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Understanding the Properties of “Numbers”

Victor J. Katz and Karen Hunger Parshall

in Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691149059
eISBN:
9781400850525
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691149059.003.0013
Subject:
Mathematics, History of Mathematics

This chapter looks at how mathematicians sought to understand the properties of “numbers” and in doing so pave the way for modern algebra. As mathematicians in the eighteenth and nineteenth centuries ... More


Mathematical Preliminaries

Partha P. Mitra and Hemant Bokil

in Observed Brain Dynamics

Published in print:
2007
Published Online:
May 2009
ISBN:
9780195178081
eISBN:
9780199864829
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195178081.003.0005
Subject:
Neuroscience, Techniques, Molecular and Cellular Systems

This chapter reviews a broad range of mathematical topics relevant to the rest of the book. It begins with a brief discussion of real and complex numbers and elementary real and complex functions, ... More


Why Is the State Complex?

Gary E. Bowman

in Essential Quantum Mechanics

Published in print:
2007
Published Online:
January 2008
ISBN:
9780199228928
eISBN:
9780191711206
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199228928.003.0010
Subject:
Physics, Condensed Matter Physics / Materials

This chapter begins with basic complex numbers, including their polar form and their representation with Argand diagrams. This leads naturally into a discussion of the phase of a complex number, and ... More


Complementarity in Mathematics

José Ferreirós

in Mathematical Knowledge and the Interplay of Practices

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691167510
eISBN:
9781400874002
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691167510.003.0004
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter presents a thesis about what it regards as a crucial complementarity of elements in mathematical practice and in the configuration of mathematical knowledge: both conceptual and ... More


Vector analysis and complex algebra

Dennis Sherwood and Jon Cooper

in Crystals, X-rays and Proteins: Comprehensive Protein Crystallography

Published in print:
2010
Published Online:
January 2011
ISBN:
9780199559046
eISBN:
9780191595028
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199559046.003.0002
Subject:
Physics, Crystallography: Physics

This chapter introduces the essential mathematics of diffraction. Initially the concept of a vector is presented along with the mathematical operations which we can perform on vector quantities ... More


Classical Arithmetization

John Stillwell

in Reverse Mathematics: Proofs from the Inside Out

Published in print:
2019
Published Online:
May 2020
ISBN:
9780691196411
eISBN:
9781400889037
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691196411.003.0002
Subject:
Mathematics, History of Mathematics

This chapter describes how one proceeds from natural to rational numbers, then to real and complex numbers, and to continuous functions—thus arithmetizing the foundations of analysis and geometry. ... More


Complex Numbers

David P. Feldman

in Chaos and Fractals: An Elementary Introduction

Published in print:
2012
Published Online:
December 2013
ISBN:
9780199566433
eISBN:
9780191774966
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199566433.003.0024
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter focuses on complex numbers, also known as imaginary numbers, and their algebra. It first considers the square root of -1 before performing basic mathematical operations with complex ... More


Complex Numbers and Grassmann Numbers

Bryan J. Dalton, John Jeffers, and Stephen M. Barnett

in Phase Space Methods for Degenerate Quantum Gases

Published in print:
2014
Published Online:
April 2015
ISBN:
9780199562749
eISBN:
9780191747311
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199562749.003.0003
Subject:
Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

This chapter sets out the basic algebraic properties of both complex numbers (c-numbers) and Grassmann numbers (g-numbers). The formal rules of addition and multiplication together with the ... More


Mathematics and Reality

Roger Penrose and Martin Gardner

in The Emperor’s New Mind: Concerning Computers, Minds, and The Laws of Physics

Published in print:
1989
Published Online:
November 2020
ISBN:
9780198519737
eISBN:
9780191917080
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198519737.003.0011
Subject:
Computer Science, Artificial Intelligence, Machine Learning

Let us imagine that we have been travelling on a great journey to some far-off world. We shall call this world Tor’Bled-Nam. Our remote sensing device has picked up a signal which is now displayed ... More


i: Imaginary numbers

Marcel Danesi

in Pythagoras' Legacy: Mathematics in Ten Great Ideas

Published in print:
2020
Published Online:
January 2020
ISBN:
9780198852247
eISBN:
9780191886959
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198852247.003.0007
Subject:
Mathematics, History of Mathematics, Educational Mathematics

What kind of number is √−1? In a way that parallels the unexpected discovery of √2 by the Pythagoreans, when this number surfaced as a solution to a quadratic equation, mathematicians asked ... More


Mathematical Background

M. Suhail Zubairy

in Quantum Mechanics for Beginners: With Applications to Quantum Communication and Quantum Computing

Published in print:
2020
Published Online:
June 2020
ISBN:
9780198854227
eISBN:
9780191888649
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198854227.003.0002
Subject:
Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

In the spirit of making this book reasonably self-contained, certain topics that may be required in understanding the foundation and the applications of quantum mechanics are discussed. Foremost are ... More


Applications of Complex Numbers and Quaternions: Historical Remarks, with a Note on Clifford Algebra

Peter Simons

in Abstractionism: Essays in Philosophy of Mathematics

Published in print:
2016
Published Online:
January 2017
ISBN:
9780199645268
eISBN:
9780191755330
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199645268.003.0016
Subject:
Philosophy, Logic/Philosophy of Mathematics

Frege required his logicist definitions of numbers to obey the applicability constraint, that all potential applications somehow be prefigured in the definitions. Can this constraint hold for ... More


Encircling Graphs

Arthur Benjamin, Gary Chartrand, and Ping Zhang

in The Fascinating World of Graph Theory

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691175638
eISBN:
9781400852000
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175638.003.0006
Subject:
Mathematics, Applied Mathematics

This chapter considers Hamiltonian graphs, a class of graphs named for nineteenth-century physicist and mathematician Sir William Rowan Hamilton. In 1835 Hamilton discovered that complex numbers ... More


Julia Sets for the Quadratic Family

David P. Feldman

in Chaos and Fractals: An Elementary Introduction

Published in print:
2012
Published Online:
December 2013
ISBN:
9780199566433
eISBN:
9780191774966
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199566433.003.0025
Subject:
Physics, Theoretical, Computational, and Statistical Physics

The Julia set for a function refers to the collection of all initial conditions that do not tend toward infinity when iterated by that function. This chapter examines Julia sets that arise from ... More


Introduction

Bryan J. Dalton, John Jeffers, and Stephen M. Barnett

in Phase Space Methods for Degenerate Quantum Gases

Published in print:
2014
Published Online:
April 2015
ISBN:
9780199562749
eISBN:
9780191747311
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199562749.003.0001
Subject:
Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

This chapter provides an overview of the phase space method for treating the physics of systems of bosons and fermions in regimes where quantum degeneracy occurs. Key concepts such as modes, creation ... More


On Frege’s Applications Constraint

Paul McCallion

in Abstractionism: Essays in Philosophy of Mathematics

Published in print:
2016
Published Online:
January 2017
ISBN:
9780199645268
eISBN:
9780191755330
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199645268.003.0015
Subject:
Philosophy, Logic/Philosophy of Mathematics

As Benacerraf famously observed, the natural numbers may be reduced to sets in many different ways and these various set-theoretic reductions seem to be equally appealing. There is often more than ... More


Electricity, Magnetism and Circuits

Patrick Magee and Mark Tooley

in The Physics, Clinical Measurement and Equipment of Anaesthetic Practice for the FRCA

Published in print:
2011
Published Online:
November 2020
ISBN:
9780199595150
eISBN:
9780191918032
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780199595150.003.0008
Subject:
Clinical Medicine and Allied Health, Anesthesiology

Electricity is a broad term that includes a variety of phenomena resulting from the flow (or presence) of electric charge. It is a complex subject, but this chapter will provide the necessary ... More


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