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 TOPICS This page:   imaginary/complex   numbers    Search   Dr. Math See also the Dr. Math FAQ:   imaginary numbers and   integers,   rational numbers,   irrational numbers Internet Library:   imaginary/complex   numbers HIGH SCHOOL About Math Analysis Algebra    basic algebra    equations/graphs/      translations    linear algebra    linear equations    polynomials Calculus Complex Numbers Calculators/    Computers Definitions Discrete Math    permutations/    combinations Exponents    Logarithms Fibonacci Sequence/   Golden Ratio Fractals Functions Geometry    Euclidean/plane      conic sections/        circles      constructions      coordinate plane      triangles/polygons    higher-dimensional      polyhedra    non-Euclidean    practical geometry    symmetry/tessellations History/Biography Interest Logic Negative Numbers Number Theory Physics/Chemistry Probability Projects Puzzles Sequences/Series Sets Square/Cube Roots Statistics Transcendental   Numbers Trigonometry Browse High School Imaginary/Complex Numbers Stars indicate particularly interesting answers or good places to begin browsing. Selected answers to common questions:     DeMoivre's theorem.     Imaginary numbers in real life. The Riemann Zeta Function [10/11/1998] What are the Riemann hypothesis and the Riemann zeta function? Roots in C [01/01/1999] How do you prove the theorem that says that every polynomial has a root in C? Roots of Unity [04/18/1997] Who devised the formula: xth root of i = (cos (pi/2x))(sin (pi/2x) i)? Simplifying and Working with Imaginary Numbers [04/11/2008] What is the rule for simplifying an expression like sqrt(50)/sqrt(-5)? Do you get i*sqrt(10) or -i*sqrt(10)? Is there a general rule for simplifying imaginary square roots with regard to handling the i? Simplifying Complex Numbers [05/23/2003] Can you explain why the Product Rule doesn't apply to the problem sqrt(-49) x sqrt(-16) ? 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 Complex Variables Equations [09/01/1998] Determine all z such that: 2z + (conjugate of z)^2 = -1+6i ... Solving Quadratics with Imaginary Roots [05/28/1998] Different ways to solve the quadratic equation 3x^2 + 2x + 5 = 0. Solving x^x = i [12/23/2000] How can I find the value of x if x^x = i? Square Root of a Negative Number Squared [01/23/1997] Is Sqrt(-6)^2 equal to 6 or -6? 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 Root of i [03/08/1999] The square root of -1 is i, but what is the square root of i? Square Root of i [12/06/2001] What is the square root of i (square root of the square root of negative one)? Square Roots, Complex Numbers [6/15/1996] x^2 = -9 : I tried taking the square root of both sides. Is this impossible? 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? Square Roots Of Complex Numbers [02/22/1999] Find the square roots of 5-12i. Square Roots of Complex Numbers [03/28/1999] Devise at least two methods for finding the square root of (a+bi). Sum of i [03/23/2002] If the sum as i goes from 1 to n of 2^i is 2^n -1, what is the sum as i goes from 1 to n of 3^i ? Sum of Two Squares [05/26/2003] Can you generate the sequence [400, 399, 393, 392, 384, 375, 360, 356, 337, 329, 311, 300]? Taking the Natural Log of e^(ki) [05/18/2000] How is the natural log defined for e^(ki)? Applying the equation e^(i*2pi) = 1 we get ln[e^(i*2pi)] = ln[1], so i*2pi = 0, which doesn't seem possible. 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) Trigonometry and Complex Numbers [08/05/1998] Simplify (sqrt 3 - i)^7 into the form a + bi using DeMoivre's Theorem. Trigonometry Without Calculators [01/06/1997] How do you find Cos[40] without a calculator? Understanding Imaginary Numbers [09/18/2002] Why is the square root of i negative, and what about when it is cubed? Why is i times i equal to -i? Also, could i be considered real? Using Imaginary Numbers [05/04/2001] Where do we use imaginary numbers in the real world? Using Polar Coordinates to Solve a Complex Number Problem [02/03/2005] Find the number of ordered pairs of real numbers (a,b) such that (a + bi) ^ 2002 = a - bi. 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... The "What?" -- and "So What?" -- of the Complex Conjugate [01/12/2011] A student wonders about the definition, and purpose, of the complex conjugate. Doctor Rick explains, and also offers an extension. What are the solutions to x^4+9 = 0 ? [03/31/2003] How to work with complex numbers when the root is higher than two? What is i^(Googol)? [05/27/2000] What is the value of i raised to the googol power? When i^n Will Be i, -i, or 1 [9/10/1996] To find the value of i, no matter the exponent, divide the exponent by 4 and the remainder will be the equivalent exponent... why does this remainder method work? Why does ln(-x) x>0 equal ln(x)+pi*i? [6/5/1996] Could you please explain why the ln(-x) x>0 equals ln(x)+pi*i? Why Do Imaginary Numbers Exist? [02/17/2001] Why do we have imaginary numbers? Are they useful for anything in the real world? Why do mathematicians like them so much? Why Is (-n)^fractional Invalid ? [05/26/2003] The problem is with negative numbers being raised to a fractional exponent. Why Multiply Two Complex Numbers? [02/20/1999] How do you graph it, and how do you see it in terms of vectors? 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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