Imagining Numbers: (particularly the square root of minus fifteen)

Sprednja platnica
Farrar, Straus and Giroux, 1. feb. 2004 - 288 strani
1 Recenzija

How the elusive imaginary number was first imagined, and how to imagine it yourself

Imagining Numbers (particularly the square root of minus fifteen) is Barry Mazur's invitation to those who take delight in the imaginative work of reading poetry, but may have no background in math, to make a leap of the imagination in mathematics. Imaginary numbers entered into mathematics in sixteenth-century Italy and were used with immediate success, but nevertheless presented an intriguing challenge to the imagination. It took more than two hundred years for mathematicians to discover a satisfactory way of "imagining" these numbers.

With discussions about how we comprehend ideas both in poetry and in mathematics, Mazur reviews some of the writings of the earliest explorers of these elusive figures, such as Rafael Bombelli, an engineer who spent most of his life draining the swamps of Tuscany and who in his spare moments composed his great treatise "L'Algebra". Mazur encourages his readers to share the early bafflement of these Renaissance thinkers. Then he shows us, step by step, how to begin imagining, ourselves, imaginary numbers.

Iz vsebine knjige

 

Zadetki 1–5 od 48

Stran
7 In Plato's Meno, Socrates asks Meno's young slave to construct a square
whose area is twice the area of a given square. Here is the diagram that Socrates
finally draws to help his interlocutor answer the question: The profile of this
diagram ...
Stran
... ordinary-sounding problem that might move a sixteenth-century mathematician
to use quantities like to effect its (theoretical, but not practical) solution. Suppose
that someone has given you the following information about an aquarium tank.
Stran
25 This is a perfectly valid conclusion, given that Chuquet was seeking “ordinary
number” solutions to his problem. To get a sense, though, of why Chuquet might
have been led to think of such expressions—deemed by him impossible—as ...
Stran
We sometimes ask questions in full expectation that the answer will be easily
given. “Do you want some more pie?” But we pose (throw out) problems for
solution only if we expect that something of a mental stretch is required to come
up with ...
Stran
... (in which you assume— contrary to fact, as it turns out—that there exists an
equation of the form √2 = A/B); to be working in a terrain, an Erewhon, that
allows for rational investigation (given its own rules) and reveals its own
impossibility.

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LibraryThing Review

Uporabnikova ocena  - rcorfield - LibraryThing

This is an interesting little book and I thoroughly enjoyed it. It sets out to help the user understand and, more importantly, visualise, imaginary numbers (i.e. the square-root of -1). The author ... Celotno mnenje

LibraryThing Review

Uporabnikova ocena  - fpagan - LibraryThing

Literally a case of "mathematics for poets." The gentlest of intros to imaginary and complex numbers. It certainly doesn't explain things like raising one complex number to the power of another. Celotno mnenje

Izbrane strani

Vsebina

Title Page
What is a square root?
NOTES
The problem of describing how we imagine
Permission
Forced conventions or definitions?
What kind of law is the distributive law?
ECONOMY OF EXPRESSION 23 Charting the plane
Back to Bombellis puzzle
Interviewing Bombelli
PUTTING GEOMETRY INTO NUMBERS 45 Many hands
Imagining the dynamics of multiplication by
Writing and singing
The power of notation
A plane of numbers
Thinking silently out loud

The geometry of qualities
The spareness of the inventory of the imagination
JUSTIFYING LAWS 26 Laws and why we believe them
Defining the operation of multiplication
The distributive law and its momentum
Virtuous circles versus vicious circles
So why does minus times minus equal plus?
PART II
BOMBELLIS PUZZLE 31 The argument between Cardano and Tartaglia
Bombellis LAlgebra
I have found another kind of cubic radical which is very different from the others
Numbers as algorithms
The name of the unknown
Species and numbers
STRETCHING THE IMAGE 37 The elasticity of the number line
To imagine versus to picture
The inventors of writing
Arithmetic in the realm of imaginary numbers
The absence of time in mathematics
Questioning answers
The complex plane of numbers
Telling a straight story
SEEING THE GEOMETRY IN THE NUMBERS 53 Critical moments in the story of discovery
What are we doing when we identify one thing with another?
Song and story
Multiplying in the complex plane The geometry
behind multiplication by by 1 + and by 1 +2
What is a number?
So how can we visualize multiplication in the complex plane?
PART III
THE LITERATURE OF DISCOVERY OF GEOMETRY IN NUMBERS 60 These equations are of the same form as the equations for cosines though ...
A few remarks on the literature of discovery and the literature of
UNDERSTANDING ALGEBRA VIA GEOMETRY 62 Twins
Dal Ferros expression as algorithm
Form and content
But
THE QUADRATIC FORMULA
BIBLIOGRAPHY
PERMISSIONS ACKNOWLEDGMENTS
Avtorske pravice

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Imagining Numbers: (Particularly the Square Root of Minus Fifteen)
Barry Mazur
Omejen predogled - 2004
Imagining Numbers: (particularly the Square Root of Minus Fifteen)
Barry Mazur
Omejen predogled - 2003
Imagining Numbers: (particularly the Square Root of Minus Fifteen)
Barry Mazur
Omejen predogled - 2004
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Pogosti izrazi in povedi

algebra allowed angle answer asked Bombelli calculation called Cardano Cartesian century complex numbers complex plane consider coordinates course cube roots cubic equations definition described discussion distributive law early equal equation example experience expression fact Ferro’s formula formula further geometric give given hand horizontal idea identified imaginary numbers imagination indicator interpret kind later least letter look magnitude mathematician mathematics means method minus multiplication nature negative negative numbers notation number line objects operation origin perhaps phrase picture plane play poem poetry polynomial positive whole numbers powers present Press problem published quantities question radicals real numbers referred rotation rules seems sense side simple solutions solve specific square root structure suggest things thought transformation translation treatise tulip understanding unknown values visualize writing written yellow

O avtorju (2004)

Barry Mazur does his mathematics at Harvard University and lives in Cambridge, Massachussetts, with the writer Grace Dane Mazur.

Bibliografski podatki

QR code for Imagining Numbers
NaslovImagining Numbers: (particularly the square root of minus fifteen)
AvtorBarry Mazur
ZaložnikFarrar, Straus and Giroux, 2004
ISBN1429931469, 9781429931465
Dolžina288 strani
  
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