Imagining Numbers: (particularly the square root of minus fifteen)Farrar, Straus and Giroux, 1. feb. 2004 - 288 strani How the elusive imaginary number was first imagined, and how to imagine it yourself |
Iz vsebine knjige
Zadetki 1–5 od 24
... . Back to Bombelli's puzzle. 44. Interviewing Bombelli. 9 - PUTTING GEOMETRY INTO NUMBERS 45. Many hands. 46. Imagining the dynamics of multiplication by : algebra and geometry mixed. 47. Writing and singing. 48.The power.
... hand, to make a few calculations (multiplying small numbers, mostly). The operation of multiplication itself is something we will be looking at. There are three standard ways of denoting the act of multiplication: by a cross x, by a ...
... Using the equation that asserts this, ? try your hand at estimating √2. Is it smaller than Do you see why √3· √5 = Square roots are often encountered geometrically, as lengths of lines. We will see shortly, for example, that √2 is.
... hand at guessing how to achieve the aim before taking a look at the proof. Prelude Suppose that √2 could be expressed as a ratio of whole numbers. That is, suppose we could write an equation of the form (2.1) where A and B are whole ...
... hand makes it easy to approximate √2 as accurately as one might wish, by fractions, and on the other makes it relatively easy to see that √2 itself is not a fraction. It would take us too far afield to develop this theme, but I can't ...
Vsebina
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 | |
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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 - 2003 |