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Browse High School Imaginary/Complex Numbers
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Selected answers to common questions:
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
- 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
- (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
- Log of a Negative Number [11/26/2002]
Can you explain how to find the log of a negative number (using
- 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, -
- 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-
- Products of Complex Conjugates [05/21/1998]
Proof that the complex conjugate of a product is equal to the product of
- 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
- 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
- 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?