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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 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.

Complex Roots of a Quadratic Equation [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?

Conjectures vs. Hypotheses [01/12/1999]
What is the difference between the terms 'conjecture' and 'hypothesis'? Should the Riemann hypothesis be the Riemann conjecture?

Conjugate Roots of Complex Numbers [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?

Convergence of Product of Sines [10/17/2003]
Prove that (sin(pi/n))*(sin(2pi/n))*...*(sin((n-1)pi/n)) = n/(2^(n-1)) for n >= 2.

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

Cube Root of 1 [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?

Cube Roots of Numbers [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?

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

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

Determining the Equation of a Circle [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)?

Disappointed by Definitions: Where's the Deduction? [10/17/2015]
Finding definitions instead of proofs, an undergraduate enamored of logico-deductive reasoning reconsiders his choice of college major — and in the process, raises existential questions about mathematics. After several exchanges that delve into imagination, intellectual maturation, and whether math is discovered or invented, Doctor Peterson pulls the disenchanted student back from the brink.

Dividing by Complex Numbers [05/04/2003]
Divide: (8+4i)/(1+2i)

e^(pi*i) = -1: A Contradiction? [8/17/1996]
I know that e^(i*Pi) = -1. But squaring and taking a natural log of both sides, you get 2*i*Pi = 0. Please explain.

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

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

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

Euler's Faith and Folly [03/25/2011]
What did Euler wrongly assume when he first derived pi^2/6 from the infinite sum of 1/n^2? Doctor Jordan reveals two missteps initially committed by the famous mathematician on this now-classic result.

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

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

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

Factoring 13 with Complex Numbers [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?

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

Finding Roots of Complex Numbers [09/01/2005]
How do you find the nth roots of a complex number a + bi?

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

Finding the Square Root of a Quadratic Function [10/30/2002]
Find the square root of 3+4i.

Find the Flaw [08/02/2001]
I don't understand where the following proof goes wrong...

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

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

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

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

Geometric Interpretation of Inequality [8/23/1996]
If z1 and z2 are complex numbers, interpret geometrically the inequality | z1 + z2 | < | z1 | + | z2 |.

Graphing Complex and Real Numbers [02/26/2003]
Since on the Cartesian plane we can only graph real zeros and real solutions, are we truly graphing the function when we omit the complex and imaginary zeros and solutions?

Graphing Complex Functions [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?

Graphing Sums of Complex Numbers [12/06/2006]
Why is it that when two complex numbers are graphed, then the sum of those two complex numbers is graphed (all of this on the same graph), and then lines are drawn to connect the parts of each graph farthest from the origin, a parallelogram is formed?

Graph of y = (-n)^x [01/17/2005]
I am curious as to what the graph of y = (-n)^(x) would look like, such as y = (-2)^x. My graphing calculator will not show the graph as anything, but displays many real values in the table of values.

History of Complex Numbers [12/12/2005]
Why were complex numbers invented?

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

i and (-1) with Multiple Powers [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?

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

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