Complex numbers and function. - a historic journey. (From Wikipedia, the free encyclopedia). Contents. Complex numbers Diophantus Italian rennaissance mathematicians Rene Descartes Abraham de Moivre Leonhard Euler Caspar Wessel Jean-Robert Argand Carl Friedrich Gauss.
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- a historic journey
(From Wikipedia, the free encyclopedia)
Made translations of Diophantus’ books
Calculated with negative numbers
Rules for addition, subtraction and multiplication of complex numbers
Lets try to put in the number 4 for x
64 – 60 – 4 = 0
We see that 4 has to be the root (the positive root)
Cardano’s formula gives:
Bombelli found that:
On the analytic representation of direction
The modulus is:
The argument is:
Then (by Wessels discovery):
Ranked beside Euler, Newton and Archimedes as one of history's greatest mathematicians.Carl Friedrich Gauss (1777-1855)
Every complex polynomial of degree n has exactly n roots (zeros), counted with multiplicity.
(where the coefficients a0, ..., an−1 can be real or complex numbers), then there exist complex numbers z1, ..., zn such that
The theorem is usually formulated for closed paths as follows: let U be an open subset of C which is simply connected, let f : U -> C be a holomorphic function, and let γ be a path in U whose start point is equal to its endpoint. Then
Let f(x + iy) = u + iv
Then f is holomorphic if and only if u and v are differentiable and their partial derivatives satisfy the Cauchy-Riemann equations