in courses on calculus or abstract algebra.
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Subsets of Fields Whose nth-Root Functions Are Rational Functions
David E. Dobbs
International Journal of Mathematical Education in Science and Technology Vol. 49, No. 6 (2018) pp. 948–958
Let R be an integral domain with quotient field F, let S be a non-empty subset of R and let n = 2 be an integer. If there exists a rational function ?: S [right arrow] F such that ?(a)[superscript n] = a for all a ? S, then S is finite. As a...
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On the Smoothness Condition in Euler's Theorem on Homogeneous Functions
David E. Dobbs
International Journal of Mathematical Education in Science and Technology Vol. 49, No. 8 (2018) pp. 1250–1259
For a function "f": [real numbers set][superscript n]\{(0,…,0)}[right arrow][real numbers set] with continuous first partial derivatives, a theorem of Euler characterizes when "f" is a homogeneous function. This note determines...
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What Function is This?
David E. Dobbs; John C. Peterson
Mathematics and Computer Education Vol. 31, No. 1 (1997) pp. 29–32
Presents several types of functions which fit a given set of data and create opportunities for classroom discussion comparing different kinds of functions and identifying some of the potential hazards associated with extrapolation from best-fit...