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Geometry of Complex Numbers is an undergraduate textbook on geometry, whose topics include circles, the complex plane, inversive geometry, and non-Euclidean geometry. It was written by Hans Schwerdtfeger , and originally published in 1962 as Volume 13 of the Mathematical Expositions series of the University of Toronto Press .
A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the "imaginary unit", that satisfies i 2 = −1.
More precisely, let f be a function from a complex curve M to the complex numbers. This function is holomorphic (resp. meromorphic) in a neighbourhood of a point z of M if there is a chart ϕ {\displaystyle \phi } such that f ∘ ϕ − 1 {\displaystyle f\circ \phi ^{-1}} is holomorphic (resp. meromorphic) in a neighbourhood of ϕ ( z ...
Lagrange's identity for complex numbers has been obtained from a straightforward product identity. A derivation for the reals is obviously even more succinct. Since the Cauchy–Schwarz inequality is a particular case of Lagrange's identity, [4] this proof is yet another way to obtain the CS inequality. Higher order terms in the series produce ...
If g in G is not in the center its conjugacy class is exactly g and εg. Thus there are 2 N − 1 + 1 conjugacy classes for N odd and 2 N − 1 + 2 for N even. G has | G / [G, G] | = 2 N − 1 1-dimensional complex representations. The total number of irreducible complex representations is the number of conjugacy classes.
In arithmetic, a complex-base system is a positional numeral system whose radix is an imaginary (proposed by Donald Knuth in 1955 [1] [2]) or complex number (proposed by S. Khmelnik in 1964 [3] and Walter F. Penney in 1965 [4] [5] [6]).
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Olariu, Silviu (2002) Complex Numbers in N Dimensions, Chapter 1: Hyperbolic Complex Numbers in Two Dimensions, pages 1–16, North-Holland Mathematics Studies #190, Elsevier ISBN 0-444-51123-7. Poodiack, Robert D. & Kevin J. LeClair (2009) "Fundamental theorems of algebra for the perplexes", The College Mathematics Journal 40(5):322–35.