Complex Numbers

• Complex Numbers and the Mandelbrot Set [Ong, 12/15/1996]
  Can you tell me about the Mandelbrot set?

• Complex Roots [Scott, 11/1/1994]
  We know it is possible to look at the graph of a polynomial and tell a great deal about its real roots by looking at the x-intercepts. What can be discovered about a polynomial's complex roots by looking at the graph? There seem to be some interesting "wiggles" at locations that appear to be related to the "average" of the complex pairs. It appears that the "wiggle" of these graphs is always influenced by the complex roots. What we are trying do is develop a graphing technique that will let us find the complex roots from the real graph. (Contributions by Profs. Conway and Maurer.)

• Defining Complex Numbers [JSnTW, 12/01/1997]
  Can you explain complex numbers simply?

• Graphing Complex and Imaginary Numbers [Liberal, 10/23/1997]
  How do you graph imaginary numbers?

• Imaginary Numbers in Real Life [Chris, 11/20/2001]
  We have been discussing when we would use the imaginary number i in real life.

• Square Root of a Negative Number [Pedlar, 01/25/1997]
  Is it possible to find the square root of a negative number and, if so, to what number system do these square roots belong?

• Visualizing Complex Numbers [Dhoot, 03/19/2001]
  What are imaginary numbers, and how are they used? How is the square root of -1 possible?

• What Are Imaginary Numbers? [Telis, 7/24/1996]
  What are imaginary numbers, what is their purpose, and how are they used?

• What is i? [Toeti, 9/24/1995]
  A student asks a question about the relationship of imaginary numbers to the SQRT of -1.

• Nonreal Roots [Thompson, 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?

• Complex Numbers [Spears, 3/18/1996]
  Why are complex numbers important?

• Polar to Rectangular Conversions [Glick, 4/19/1996]
  How do I convert "8 cis 30" into rectangular coordinates?

• Why does ln(-x) x>0 equal ln(x)+pi*i? [Tolbert, 6/5/1996]
  Could you please explain why the ln(-x) x>0 equals ln(x)+pi*i?

• Complex Equations [van Schaik, 6/14/1996]
  Let z be an element of the complex numbers...

• Square Roots, Complex Numbers [Ken, 6/15/1996]
  x^2 = -9 : I tried taking the square root of both sides. Is this impossible?

• Imaginary Number Manipulations [Gleyzer, 7/10/1996]
  Through some of my manipulations of the imaginary number (i), I somehow demonstrated that -i = i. What is going on here?

• Functions of Imaginary Numbers [Pooh, 7/31/1996]
  Does (ln i) itself exist? Where does e^iA = cos A + i sin A come from?

• Manipulation of (Imaginary?) Roots [Michael, 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...

• Multiplying and Simplifying Complex Binomials [Hatcher, 8/19/1996]
  Why (2+3i)(5-i) is 13 + 13i and not 10 + 13i - 3i^2?

• Natural log of complex numbers [Halpert, 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?

• Complex Numbers: What and Why? [Martin, 9/1/1996]
  What is a complex number? How does it work? What sort of problems do complex numbers solve? What are some examples?

• When i^n Will Be i, -i, or 1 [Guito, 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?

• Complex Numbers in Second Degree Equation [Ekdahl, 9/15/1996]
  How do I solve z^2+(4-2i)z-8i = 0?

• Complex Numbers and Euler's Equation [Chu, 11/10/1996]
  If x = Cos A + iSin A and y = Cos B + iSin B, show that [(x+y)(xy-1)]/[(x-y)(xy+1)] = (Sin A + Sin B)/(Sin A - Sin B).

• Maximizing Output of a Restricted Function [Lovelace, 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.

• Complex Polynomial [Johansson, 11/22/1996]
  How do I find a solution to this complex equation...

• Complex Numbers and Trigonometry [Smith, 12/25/1996]
  If x is a real number, ArcSin(Sin(x)) = x. If z is a complex number, ArcSin(Sin(z)) does not equal z. Why?

• Trigonometry Without Calculators [Gollapalli, 01/06/1997]
  How do you find Cos[40] without a calculator?

• Fractals, Complex Numbers, and Chaos [Ngee, 01/20/1997]
  Do fractals have anything to do with complex numbers? Do they have something to do with chaos?

• Euler Equation [Davis, 01/21/1997]
  What is the meaning behind e^(pi*i) = -1?

• Square Root of a Negative Number Squared [Smith, 01/23/1997]
  Is Sqrt(-6)^2 equal to 6 or -6?

• i^i [Solovey, 04/03/1997]
  What is i to the power of i ?

• Proof of e^(ix) = cos(x) + isin(x) [Graf, 04/07/1997]
  I would like to see a rigorous proof that e^(ix) = cos(x) + isin(x) for x = Pi.

• Roots of Unity [Volante, 04/18/1997]
  Who devised the formula: xth root of i = (cos (pi/2x))(sin (pi/2x) i)?

• Proof of DeMoivre's Theorem [Liya, 05/01/1997]
  A typical induction proof: DeMoivre's theorem.

• The Square Root of i [Leif, 05/25/1997]
  What is the square root of i?

• Euler Equation [Dale, 09/13/1997]
  Does the Euler equation still work if we decide to work in degrees? Is it arbitrary?

• Complex Numbers, Trig Functions and Roots of 1 [Knobler, 10/30/1997]
  I'm convinced that, for an arc of length x in radians: (cos x + i sin x)^(2 pi / x) = 1... It's cool but why does it work?

• Cube Roots of Numbers [Martin, 11/05/1997]
  If you take i (sqrt(-1)), the cube root is -i, but since x^3 = i is degree three there should be three different values of x. What are they?

• Square Roots in Complex Numbers [Jurd, 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?

• Complex Numbers [Zykorie, 11/23/1997]
  Do all complex numbers have a multiplicative inverse?

• Applications of Complex Numbers [Stewart, 12/06/1997]
  I need to find some common applications of complex numbers, like the square root of negative one, in our society today.

• Complex Numbers Problems [Abecina, 12/23/1997]
  On an Argand diagram the points P and Q represent the numbers z1 and z2 respectively...

• Cube Root of 1 [Laxer, 01/07/1998]
  The cube root of 1 has three roots. I know one is +1. Can you show me the steps to find the other two?

• Euler's Formula [Wolf, 01/27/1998]
  Can you derive trig formulas using a combination of Euler's Equation and the unit circle?

• Polynomial Degrees and Definition of a Field [Metelli, 03/02/1998]
  The degree of polynomials added together, and definition of a field.

• Real Life Applications of Imaginary Numbers [Shah, 03/08/1998]
  Who uses imaginary numbers and why? Why are they so important?

• Exploring i [Lilley, 03/16/1998]
  Does i^0 equal 1? What is i to any power?

• The Fourth Root of -1 [Sweeny, 03/27/1998]
  How do you find the fourth root of -1? The square root of i?

• The Absolute Value of a Complex Number [Nadel, 05/06/1998]
  Why is |a + bi| equal to the square root of a^2 + b^2?

• Products of Complex Conjugates [Garrett, 05/21/1998]
  Proof that the complex conjugate of a product is equal to the product of the conjugates.

• Solving Quadratics with Imaginary Roots [Fernandez, 05/28/1998]
  Different ways to solve the quadratic equation 3x^2 + 2x + 5 = 0.

• Multiplying and Dividing Complex Numbers [Michael, 07/16/1998]
  How do you calculate (a+bi)*(c+di) and (a+bi)/(c+di)?

• Polar Coordinates From Cartesian Coordinates [Cherry, 07/27/1998]
  How do you find the polar coordinates from the Cartesian coordinate (3, -3 sqrt(3))?

• Imaginary Numbers in Electricity [Munoz, 07/31/1998]
  How are imaginary numbers used in measuring electricity flow and AC analysis?

• Trigonometry and Complex Numbers [Wang, 08/05/1998]
  Simplify (sqrt 3 - i)^7 into the form a + bi using DeMoivre's Theorem.

• Factoring 13 with Complex Numbers [Scomazzon, 08/11/1998]
  How do you show that 13 is not prime using imaginary numbers? We know that 13 = (3 + 2i)(3 - 2i), but how do you do this in general?

• Graphing Complex Functions [Liddle, 08/11/1998]
  In the quadratic equation y = x^2 + 5x + 12, when y = 0 has no solutions, where (if anywhere) do these numbers lie on the graph of this equation?

• DeMoivre's Formula [Lukens, 08/13/1998]
  Can you explain DeMoivre's Theorem?

• Solving Complex Variables Equations [Zumpolle, 09/01/1998]
  Determine all z such that: 2z + (conjugate of z)^2 = -1+6i ...

• Meromorphic Functions [Desabrais, 09/18/1998]
  What is a meromorphic function?

• The Riemann Zeta Function [Bandel, 10/11/1998]
  What are the Riemann hypothesis and the Riemann zeta function?

• The Sin(z) Mapping [Hull, 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?

• CADAEIBFEC and Other NCTM Questions [Moreno, 10/27/1998]
  CADAEIBFEC is a mnemonic for an important piece of mathematical information. What is it?

• Transformations in the Complex Plane [Tootsie, 12/07/1998]
  I have questions about complex transformations. Which set in the complex plane is defined by Im(1/z) < -1/2? ...

• Complex Analytic Functions [Ecgal, 12/08/1998]
  I'm trying to find out if abs(z)*(conjugate z) is analytic using the Cauchy-Riemann equations.

• Roots in C [Fraiman, 01/01/1999]
  How do you prove the theorem that says that every polynomial has a root in C?

• Complex Numbers - Finding Values [Wilson, 02/13/1999]
  I need some help with complex numbers.

• Polar Number Multiplication and Division [Matt , 02/13/1999]
  Proving polar number multiplication and division rules.

• (i)th Root and (i)th Power [Evan, 02/13/1999]
  How do you simplify x to the power of i (and 1/i), where x could be any number?

• Find a Point Above a Segment [Noam, 02/16/1999]
  Compute segment rotation in the complex coordinates by multiplying vectors.

• Why Multiply Two Complex Numbers? [Angkit, 02/20/1999]
  How do you graph it, and how do you see it in terms of vectors?

• Square Roots Of Complex Numbers [Max, 02/22/1999]
  Find the square roots of 5-12i.

• Square Root of i [Nadine, 03/08/1999]
  The square root of -1 is i, but what is the square root of i?

• Square Roots of Complex Numbers [Jaimee, 03/28/1999]
  Devise at least two methods for finding the square root of (a+bi).

• Euler Equation and DeMoivre's Theorem [Anthony, 05/18/1999]
  Do you have a proof of the equation e^(i*Pi) + 1 = 0?

• Proof that e^i(pi) = -1 [White, 06/02/1999]
  How can it be proven that e^[i(pi)] = -1? And why does it matter?

• Fractional Exponents and Complex Roots [Hanson, 06/11/1999]
  Does z^(a/b) = (z^a)^(1/b) or (z^1/b)^a?

• Evaluating e^(i*pi) and i^i [Balen, 07/25/1999]
  How can I evaluate e^(ipi) and i^i?

• The ith Root of -1 [Ford, 09/16/1999]
  Why does the ith root of -1 equal 23.14069...?

• Is the Set of Complex Numbers Open or Closed? [Ricardo, 09/20/1999]
  Are the null set and C (the set of complex numbers) open sets, closed sets, both, or neither?

• Complex Roots of a Quadratic Equation [Aoun, 10/25/1999]
  If 1+i is a root of the equation z^2 + (a+2i)z + 5+ib = 0, and a and b are real numbers, how can I determine the values of a and b?

• Complex Numbers in Quadratic Equations [Oya, 11/09/1999]
  How are imaginary numbers used in solving quadratic equations? How can solutions of this type be represented graphically?

• Determining the Equation of a Circle [Cody, 05/12/2000]
  How can I determine which of five equations describes the set of all points (x,y) in the coordinate plane that are a distance of 5 from the point (-3,4)?

• Absolute Values and Imaginary Numbers [Khaine, 05/17/2000]
  Could the solution to |x|= -8 be an imaginary number? Since no absolute value can be negative, this [like sqrt(-1)] cannot be solved.

• Taking the Natural Log of e^(ki) [Thomas, 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.

• What is i^(Googol)? [Feest, 05/27/2000]
  What is the value of i raised to the googol power?

• Imaginary Numbers, Division By Zero [Marc, 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?

• Multiplying Radicals of Negative Numbers [Amy, 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?

• Complex Numbers to Complex Powers [Lee, 10/19/2000]
  Can Euler's equation be used to find any number raised to a complex power? How is it possible that all real numbers raised to an imaginary power map to the complex unit circle?

• Conjugate Roots of Complex Numbers [Bailey, 12/01/2000]
  If you take the nth root of a complex number, is there a way to tell if there will be any conjugate roots among the n answers?

• Pi-th Root of -1 [Mraz, 12/15/2000]
  How can you find the pi-th root of -1?

• Solving x^x = i [Pledger, 12/23/2000]
  How can I find the value of x if x^x = i?

• Complex Conjugate Roots of Real Polynomials [Swank, 01/11/2001]
  How can I prove that if a polynomial p(x) with real coefficients has a complex number as a root, then its complex conjugate must also be a root?

• Non-Real Cube Roots [Nicole, 01/28/2001]
  Find the two non-real cube roots of -8.

• Why Do Imaginary Numbers Exist? [Cara, 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?

• History of Imaginary Numbers [Raley, 03/09/2001]
  Who invented imaginary numbers?

• Square Root of i [Javier, 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).

• Complex Numbers [Rosie, 04/08/2001]
  What exactly is the complex number system comprised of?

• Closed Operations for Negative Irrationals [Lisa, 04/28/2001]
  What set of operations is closed under negative irrational numbers?

• Using Imaginary Numbers [Srini, 05/04/2001]
  Where do we use imaginary numbers in the real world?

• Complex Numbers: Subtraction, Division [April, 08/12/2001]
  How do you divide imaginary numbers like a+bi/a-bi?

• i and (-1) with Multiple Powers [Kenneth, 08/09/2001]
  What is the order in which i and (-1) should be raised when using multiple powers? Why is it possible to obtain so many different values?

• The Imaginary Number J [Sam, 09/14/2001]
  One of my teachers says you cannot find the square root of a minus number, especially minus one. I say that the square root of minus one equals J and is an imaginary number....

• Finding Roots of Polynomials with Complex Numbers [Ed, 09/27/2001]
  I read in the archives that you can find the roots of 3rd or higher- degree polynomials with complex numbers...

• Complex Powers [Bill, 09/28/2001]
  How do I show that abs(z^i) is less than exp^pi where z is a complex number not equal to 0?

• Operations and Complex Numbers [Matt, 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?

• Square Root of i [Morgan, 12/06/2001]
  What is the square root of i (square root of the square root of negative one)?

• -i Not a Negative Number [Mark`, 12/12/2001]
  Proof that both i and -i are square roots of -1.

• Imaginary Numbers Raised to Imaginary Numbers [Dan, 12/29/2001]
  I input i^i into my TI-89 graphing calculator, and the calculator returned e^(-pi/2). Why?

• Are Exponents Associative? [Lucas, 02/05/2002]
  How Much Is 2^(i) or X^(i) ?

• Derivation of Sum/Difference of Sine, Cosine, Tangent [Meredith, 02/16/2002]
  How can I find the derivation of the sum/difference of sine, cosine, and tangent?

• Was Euler wrong? 2*Pi=0? [Warren, 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...

• Sum of i [Julie, 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 ?

• Cosine 20 Degrees [Matthew, 03/24/2002]
  What is the exact value of cosine 20 degrees?