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.

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  1. PeteyCoco

    Programming Function With Cut On Negative Imaginary Axis

    The Function To Be Programmed \sigma_m=\frac{4(n_r^2 -1)J_m(n_r k R)}{\pi^2kD_m^+(kR)D_m^-(kR)} where D_m(z)=n_rJ'_m(n_rz)H_m(z)-J_m(n_rz)H'_m(z). The '+'/'-' superscripts indicate the limits as z approaches the branch cut, which lies along the negative imaginary half axis, from the positive...
  2. 159753x

    Intuition for imaginary part of Fourier Transformation?

    Hi there, I'm having trouble understanding the Fourier transform of a function where the result in the frequency domain has imaginary components. For example, if you take the Fourier transform of Sin[t] , the result is I Sqrt[\[Pi]/2] DiracDelta[-1 + \[Omega]] - I Sqrt[\[Pi]/2]...
  3. B

    MHB Evaluating the log of an imaginary number

    needed help to solve this math home work? Please help.. What is the value of log(i*pi/2) ? I know the answer is "i*pi/2", but don't know the procedure to solve it. Please help me. Thanks a lot in advance.
  4. D

    MHB Complex numbers finding the Imaginary part

    is there a way to solve this without performing the tedious expansion of $(1+j)^8$? here's the problem $\text{Im}[(1+j)^8(x+jy)]$
  5. powerof

    Help with matrix form of the imaginary unit, i

    While investigating more about complex numbers today I ran across the 2x2 matrix representation of a complex number, and I was really fascinated. You can read what I read here. As I understand it, you write z in its binomial form but instead of "1" you use the identity matrix, I, and for i...
  6. kq6up

    What Went Wrong with Imaginary Eigenvalues?

    Homework Statement Multiply the matrices to find the resultant transformation. $$x\prime =2x+5y\\ y'=x+3y $$ and $$ x\prime \prime =x\prime -2y\prime \\ y\prime \prime =3x\prime -5y\prime $$ Homework Equations $$Mr=r\prime$$ The Attempt at a Solution I get imaginary eigenvalues of -i and...
  7. S

    Possible complex angles with no imaginary periodicity

    Trying the already known equation, sin^2(x) + cos^2(x) = 1 i wondered what would happen if i took that either sin(x) or cos(x) squared equalled a number greater than 1, so when i plugged in sin(x) as 5/3 i got cos(x) 4i/3 , went to euler's equation and added the results, then put the...
  8. dwn

    Imaginary Numbers/Mesh Analysis

    Homework Statement Image Attached Homework Equations Impedance/Reactance Mesh Analysis The Attempt at a Solution KVL -V + 2I1 + 4.7(I1 - I2) + j2(I1 - I3) = 0 4.7(I2 - I1) - j0.056(I2) + 2(I2 - I3) = 0 j2(I3 - I1) + 2(I3 - I2) + I3 = 0 (4.7 + j2)I1 - 4.7I2 - j2(I3) = 4V 4.7(I2-I1) -...
  9. J

    Is the Hypothesis of Real and Imaginary Components for F(ω) True?

    I never see the following hypothesis but I believe that they are true... ##\text{Re}(\hat f (\omega)) = a(\omega)## ##\text{Im}(\hat f (\omega)) = b(\omega)## where: ##f(t) = \int_{-\infty}^{+\infty}\hat f(\omega) \exp(i \omega t) d\omega = \int_{0}^{\infty} a(\omega) \cos(\omega...
  10. S

    Imaginary free energy and decay rate

    In euclidean quantum field theory, the imaginary part of the free energy, defined as the logaritm of the partition function, is it connected to the decay rate?
  11. S

    Imaginary Partition function

    I have a partition function in euclidean quantum field theory. I have a parameter, let's say a charge, that I can change in the action that define the partition function. I found that for small charge the partition function is positive, but there is a critical charge, above the one the...
  12. H

    Nodal Analysis: Imaginary Numbers

    Homework Statement I have to find the Thevinin Equivalent for the following circuit. I am assuming the current is going out of the node. V= node between inductor and capacitor V0 = V[40/(40-150j)] (V-75)/(600+150j) + (-0.02V0) + V/(40-150j) = 0 The only problem I have is with the last...
  13. C

    Calculate imaginary part if real part is following

    Homework Statement f:\ v(x,y)=4xy+2x The task is to calculate the imaginary part. Homework Equations The Attempt at a Solution I have no idea what to do because in my opinion u(x,y) can be anything. For example: f(x,y)=4xy+2x+(3x-4y)\text i. But I must be wrong. I would...
  14. T

    A deeper understanding of the imaginary number

    I know what a complex number is. Learned it way back when I took college classes. I know it is a number that has a real and imaginary part of the form a + bi. What I have always failed to understand is what conceptually does it mean. I know what i is , it's the square root of -1. I just could...
  15. Radarithm

    Understanding Numbers Raised to Imaginary Powers: Exploring λi and e^lnx

    Homework Statement How would you define a number that is raised to an imaginary power? Homework Equations λi= ? λ = 6+4i The Attempt at a Solution eln x = x Other than that I have absolutely no idea how to go about solving this.
  16. S

    Imaginary experiment on gravity.

    I am not good at maths but very passionate about physics (I know is sad). I am trying to imagine how strong (or weak) the force of gravity is, so I have this imaginary experiment: [BWe have to stones of, let's say, one kilo each floating in a void, separated by let's say one meter. In absence...
  17. A

    Algebra with Complex Numbers & Imaginary Unit

    When i is defined by an equation which has 2 solutions, how does it make sense to do algebra with complex numbers?
  18. C

    Find the real and Imaginary parts of sin(3+i)

    Homework Statement Find the real and Imaginary parts of sin(3+i) Homework Equations sin(x+y)= sinxcosy+sinycosx The Attempt at a Solution I think I am right in saying that you use the sine addition formula but then i get stuck from there. Is it something to do with exponential form?
  19. P

    Modified Bessel function with imaginary index is purely real?

    I'm trying to decide if the modified Bessel function K_{i \beta}(x) is purely real when \beta and x are purely real. I think that is ought to be. My reasoning is the following: \left (K_{i \beta}(x)\right)^* = K_{-i \beta}(x) = \frac{\pi}{2} \frac{I_{i \beta}(x) - I_{-i \beta}(x)}{\sin(-i...
  20. D

    Contour integral along the imaginary axis

    I'd like to evaluate the integral, \int^{i\infty}_{-i\infty} \frac{e^{iz}}{z^2 + a^2}dz along the imaginary axis. This function has poles at z = \pm ia , with corresponding residues \textrm{res}(\frac{e^{iz}}{z^2 + a^2},\pm ia) = \pm\frac{e^{\mp a}}{2ai} My question is - I'm not sure...
  21. T

    Finding the real and imaginary part

    Homework Statement Determine the real part, the imaginary part, and the absolute magnitude of the following expressions: tanh(x-ipi/2) cos(pi/2-iy) Homework Equations cos(x) = e^ix+e^-ix tanh(x) = (1-e^-2x)/(1+e^-2x) The Attempt at a Solution for cos(pi/2-iy)=...
  22. M

    Find beats formula using imaginary parts

    Homework Statement Using the imaginary parts When using complex representation, it is customary to use the real parts. Instead use the imaginary part of e^{j\theta} to calculate an expression for the sum: \sin(\omega t) + \sin((\omega + \Delta \omega)t) Remember, it should come out to...
  23. B

    DFT/FFT Imaginary vs real values

    I have a working DFT and FFT now that I coded in a program .. now from testing I can see that with both the FFT and the DFT if I just graph the Imaginary number I will get the right frequency for example: F[t] = 10 Sin((2 * PI * 2000 *t)/8000) 0 <t <1024 will get me a frequency 2000...
  24. G

    Substitution of imaginary variables in integral?

    I wanted to do this integral $$\int_a^b \frac{dx}{1-x^2} $$ and I was able to get the right answer with the substitution u=ix, where i is the square root of -1. But is this a valid mathematical procedure? $$\int_a^b \frac{dx}{1-x^2}=i \int_{-ia}^{-ib} \frac{du}{1+u^2}$$ Do those limits...
  25. S

    MHB Finding the real and imaginary parts of a function

    If f:C-->C is holomorphic and , find the real and imaginary parts ug and vg of g in terms of the real and imaginary parts uf and vf of f.
  26. S

    Square matrix to the power of a imaginary unit

    Hi; How to raising a square matrix to the power of a complex number? for example: [1 2;3 4]^(1+i) or mathematics software such as Scilab how solve such problems? -->[1 2;3 4]^(1+%i) ans = - 0.1482039 - 0.2030943 - 0.3046414 - 0.4528453 Thanks in advance...
  27. S

    MHB Evaluating imaginary expression

    Find all possible Values of: 2(-i)
  28. B

    Exploring the Properties of Vectors and Imaginary Numbers

    Homework Statement Consider a vector z defined by the equation z=z1z2, where z1=a+ib, z2=c+id. (a) show that the length of z is the product of the lengths of z1 and z2. (b) show that the angle between z and the x-axis is the sum of the angles made by z1 and z2 separately. The Attempt at a...
  29. C

    Integral of a real function multiplied by an imaginary function.

    So I'm reviewing some mathematics for quantum mechanics and this came equation came up \int_{-\infty}^{\infty} a \left( k \right)^{*} i \dfrac{d\,a\left(k\right)}{dk}dk. If a \left( k \right) is constrained to be real then this integral is zero or so the text says. Why is this the case...
  30. A

    Imaginary number's definition misunderstanding

    Homework Statement I'm in this self-learning course. I came on this problem I thought of. So, i^2=-1. But, isn't i=sqroot of -1? If so, the product of the two minus -1 and the square root of that should give 1. Am I not getting something? I searched the web with the keywords of my...
  31. T

    Diff EQ - Imaginary identities

    Homework Statement Determine the general solution of: y(6) + y''' = t The Attempt at a Solution Ok, r = 0, 0, 0, 1/2 +- 3i/√2, -1/2 + 3i/√2 What do I do with that last r value? It turns into ce-t somehow, but I don't see it. edit: typed a number in wrong, fixed now~
  32. T

    Imaginary prime number divisor

    What would be the implications of assuming the existence of an imaginary number that can divide a prime number and is related to the number it is dividing? By imaginary I mean a number that is just in our imagination and not the imaginary number "i".
  33. F

    Imaginary factor in WAVE guide TE field

    Hey guys. I'm trying to comprehend the TEmn EM fields in wave guides. I've gone through the derivation, using Pozar's microwave textbook, and for the most part it's straight forward. I am having a hard time though determining what the effect of the imaginary factor in the field equations are...
  34. C

    Modern physics, imaginary particle

    Homework Statement The energy level scheme for the mythical one-electron element crazyidium(the names not really relevant). The potential energy of an electron is taken to be zero at an infinite distance from the nucleus (a) How much energy does it take to ionize an electron from the ground...
  35. P

    MHB What is the imaginary part of the given function?

    Show that, \[\mbox{Im}(f(z))=\frac{1-|z|^2}{|z-i|^2} \mbox{ where }f(z)=\frac{z+i}{iz+1}\] \begin{eqnarray} \mbox{Im}(f(z))&=&\frac{1}{2i}(f(z)-\overline{f(z)})\\ &=&\frac{1}{2i}\left(\frac{z+i}{iz+1}-\frac{\overline{z}-i}{-i\overline{z}+1}\right)\\...
  36. T

    Imaginary Time: Parallel Universes & Mass?

    At about 7:00 he talked about imaginary time. Does that account for parallel universe? Also is there an imaginary mass as well?
  37. P

    What inspired mathematicians invent imaginary numbers?

    Let me start by writing about the natural or counting numbers. The story of how, where and when we invented them is lost in the misty dawn of history. But perhaps our African ancestors, like our living simian cousins (and some other animals) evolved the ability to distinguish between few and...
  38. S

    Any real world use of imaginary numbers?

    Everybody says that it is used in engineering or somewhere but how can you use it. in real world it is impossible to take square of any number and get negative answer. how can it have any use when it does not even exist. and people talk about imaginary plane, what is it? Thanks for helping...
  39. C

    Imaginary momentum and virtual particles

    There is a type of exchange of particles which is generalised by a type of potential: \frac{e^{-\alpha\r}}{R} This potential is used to explain the exchange of bounded particles (e.g a poin between neutron and proton) between two possible configurations. The potential comes from the fact that...
  40. LarryS

    QHO Solutions: What is Imaginary Part?

    The solutions, in the position basis, of the Schrodinger Equation for the Quantum Harmonic Oscillator are a family of functions based on the Hermite Polynomials. The Wikipedia link for this subject is http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator . But this Wikipedia article and...
  41. cocopops12

    Do all complex functions have orthogonal real and imaginary parts?

    z = h(x) + ig(x) True or False: By the definition of the complex plane, h(x) and ig(x) will always be orthogonal. If this was true, wouldn't that mean that we can find a 'very general' Fourier series representation of any function f(x) as an infinite series of An*h(x) + infinite series of...
  42. A

    Surds in polar form of imaginary number

    Homework Statement Find the polar form for zw by first putting z and w into polar form. z=2√3-2i, w= -1+i Homework Equations Tan-1(-√3/3)= 5∏/6 The Attempt at a Solution r= √[(2√3)2+(-2)2]=4 tanθ= -2/(2√3)=-1/√3=-√3/3=> acording to above... tan-1(-√3/3)= 5∏/6 so, in polar form z should be...
  43. J

    Proving that an integral is a pure imaginary

    Homework Statement Show that \int_{\gamma}\ f^*(z)\ f'(z)\ dz is a pure imaginary for any piecewise smooth closed curve \gamma and any C^1 function f whose domain contains an open set containing the image of \gamma 2. The attempt at a solution I have tried to approach it from some...
  44. ElijahRockers

    Imaginary Numbers Derivation

    Homework Statement Derive the following relation, where z1 and z2 are arbitrary complex numbers |z1z2*+z1*z2| ≤ 2|z1z2| The Attempt at a Solution I found the expression |z1z2*+z1*z2| = |2(a1a2+b1b2)| = √(4[a12a22 + 2a1a2b1b2 +b12b22]) But that is where I get stuck. How does the...
  45. C

    Imaginary components of real integrals

    Why does the incomplete gamma function have an imaginary component, when the exponential integral does not? \Gamma(0,z,\infty)\equiv\int^\infty_z \frac{e^{-t}}t dt Ei(z)\equiv-\int^\infty_{-z} \frac{e^{-t}}t dt Looking at how these integrals are usually defined I would have expected them to...
  46. L

    Solving equation with two imaginary roots

    1. -xn2+2(k+2)x-9k=0 Has two imaginary roots, what are the values of k? Attempted to break it down and use the quadratic formula but wasn't able to do it. Would like a pointer in the right direction of where to begin to solve it. Thanks
  47. P

    What should be the influence of the imaginary part on a complex number?

    Hi, What should be the influence of the imaginary part on a complex number? I am asking because I am running a simulation model where the input is a complex number; say z=a+ib Now the problem is that I get the same result when I put a=0 and give some high value to b, as when I do the...
  48. P

    MHB Imaginary Part of a Complex Function: How to Find It?

    How do I find the imaginary part of $\displaystyle \frac{1}{i}xe^{-ix}+e^{ix}$?
  49. M

    Is an imaginary electromagnetic gauge field something physical?

    Hi, my question is, if there is an interpretation for electromagnetic gauge fields, whose components are imaginary. This would lead to an imaginary magnetic field... Does anything like this exist? Or is it forbidden ny some first principal arguments? Thank you in advance for every input! Melvin
  50. C

    Taking imaginary integral and derivative

    Homework Statement when I am solving quantum problem, i see an equation like e^(-kx) e^(icx) i is imaginary. how can i take the integral and derivative of this function Homework Equations e^ix ) cosx + isinx The Attempt at a Solution actually i tried e^x(-k+ic) and i said the...
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