What is Imaginary: Definition and 362 Discussions

An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i2 = −1. The square of an imaginary number bi is −b2. For example, 5i is an imaginary number, and its square is −25. By definition, zero is considered to be both real and imaginary.Originally coined in the 17th century by René Descartes as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of Leonhard Euler (in the 18th century) and Augustin-Louis Cauchy and Carl Friedrich Gauss (in the early 19th century).
An imaginary number bi can be added to a real number a to form a complex number of the form a + bi, where the real numbers a and b are called, respectively, the real part and the imaginary part of the complex number.

View More On Wikipedia.org
Recent contents Top users
  1. A

    Convolution properteis and the imaginary unit

    finding the FT of x(t)=sin(πt) sin(50πt) : ( '*' is the convolution operator) its FT X(Ω)=(1/2π) F(sin(πt)) * F(sin(50πt)) = (1/2π) jπ(δ(Ω+π)-δ(Ω-π)) * (jπ (δ(Ω+π)-δ(Ω-π)) ) (a) from my professor's solution it next goes: = (π/2) (-δ(Ω+π)+δ(Ω-π)) * ((-δ(Ω+π)+δ(Ω-π)) )...
  2. N

    Imaginary Time: Explained for High School Students

    I recently read the book "A Brief History of Time"by Stephen Hawking, and in it he described the concept of imaginary time. It had something to do with the squares of numbers being equal to negative numbers, which were called imaginary numbers. Also, he mentioned being able to travel in...
  3. P

    Why does (e^{i\alpha})^2 always equal 1?

    In solutions to a problem I was working on, I saw that when an expression such as e^{i\alpha} , alpha being an angle (in polar coordinates), was squared, the expression goes to unity, ie (e^{i\alpha})^2=1 But I see no reason to think that \alpha is a multiple of \pi. Could there be any other...
  4. A

    Relationship between Imaginary Time Green's function and Average Occupancy

    Hello everyone, In Fermi Liquid Theory, I'm trying to understand what the relationship is between the Green's function and the average occupancy of levels. In my lecture they gave the relation \left\langle n_k \right\rangle = G(k,\tau\rightarrow 0^+) Anyone know where this comes from...
  5. B

    Imaginary Geometry in Control Systems

    So I have a control systems mid-semester exam coming up and the lecturer has posted up a formula sheet for us. However it is different to past years exams and has a geometry section with the following equations: e^(±jθ)=cos(θ)±jsin(θ) cos(θ)=(e^jθ+e^-jθ)/2 sin(θ)=(e^jθ-e^-jθ)/2j Now I've...
  6. N

    Solving sin z=2: Equating Real and Imaginary Parts

    Homework Statement Solve sin z=2 by (a) equating the real and imaginary parts (b) using the formula for arcsin z. Homework Equations (a) sin z = sin x * cosh y + i * cos x * sinh y arccosh z = log[z + sqrt(z^2 - 1)] (b) arcsin z = -i * log [i * z + sqrt(1 - z^2)] The...
  7. A

    Imaginary Numbers: My Number System & Research Paper

    Well I have developed a number system which allows the existence of imaginary numbers. Please visit it at : http://www.scribd.com/doc/46064105/Math-Paper. An intro of these ideas is presented at :http://www.scribd.com/doc/46117043/Introduction-to-My-Research-Paper Please provide me feedback...
  8. C

    Linearizing imaginary functions

    so i have the function z=(2+i)/(i(-3+4i)) and i need to linearize it to find the Im(z) and Re(z) I get down to z= (-6 +8i -3i -4 )/ (9i +12 +12 -16i) which i then simplify down to z= (5i -10)/(-5i+24) However when solve it i get a different answer from wolfram (from when i plugged...
  9. C

    Behavior of imaginary part

    does the behavior the imaginary part behave in anyway similar to the real part of a holomorphic function. say if the real part if bounded or positive, what can you conclude about the imaginary part.
  10. Z

    Conjugate transpose/real and imaginary parts

    In my linear algebra text it says it's possible to define (for nxn matrix A) A_1^* =\frac{A+A^*}{2} A_2^* =\frac{A-A^*}{2i} so A=A1+iA2 It then asked if this was a reasonable way to define the real and imaginary parts of A. Is there a specific convention to define the real and imaginary parts...
  11. C

    Imaginary Part of Dielectric function

    Can someone please explain the concept of optical losses and its correlation with the imaginary part of the dielectric function in elementary terms. I am confused.
  12. N

    Exponents and Imaginary Numbers

    Hello, I did the integral of a Fourier Transform which resulted in this: A(je^(-jwe^(To+t/2) - je^-jw(T0-t/2))/(1/w) Where A is the amplitude, j the imaginary number, and w is omega or 2*pi*f. My question is, how can this be further simplifier. I am forgetting how to simplify...
  13. D

    A tachyon is a hypothesized particle that has imaginary mass

    A tachyon is a hypothesized particle that has imaginary mass (imaginary numbers) and moves faster than light speed. I don't believe in it because it can't have imaginary mass, what about you?
  14. M

    Can speed be a imaginary number validity of work energy theoram in 1D

    Consider mass m_{1}and m_{2}with position vector (from an inertial frame) \overrightarrow{x_{1}} and \overrightarrow{x_{2}} respectively and distance between them be x_{0}. m_{1}\frac{d^{2}}{dt^{2}}\overrightarrow{x_{1}}=\overrightarrow{F} \Rightarrow...
  15. S

    Thermodynamics imaginary heat machine?

    Homework Statement given the following diagram : http://www.freeimagehosting.net/uploads/… a)mention for every step if it is isothermal / adiabatic / else. does the system receive heat or emit heat. b)given this gas is ideal gas, sketch a diagram with respect to P and V c)calculate work done...
  16. C

    Integrating along the imaginary axis

    I'm really confused with how to prove this result...could anybody help please? Let I_{1} be the line segment that runs from iR (R>0) towards a small semi-circular indentation (to the right) at zero of radius epsilon (where epsilon >0) and I_{2} a line segment that runs from the indentation...
  17. H

    Is an irrational root of a real number imaginary or real?

    We can easily comment the result of a root operation just by the information if the degree of the root is odd or even. But what if the degree of the root (or power) is irrational? For example; -64 ^ \frac{1}{2} \, = \, j8 \,\,\,\,\, (imaginary) -64 ^ \frac{1}{3} \, = \, -4 \,\,\,\,\...
  18. K

    How to eliminate imaginary parts of complex expression?

    Hi, I have a problem on how to convert the imaginary parts of expression into all real parts. For example: x1 = - (a + ib) x2 = (a + ib) x3 = - (a - ib) x4 = (a - ib) My question is that how to express x1, x2, x3 and x4 in terms of real parts only without imaginary parts. I have used...
  19. F

    2nd order nonlinear imaginary partial dif eqn

    Hi all, I am having a hard time solving a partial second order differential equation with an imaginary part. I basically took a much bigger function with real and imaginary parts and simplified it down to this. I also know the solution to a similar equation (shown in image). Any help would...
  20. P

    Imaginary result of integral

    Why and how does this definite integral result in an imaginary solution ? At wolframalpha ... definite integral 1 / [e^x arcsin x] dx from 1 to 10 = 0.156 + .09i Area under such a function should be positive or negative but how does it become imaginary ? Thanks
  21. V

    Solve Imaginary Numbers: 2*EXP(i*pi/3) -> 1+sqt(3)i

    Homework Statement Write 2*EXP(i*pi/3) in the form \alpha + i\beta Answer is given = 1 + sqt(3)i Homework Equations The Attempt at a Solution I'm supposed to turn this exponential form of imaginary number into a standard form in order to solve an ODE. I have no idea how they got 1+sqt(3)i...
  22. R

    Imaginary Field Lines vs True Iron Filings

    My (high school, gah!) textbook gives an experiment: I take a straight current carrying conductor, a cardboard sheet is placed perpendicular to the conductor, so that the conductor passes straight through the sheet, remains perpendicular. Then I use a salt sprinkler to sprinkle iron filings on...
  23. T

    Find the real and imaginary part of sin(4+3i)

    Homework Statement Find the real and imaginary part of sin(4+3i) Homework Equations sinx = \frac{e^z - e^(-z)}{2i} cosx = \frac{e^z + e^(-z)}{2} sin(iy) = i\frac{e^y - e^(-y)}{2} cos(iy) = \frac{e^y + e^(-y)}{2} various trig identities The Attempt at a Solution So I used sin(x+y) trig...
  24. F

    Are there imaginary numbers other than i?

    Are there "imaginary" numbers other than i? I'm taking a class in complex analysis and the professor wrote the textbook so I'm getting most of it. There is one elephant in the room though, and I haven't been able to make office hours to clear it up. Are there "imaginary" numbers other than...
  25. U

    How can imaginary numbers be integrated on the Argand plane?

    How is this possible? \int_{i\infty}^\pi e^{ix} dx = i I mean, I understand that the integral of exp(ix) is -i exp(ix) and then you evaluate that from π to i∞ — but that's exactly it, how does one "draw a line" from (π, 0) on the Argand plane to (0, ∞)? (assuming Argand plane tuples (a, b) ↔...
  26. F

    Exploring the Inner Mechanics of an Imaginary Collapsing Universe

    Imagine an empty universe, where nothing exist and time stands still. Then add lots of stars of equal size, distributed in a symmetry around a spot that we call the center of our universe. Since time has not passed, no curvature (gravity) has propagated to affect any of the other stars. No...
  27. N

    Imaginary Velocity: Exploring Mass & Energy

    So, according to my understanding, m= m_o/√(1-(v^2/c^2 )) gives the mass of an object in respect to the object's original mass and its velocity. I wondered what happened if the mass of an object became lower than the rest mass? [I have no idea how this would happen, but it was a, what if...
  28. C

    Imaginary Numbers and Properties: A Puzzling Case

    -1/1=1/-1 sqrt(-1/1)=sqrt(1/-1) i/1=1/i i*(i/1)=i*1/i i^2/1=i/i -1/1=1 -1=1 <-- Well my conclusion is that properties don't work with imaginary numbers or did i do something wrong?
  29. R

    Imaginary part of dielectric constant.

    in ac fields permittivity becomes complex quantity and has real and imaginary parts. in metals (may be few exceptions but i don't know) imaginary part is always positive and represents loss factor or energy absorbed. why the plot of imaginary part of dielectric constant as function of energy is...
  30. N

    Lowering the imaginary power

    i need to find the equation for the capacitator to lower the imaginary power how they got thered marked equation
  31. C

    Dividing/Multiplying Imaginary Numbers

    Is there a more convenient way to multiply and divide imaginary numbers than converting back and forth from phasors? (I guess I should say "when also having to add and subtract them") I always find AC circuit calculations to be tedious and problem filled when I do it that way. For example...
  32. T

    Imaginary Numbers in a general homogenous solution for a differential equation

    Homework Statement Find the general solution for: y''+2y'+5y=3sin2t The attempt at a solution y''+2y'+5y=3sin2t First step is to find the general solution to the homogenous equation, so skipping 2 steps (letting y=e^rt and dividing) R^2+2r+5 (-2+/- sqroot(4-4*5))/2 =-1 +/- 2i...
  33. S

    Can imaginary numbers have real world applications?

    Why do we have an imaginary number? I don't see it's usefulness. Why dos it matter if we can make up a number that satisfies this equation (\ x^{2}+1=0 )? It must have real world applications that I'm unaware of.
  34. LarryS

    Wave Function: Real vs Imaginary Part

    Wave functions are, of course, almost always complex-valued. In all of the examples that I have seen (infinite square well, etc.), the real part of the wave function and the imaginary part of the wave function are basically the same function (except for a phase difference and possibly a sign...
  35. F

    Extract Real and Imaginary Equation with MATLAB

    Extract Real and Imaginary Equation with MATLAB ! Hi all, How to write M-files that can extract the real and imaginary components, or the magnitude and phase, of a symbolic expression for a complex signal with MATLAB. x(t) = e^j*2*pi*t/16 + e^j*2*pi*t/8 <<< equation example Thanks In...
  36. B

    Imaginary part of complex number (first post)

    Homework Statement C=A*e^(-i*wt)*sin(k*x); A,w,t,k,x are real numbers. Find imaginary part. Homework Equations The Attempt at a Solution Im(C)=cos(wt)-i*sin(wt)
  37. S

    Can the limits of a function be imaginary?

    I was just doing some homework, and I got to thinking about this. So if the limit of a function is an imaginary number, does that mean that the limit does not exist? Or that it does not exist on the xy-plane, or what? I mean...imaginary and complex numbers exist, we just can't graph them...
  38. L

    Evaluating Double Integral Involving Imaginary Error Function

    Homework Statement Evaluate \int\int x^{2}e^{x^{2}y} dx dy over the area bounded by y=x^{-1}, y=x^{-2}, x=ln 4 Homework Equations The Attempt at a Solution \int^{1}_{(ln 4)^{-2}}\int^{y^{-1}}_{y^{\frac{-1}{2}}}x^{2}e^{x^{2}y}dx dy I got this far before I realized that this wasn't a...
  39. C

    Imaginary number -i raised to negative power

    Homework Statement I came across this expression in homework and for the life of me I can't figure out how this evaluates to 0: 1 - ( -i )^-4 = 1 - 1 = 0 I know that i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. I'm just not sure how to treat the negative on the i. Do I just treat i as if it...
  40. P

    Real and imaginary parts of an expression

    can anyone tell me how to get the real and imaginary parts of the following function : (x+ i y)* Log( a+i b) where x, y a and b are all real numbers and i =sqrt (-1). Thanks very much
  41. genphis

    Imaginary numbers negative confusion

    i know this must seem real stupid but if 1 x 1 =1 ( square root wise) how can -1x-1=+1 again square root wise. i am reading fermats last theorum to me if you times negative you increase the negative. i don't see why the imaginary numbers need to be invoked. i understand the argument for...
  42. T

    Imaginary Numbers to Polar form

    Homework Statement (1+i)i = reiθ Find the real values of r and θ. The Attempt at a Solution Well, after doing a similar(ish) question I decided taking logs would be a good start: i loge(1+i) = loger + iθ From here, I have no idea where to go. Using a power of i is killing me...
  43. R

    How do you determine the cube root of (-1+i) in complex numbers?

    I'm doing some practice problems for my mechanics exam tomorrow (good ol' SHM) and I can't solve this for the life of me: Determine: (-1+i)^(1/3) Any help would be greatly appreciated.
  44. S

    Real and imaginary parts of wave function

    A very general question: What do the real and imaginary parts of a wave function correspond to physically? Cheers
  45. S

    Implications of Imaginary Numbers?

    Hello, I have a quick question that I imagine anyone who has studied physics or math at a university can answer rather easily. If not, I apologize in advance for the effort! What is the physical significance of imaginary numbers? I have heard repeatedly that imaginary numbers are relevant...
  46. T

    Imaginary numbers and Imaginary Time

    Imaginary numbers are a lot less mysterious than they sound. They are the result from trying to take the square root of a negative number. They are called “imaginary” because they don’t exist in the normal number system, normally you can’t take the square root of a negative number because the...
  47. LarryS

    Imaginary Time and Path Integrals?

    Steven Hawking writes in A Brief History of Time that time itself must sometimes have an imaginary component in order for Feynman's Sum-Over-Histories approach to work. Why, in a nutshell, is this so? Thanks in advance.
  48. O

    Curved spacetime and imaginary coordinate

    In Misner, Thorne, Wheeler: "Gravitation" it is stated on that "no one has discovered a way to make an imaginary coordinate work in the general curved spacetime manifold" (p.51). Can anyone elaborate on this? Right now, I don't get why it wouldn't work and nothing more is said in the book.
  49. M

    Imaginary parts of GAMMA(1/2+I*y)

    Hi: Does anyone know of an explicit formula for the Real and Imaginary parts of GAMMA(1/2+I*y) as functions of y ? I know about |GAMMA(1/2+I*y)|^2 =Re(GAMMA(1/2+I*y))^2+Im(GAMMA(1/2+I*y))^2= Pi/cosh(Pi*y) but can't find anything about each of the Real and Imaginary terms...
  50. H

    Imaginary parts of roots of unity

    Hi all, What happens when we take the product of the imaginary parts of all the n-roots of unity (excluding 1)? I read somewhere that we get n/(2^(n-1)). How can we prove this? Thanks!
Back
Top