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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. Imaginary Number Manipulations [7/10/1996] Through some of my manipulations of the imaginary number (i), I somehow demonstrated that -i = i. What is going on here? Imaginary Numbers, Division By Zero [07/03/2000] If we can create a number system using the square root of -1, why can't we do the same with division by 0? Could we define a number to be equal to 1/0? Also, do imaginary numbers have any real-life uses? Imaginary Numbers in Electricity [07/31/1998] How are imaginary numbers used in measuring electricity flow and AC analysis? Imaginary Numbers Raised to Imaginary Numbers [12/29/2001] I input i^i into my TI-89 graphing calculator, and the calculator returned e^(-pi/2). Why? Inconsistency in Complex Logarithms [06/08/2004] I know that ln(1) = 0, but if I evaluate it as ln(-1 * -1) I find that it equals 2pi*i, not 0. How can that be? Infinity and Imaginary Numbers [06/15/2003] Complex variables: dealing with signs and infinity. -i Not a Negative Number [12/12/2001] Proof that both i and -i are square roots of -1. Inverse of arg(z) [10/10/2003] What is the inverse of the function arg(z)? i Power Seen through with ln() [11/25/2010] A student wonders about 1 raised to the square root of negative one. Starting with natural logarithms, Doctor Ali provides some hints to evaluating this quantity raised to an imaginary exponent. Is the Set of Complex Numbers Open or Closed? [09/20/1999] Are the null set and C (the set of complex numbers) open sets, closed sets, both, or neither? Is the Square Root of i^4 Equal to 1 or -1? [02/24/2004] If you take the square root of i to the fourth power, does that equal i to the second power, which is equivalent to -1? Or can you simplify under the radical first and say i to the fourth power is 1 and the square root is then 1? Which approach is correct? Is Zero Considered a Pure Imaginary Number (as 0i)? [12/02/2003] In the complex plane, zero (0 + 0i) is on both the real and pure imaginary axes. Is 0 therefore a pure imaginary number as well as a real number? (i)th Root and (i)th Power [02/13/1999] How do you simplify x to the power of i (and 1/i), where x could be any 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. Manipulation of (Imaginary?) Roots [8/18/1996] Let r,s, and t be the roots of x^3-6x^2+5x-7=0. Find 1/r^2+1/s^2+1/t^2... Maximizing Output of a Restricted Function [11/1/1996] Create a function whose domain is restricted to complex numbers but whose range is real, that is, non-constant, has no constant term, and contains no number greater than 3. Meromorphic Functions [09/18/1998] What is a meromorphic function? Multiplying and Dividing Complex Numbers [07/16/1998] How do you calculate (a+bi)*(c+di) and (a+bi)/(c+di)? Multiplying and Simplifying Complex Binomials [8/19/1996] Why (2+3i)(5-i) is 13 + 13i and not 10 + 13i - 3i^2? Multiplying Radicals of Negative Numbers [07/12/2000] Why do the book and I get different answers for i * sqrt(-98) - sqrt (98)? Can you multiply square roots of negative numbers? The Natural Log of -1 [12/13/2004] I was playing with my calculator, and I found that the natural log of -1 is equal to pi*i. Can you explain why? Natural log of complex numbers [8/31/1996] If I take the equation e^i*Pi=isin(Pi) + cos (Pi) = -1, square both sides, and then take the natural log, I get 2i*Pi=0. How can that be? Non-Real Cube Roots [01/28/2001] Find the two non-real cube roots of -8. Nonreal Roots [2/12/1996] What's a good way of presenting nonreal roots of systems of equations? Would you use a 3d graph with i as the z axis? Operations and Complex Numbers [12/04/2001] How does one do the standard operations such as addition and multiplication? Why was "i" invented and what are its real life uses? What exactly is a complex number? Pi-th Root of -1 [12/15/2000] How can you find the pi-th root of -1? Polar Coordinates From Cartesian Coordinates [07/27/1998] How do you find the polar coordinates from the Cartesian coordinate (3, - 3 sqrt(3))? Polar Number Multiplication and Division [02/13/1999] Proving polar number multiplication and division rules. Polar Representation of Complex Numbers [11/04/2002] Questions about polar and geometric representations of complex numbers. Polar to Rectangular Conversions [4/19/1996] How do I convert "8 cis 30" into rectangular coordinates? Polynomial Degrees and Definition of a Field [03/02/1998] The degree of polynomials added together, and definition of a field. A Primer on Complex Arithmetic [10/29/2002] How do I do problems like (4-10i)(4+10i)? Or problems like this: w=3- 4i; z=3+4i? Products of Complex Conjugates [05/21/1998] Proof that the complex conjugate of a product is equal to the product of the conjugates. Proof of DeMoivre's Theorem [05/01/1997] A typical induction proof: DeMoivre's theorem. Proof of e^(ix) = cos(x) + isin(x) [04/07/1997] I would like to see a rigorous proof that e^(ix) = cos(x) + isin(x) for x = Pi. Proof that e^i(pi) = -1 [06/02/1999] How can it be proven that e^[i(pi)] = -1? And why does it matter? Raising a Negative to an Irrational Power? It Depends [03/16/2011] Why can't you evaluate a negative number raised to an irrational power? Doctor Vogler explains how this depends on definitions of exponentiation, then describes three of them. Real Life Applications of Imaginary Numbers [03/08/1998] Who uses imaginary numbers and why? Why are they so important? Real Plane, Complex Plane [09/16/2002] When does Az + Bz + c = 0 become a straight line? Page: []

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