See also the
Dr. Math FAQ:
Browse High School Imaginary/Complex Numbers
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
Imaginary numbers in real life.
- The Imaginary Number J [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....
- 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.
- Making Sense of the Imaginary [05/30/2017]
Confident in the product rules, a middle school student tries to make sense of square
roots of negative numbers. With analogies to other ways in which she has already
extended the number system — to accommodate negatives and fractions
— Doctor Peterson paves the way to the complex plane.
- 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?
- The Oddity of Negative Bases Raised to Fractional Powers [11/29/2016]
An adult seeks greater understanding of the imaginary numbers that arise when
multiplying identical roots of negative bases. Doctor Ali clears up the ambiguity by
switching to polar notation, then introduces another imaginary thought experiment.
- 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?