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 TOPICS This page:   imaginary/complex   numbers    Search   Dr. Math See also the Internet Library:   imaginary/complex     numbers COLLEGE Algorithms Analysis Algebra    linear algebra    modern algebra Calculus Definitions Discrete Math Exponents Geometry    Euclidean/plane      conic sections/        circles      constructions      coordinate plane      triangles/polygons    higher-dimensional      polyhedra    non-Euclidean Imaginary/Complex   Numbers Logic/Set Theory Number Theory Physics Probability Statistics Trigonometry Browse College Imaginary/Complex Numbers Stars indicate particularly interesting answers or good places to begin browsing. Selected answers to common questions:     DeMoivre's theorem. Linear Congruences of Gaussian Integers [04/11/2003] When does the linear congruence zx congruent to 1 (mod m), for z, x, and m all Gaussian integers, have a solution? Also, when do we say that two Gaussian integers are relatively prime? The Logarithm of a Complex Number [07/11/2007] I'm wondering how to calculate the log of a complex number. More specifically, I need to calculate arccosh(x), where 'x' can be any negative number. Log of a Negative Number [11/26/2002] Can you explain how to find the log of a negative number (using complex numbers)? Logs of Complex Numbers [02/11/2004] Give an example showing that Log(z1/z2) does not equal Log(z1) - Log(z2) where z1 and z2 are complex numbers. Matrix Representation of Complex Numbers [11/15/2000] How can I find matrices A and B such that (A+B)^-1 = A^-1 + B^-1? Modular Functions [06/23/1997] What are modular functions? How do they relate to the proof of Fermat's Last Theorem? A More Formal Definition of the Imaginary Unit i [01/12/2004] Many students are taught that the imaginary unit i is equal to the square root of -1. In fact, this informal definition often leads to confusion. Here's a more formal definition of i which goes a long way towards clearing up the misconceptions. On the Complex Differentiability of the Hyperbolic Secant [09/24/2010] A student wonders if sech(z) is complex differentiable, and where. Picking up on the student's familiarity with the Cauchy-Riemann equations, Doctor Jordan uses trigonometric identities to examine the function's real and imaginary parts separately, and reveals the conditions under which the function is holomorphic. Overview of Riemann's Zeta Function and Prime Numbers [04/11/2006] Can you please give an overview of the importance of the Zeta function and finding prime numbers? Why is the Zeta function such a hot topic in the field of looking for prime numbers? Pi-th Root of -1 [12/15/2000] How can you find the pi-th root of -1? Plotting Complex Numbers [07/22/1997] I cannot figure out (1-i)^2i = 2^ie^1.570796. Polynomial Degrees and Definition of a Field [03/02/1998] The degree of polynomials added together, and definition of a field. Quarternions [2/17/1996] Can someone tell me what quarternions are? Raising Integers to Imaginary Powers or Exponents [07/29/2004] I would like to express 2^i in the form a + bi. Real Plane, Complex Plane [09/16/2002] When does Az + Bz + c = 0 become a straight line? Roots in C [01/01/1999] How do you prove the theorem that says that every polynomial has a root in C? Roots of Complex Numbers in Polar Form [09/15/2004] A discussion of why the polar form of a complex number with z not equal to 0 will have two square roots and n distinct nth roots. The Sin(z) Mapping [10/13/1998] If z is a complex number, show that the function z to sin(z) maps the line y = 1 into an ellipse. What does the line x = 1 map to? Sketch a Graph [6/1/1996] z=(3-2i)^1/2, then find z^. Solving x^x = i [12/23/2000] How can I find the value of x if x^x = i? The Square Root of i [05/25/1997] What is the square root of i? Square Root of i [03/30/2001] Our algebra teacher asked us to find the square root of i. I applied the properties of exponents and got (-1)^(1/4). Square Roots in Complex Numbers [11/06/1997] Why in the complex number system does every number have two square roots, when in the real number system we teach that the square root of any positive number is by definition POSITIVE? Strange Result with Euler's Formula--Is There an Error? [11/29/2005] If f is real and not an integer, then Exp(2*pi*i*f) = Exp(2*pi*i)^f = 1^f = 1. Can you tell me where the error is in that work? Does the rule Exp(a*b) = Exp(a)^b not hold in all cases? Sum of Two Squares [05/26/2003] Can you generate the sequence [400, 399, 393, 392, 384, 375, 360, 356, 337, 329, 311, 300]? Transformations in the Complex Plane [12/07/1998] I have questions about complex transformations. Which set in the complex plane is defined by Im(1/z) < -1/2? ... Trigonometric Functions and Complex Numbers [6/27/1995] Is there a solution to the following equation? Sin(a) = 5, where a = x + iy (complex value) Using Complex Conjugate Numbers [09/14/1999] In a program to compute the impedance of a cable, I see equations where a real number is multiplied by a complex number divided by the complex number's conjugate, r*(a+bi)/(a-bi). Can you explain? Was Euler wrong? 2*Pi=0? [03/13/2002] While I was surfing the Internet, I found a site with an interesting proof that shows that 2*Pi = 0 by using Euler's famous equation... Why Is (-n)^fractional Invalid ? [05/26/2003] The problem is with negative numbers being raised to a fractional exponent. y to the x Power [06/10/2003] How can I calculate the real and imaginary parts of any non-integer powers of negative numbers? Z Values in the Mandelbrot Set [04/27/2006] Could you please explain why for the Mandelbrot set the modulus for the resulting z value must remain less than 2? Page: [

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