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2014
For example, the polynomial x 2 + 1 does not have any real roots. A new number, called i, is introduced as a root of that polynomial. The complex numbers are all the numbers of the form a + bi where a and b are real numbers. It is a remarkable fact that every polynomial with real coefficients that is not a constant has a complex root.
Peter Rosenthal +2 more
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For example, the polynomial x 2 + 1 does not have any real roots. A new number, called i, is introduced as a root of that polynomial. The complex numbers are all the numbers of the form a + bi where a and b are real numbers. It is a remarkable fact that every polynomial with real coefficients that is not a constant has a complex root.
Peter Rosenthal +2 more
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On the complexity of numbering operators
Discrete Mathematics and Applications, 1996This is a paper in the classical style of the Russian school concentrated on the estimators of Shannon functions for different complexity measures and function classes. In this case the computing model is a Boolean circuit and the set of functions is the set of all enumeration functions for some subsets of \(\{0,1\}^n\) of restricted size.
I. A. Vikhlyantsev, A. E. Andreev
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1988
Publisher Summary This chapter provides an overview on complex number. These are represented pictorially on rectangular or Cartesian axes. The horizontal (x) axis is used to represent the real axis and the vertical (y) axis is used to represent the imaginary axis. Such a diagram is called an Argand diagram.
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Publisher Summary This chapter provides an overview on complex number. These are represented pictorially on rectangular or Cartesian axes. The horizontal (x) axis is used to represent the real axis and the vertical (y) axis is used to represent the imaginary axis. Such a diagram is called an Argand diagram.
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2010
Numbers of the form \( a + b\sqrt { - 1} \), where a and b are real numbers—what we call complex numbers.appeared as early as the 16th century. Cardan (1501–1576) worked with complex numbers in solving quadratic and cubic equations. In the 18th century, functions involving complex numberswere found by Euler to yield solutions to differential equations.
Donald J. Newman, Joseph Bak
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Numbers of the form \( a + b\sqrt { - 1} \), where a and b are real numbers—what we call complex numbers.appeared as early as the 16th century. Cardan (1501–1576) worked with complex numbers in solving quadratic and cubic equations. In the 18th century, functions involving complex numberswere found by Euler to yield solutions to differential equations.
Donald J. Newman, Joseph Bak
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1989
The fundamental theorem of algebra—that a polynomial of degree k has exactly k complex roots—enables us to get the “right” number of intersections between a curve of degree m and a curve of degree n. However, it is not enough to introduce complex coordinates: getting the right count of intersections also requires us to adjust our viewpoint in two other
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The fundamental theorem of algebra—that a polynomial of degree k has exactly k complex roots—enables us to get the “right” number of intersections between a curve of degree m and a curve of degree n. However, it is not enough to introduce complex coordinates: getting the right count of intersections also requires us to adjust our viewpoint in two other
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1989
The insight into algebraic curves afforded by complex coordinates—that a complex curve is topologically a surface—has important repercussions for functions defined as integrals of algebraic functions, such as the logarithm, exponential, and elliptic functions.
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The insight into algebraic curves afforded by complex coordinates—that a complex curve is topologically a surface—has important repercussions for functions defined as integrals of algebraic functions, such as the logarithm, exponential, and elliptic functions.
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Complex Numbers in Algebra [PDF]
, 1989 The next three chapters revisit the topics of algebra, curves, and functions, observing how they are simplified by the introduction of complex numbers. That’s right: the so-called “complex” numbers actually make things simpler. In the present chapter we see where complex numbers came from (not from quadratic equations, as you might expect, but from ...
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Fuzzy Sets and Systems, 1989
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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An Approach to Complex Numbers
The Mathematical Gazette, 1970The debate about how best to introduce complex numbers is no doubt perennial. (See for instance [1] for a brief survey of some of the possibilities.) One flaw which many presentations have is that they appear somewhat arbitrary. For example one can define complex numbers as polynomial residues to the modulus y2 +1: this definition ought to provoke the ...
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1999
The field of complex numbers ℂ is the set of all ordered pairs (a, b)where a and b are real numbers and where addition and multiplication are defined by: $$\begin{gathered} (a,b) + (c,d) = (a + c{\text{, }}b + d) \hfill \\ (a,b)(c,d) = (ac - bd,{\text{ }}bc + ad). \hfill \\ \end{gathered}$$ We will write a for the complex number (a, 0). In fact,
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The field of complex numbers ℂ is the set of all ordered pairs (a, b)where a and b are real numbers and where addition and multiplication are defined by: $$\begin{gathered} (a,b) + (c,d) = (a + c{\text{, }}b + d) \hfill \\ (a,b)(c,d) = (ac - bd,{\text{ }}bc + ad). \hfill \\ \end{gathered}$$ We will write a for the complex number (a, 0). In fact,
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