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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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