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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.



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

Complex Numbers: Graphing Functions with Imaginary Numbers [01/21/2003]
How can you tell if a graph includes imaginary numbers?

Complex Roots [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 [12/01/1997]
Can you explain complex numbers simply?

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

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

Square Root of a Negative Number [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 [03/19/2001]
What are imaginary numbers, and how are they used? How is the square root of -1 possible?

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

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

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

Absolute Values and Imaginary Numbers [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.

An Algebraic Derivation of the Square Root of i [10/18/2003]
A series of algebraic steps show how to find the square root of i and why it is equal to +/- [(sqrt(2)/2) + i(sqrt(2)/2)].

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

Are Exponents Associative? [02/05/2002]
How much is 2^(i) or x^(i) ?

Asin/acos/atan for Complex Numbers [3/27/1996]
How do you find asin(x+iy), acos(x+iy), and atan(x+iy)?

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

Can an Imaginary Number Be a Valid Answer? [01/12/2006]
The solution to x^2 + 1 = 0 is +/- SQRT(-1) or +/- i. But if i does not exist, how can it be an answer?

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

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

Complex Conjugate Roots of Real Polynomials [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?

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

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

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

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

Complex Numbers [03/11/2003]
z^4 + z^3 + z^2 + z + 1 = 0

Complex Numbers: 0^i [11/26/2002]
What does 0^i equal?

Complex Numbers and Euler's Equation [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).

Complex Numbers and Trigonometry [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?

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

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

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

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

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

Complex Numbers to Complex Powers [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?

Complex Numbers, Trig Functions and Roots of 1 [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?

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

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

Complex Powers [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?

Complex Powers [04/10/2002]
Given e^(2*pi*i/2*pi*i) = e^(1) = e ... 1^(1/2*pi*i) has to be equal to e. I am having trouble proving this last step.

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