Imaginary Definition and 362 Threads

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. Kenneth Adam Miller

    I Expectation value with imaginary component?

    Hello, I'm a beginner at quantum mechanics. I'm working through problems of the textbook A Modern Approach to Quantum Mechanics without a professor since I am not going to college right now, so I need a brief bit of help on problem 1.10. Everything else I have gotten right so far, but I am...
  2. I

    MHB Imaginary numbers and real numbers?

    z is either a real, imaginary or complex number, and z^12=1 and z^20 also equals 1. What are all possible values of z? I know 1 and -1 are them, and I think its also i and -i?
  3. Heisenberg1993

    A Real parameters and imaginary generators

    I was reading some lecture notes on super-symmetry (http://people.sissa.it/~bertmat/lect2.pdf, second page). It is stated that ". In order for all rotation and boost parameters to be real, one must take all the Ji and Ki to be imaginary". I didn't understand the link between the two. What does...
  4. TheChemist_

    Determining graphical set of solutions for complex numbers

    Homework Statement So we have been doing complex numbers for about 2 weeks and there is this one equation I just can't solve. It's about showing the set of solutions in graphical form (on "coordinate" system with the imaginary and the real axis). So here is the equation: Homework Equations...
  5. FallenApple

    I What was the historical problem with Imaginary Numbers?

    I don't see why imaginary numbers were necessarily so difficult among top mathematicians back then. From pleano's axioms, we can derive the fact that any negative natural number times another negative natural number must be positive. Then this result extends to the reals, using theorems derived...
  6. N

    B If an imaginary quasiparticle has a mass depending on speed

    ...becoming infinite at rest, doesn't that mean it has infinite mass whatever velocity it has? How could that quasiparticle be at rest to achieve infinite mass?
  7. Jules Winnfield

    How come escape velocity isn't imaginary?

    Going through several definitions, it appears that escape velocity is equal to the potential energy. That is:$$\frac{1}{2}m v^2=-\frac{G M m}{r}$$but if I solve for velocity, $v$, I get:$$v=\sqrt{-2\frac{G M}{r}}$$So how do I get an escape velocity that isn't imaginary?
  8. D

    I Integrating imaginary units and operators

    When integrating terms including the imaginary unit i and operators like position and momentum, do you simply treat these as constants?
  9. Tspirit

    I Can the expectation of an operator be imaginary?

    Assume ##\varPsi## is an arbitrary quantum state, and ##\hat{O}## is an arbitrary quantum operator, can the expectation $$\int\varPsi^{*}\hat{O}\varPsi$$ be imaginary?
  10. C

    Find the real and imaginary parts of (1-z)/(i+z)

    Member advised that the homework template is required. Hey there! Need help figuring this out: Find the real and imaginary parts of \frac{1-z}{i+z} What I've tried was to notice that z\bar{z}=|z|^2, thence...
  11. L

    A Imaginary parts of amplitues (Schwartz QFT text)

    From Schwartz http://isites.harvard.edu/fs/docs/icb.topic521209.files/QFT-Schwartz.pdf p. 257 or his qft book p. 455 1. Why and how does the integral in (24.24) go imaginary, when M > 2m? Is it because the logarithms can not take negative real numbers, thus we have to switch to complex...
  12. M

    A If the axiom of induction were extended to include imaginary numbers....

    If the axiom of induction was extended to include imaginary numbers, what effect would this have? The axiom of induction currently only applies to integers. If this axiom and/or the well ordering principle was extended to include imaginary numbers, would this cause any currently true statements...
  13. S

    A Imaginary time propagation to find eigenfunctions

    Hi, I have been trying to use imaginary time propagation to get the ground state and excited states eigen function but the results I got is different from the analytical solution. I know that to get excited states, I should propagate 2 or more orthogonal functions depending on the number of...
  14. T

    B Simplifying the factors of a complex number's imaginary part

    My question boils down to wondering if there is a way to simplify the imaginary part of a complex-valued function composed of n factors if the real and imaginary component for each of the factors is known but the factors may take on the value of their conjugate as well. For example, is there a...
  15. S

    I Imaginary sinusoidal exponential

    I need to convince myself that ##e^{ikr\text{cos}\ \theta}=\frac{\text{sin}\ \theta}{kr}##. Could you please help me with it?
  16. Mr Davis 97

    B Imaginary number manipulations

    I know that this is probably a very commonly asked question with students, but say that we have ##\sqrt{(-1)^2}##. If we performed the innermost operation first, then we have ##\sqrt{(-1)^2} = \sqrt{1} = 1##. However, according to rules for radicals, we can do ##\sqrt{(-1)^2} = (\sqrt{-1})^2 =...
  17. T

    B Can Complex Numbers Extend Beyond Two Dimensions?

    Complex numbers ##a+bi## can be thought of as a second dimension extension of the real number line. Is there a third dimension version of this? Are there even more complex numbers that not only extend into the y-axis but also the z axis? tex
  18. Jamesavery88

    Imaginary Planetary System

    I had a dream about groups of nomads constantly traveling around a planet in one direction to stay in the 'dusk/dawn' zone of their planet as if they strayed too far behind they would freeze and if they ventured too far ahead they would cook and vice versa. What might a planetary system look...
  19. I

    B Exploring the Complexity of 0: Real, Imaginary, or Both?

    I was just thinking about this question, and I see 3 possible answers: 1) 0 is a purely real complex number. This seems to be the most intuitive, however the one problem is that it shows up on the imaginary numberline. 2) 0 is not real nor imaginary. I understand this one, but I have found one...
  20. D

    A quick question on imaginary number

    Homework Statement Since i is defined by sq(-1),and we can also write it as (-1)^(1/2) Therefore,(-1)^(1/2) is equivalent to (-1)^(4/8),so it becomes [(-1)^4]^(1/8), so we have 1^(1/8) = 1,which is clearly absurd... Besides,since (i^5)^3 = i^15 = -i,the multiplication of power rule still holds...
  21. evinda

    MHB How do we get the imaginary part?

    Hello! (Wave) We have this: $y''+y=\frac{1}{\cos x} , y'(0)=0, y(\pi)=0$. Using the Green function I got that $y(x)= x \sin x+ \cos x( -\ln |\cos \pi|+ \ln |\cos x|)=x \sin x+ \cos x (\ln |\cos x|)$. But according to Wolfram: y'''''''+'y'='1'/'cosx , y''''('0')''='0, y'('pi')''='0 -...
  22. G

    Imaginary Transverse Space of Superluminal Lorentz Transform

    I was reading this paper: http://dinamico2.unibg.it/recami/erasmo%20docs/SomeOld/RevisitingSLTsLNC1982.pdf It is on superluminal Lorentz transformations and is too advanced for me. :confused: But anyway, take a look at equation(s) (11). For the y' and z' transformations, there is an imaginary...
  23. DaveC426913

    Explaining imaginary numbers to laypeople

    I've had discussions with laypeople (of which, I am one) about real-world manifestations of imaginary numbers. We can never seem to find a satisfactory, concise example. I know they are used in real-world calculations for things like EM wavelengths in electronics, but if you aren't into...
  24. preitiey

    Wave function -- Why is there an imaginary part?

    If wave is a real concept, then why we have a complex(imaginary) part associated with the wave function?
  25. F

    How to write imaginary coefficient in excel 2013

    i have equation: ur=2.c(1-v2)/(w.d.i)(1+v2) i don't know how to write imaginary coefficient in excel.please help:(
  26. A

    Are All Aspects of Quantum Mechanics Real or Imaginary?

    Homework Statement The real world is "real" and not "imaginary", what does this imply? 1. wave functions are real quantities 2. expectation values are real quantities 3. operators are real 4. Ψ* = Ψ, where Ψ denotes a wave function 5. energy levels are real 6. wave functions cannot have an...
  27. B

    Normalising Imaginary Eigenvector

    Hello, whilst solving a system of coupled differential equations I came across an eigen vector of ##\vec{e_{1}} = (^{1}_{i})##. Assuming that this is a correct eigenvector, how do I normalise it? I want to say that ##\vec{e_{1}} = \frac{1}{\sqrt{2}} (^{1}_{i})## but if I sum ##1^{2} + i^{2}##...
  28. T

    Understanding Meson Mixed States & Why They Don't Use Imaginary Phase

    Why mesons mixed states are defined as SOMETHING +/- SOMETHING [+/- SOMETHING] normalized by 1/sqrt(2) or 1/sqrt(3), So the sum uses quotient +1 or -1. But in electroweak symmetry breaking charged W boson is defined as W1 (+/-) i*W2, so the quotient is +i or -i. So why mesons never use...
  29. thegirl

    Cross product imaginary numbers

    Hi, I was just wondering if you have a cross product can you multiply out the constants and put them to one side. So ik x ik x E is equal to i^2(k x k x E) therefore is equal to -k x k x E. Is that correct?
  30. S

    Exploring Imaginary Numbers for Beginners

    What are imaginary numbers? Does anyone know a good book for it?
  31. Ien Cleary

    Why does imaginary time behave like space?

    I know what imaginary numbers are, but I'm struggling to understand why the Lorentz transformation makes a time-like dimension space-like. I suppose what I'm really asking is what is the difference between time-like and space-like. I've read that it has something to do with special relativity...
  32. A

    Probability function being imaginary?

    Suppose ψ1 and ψ2 are two eigenfunctions of a particle and ε1 and ε2 are the corresponding eigenvalues. If the state is in the superposition Ψ = αψ1 + βψ2 at time t=0, it evolves in time by the equation Ψ = αψ1ei ħ/ε1 t + βψ2ei ħ/ε2 t. I am trying to understand the probability amplitude Ψ*Ψ.If...
  33. K

    MHB Proving Prime Numbers in Quadratic Imaginary Fields

    Hi, I need your help with the next two problems: 1) If p is a prime number such that p\equiv{3}\;mod\;4, prove that \sqrt{-p} is prime in \mathbb{Z}[\sqrt[ ]{-p}] and in \mathbb{Z}[\displaystyle\frac{1+\sqrt[ ]{-p}}{2}] too. 2) 2) We have d > 1 a square-free integer. Consider the quadratic...
  34. J

    Massive spin 1 propagator in imaginary time formalism

    Homework Statement I have the following massive spin-1 propagator- $$ D^{\mu\nu}(k)=\frac{\eta^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{m^2}}{k^2 - m^2} $$ I want to write down the propagator in the imaginary time formalism commonly used in thermal field theories. Homework EquationsThe Attempt at a...
  35. T

    Relation between adiabatic approximation and imaginary time

    Regarding interacting green's function, I found two different description: 1. usually in QFT: <\Omega|T\{ABC\}|\Omega>=\lim\limits_{T \to \infty(1-i\epsilon)}\frac{<0|T\{A_IB_I U(-T,T)\}|0>}{<0|T\{U(-T,T)\}|0>} 2. usually in quantum many body systems...
  36. J

    Getting from non-imaginary to imaginary

    I was reading the derivation of capacitor reactance and I understand it up to the point where it is converted to polar coordinates. How do you get from X=\frac{sin(wt)}{wCcos(wt)} to X=\frac{1}{jwC} This implies that \frac{sin(wt)}{cos(wt)}=-j And I'm confused how that is derived. Thanks...
  37. K

    Is imaginary time a scientifically serious proposal?

    I've seen Stephen Hawking mention it. there's an article on it. Is imaginary time a scientifically serious proposal? what are the ramifications to physics, including general relativity and gravity, if we accept imaginary time? what about imaginary space? what about quantum gravity theories like...
  38. M

    What will happen if light is longitudinal wave?

    Topic edited by mfb (original was "What will happen if light is transversal wave? ") Will people see the world in a completely different view?
  39. samsara15

    Imaginary Volumes: Exploring Complex Numbers & Algebras

    Imaginary numbers enable one to envision a lot of ideas. But what kind of numbers/algebras would enable us to work with imaginary volumes? Volumes, by definition, always seem to be positive, since any cubes are. What kind of numbers would give/allow a more complex picture?
  40. lahanadar

    One-dimesional system non-existence fixed points

    Homework Statement First things first, this is not a HW but a coursework question. I try to understand a concept. Assume we have a one-dimensional dynamic system with: x'=f(x)=rx-x^3 Homework Equations Fixed points are simply calculated by setting f(x)=0. The Attempt at a Solution If I...
  41. T

    Can Numbers Only be Graphed Using the Z Axis and What are They Called?

    If I understand correctly an imaginary number can be graphically shown in a x/y axis graph. Are there numbers that can only be graphed by using the third z axis? What are they called? tex
  42. Invutil

    Are photons imaginary particles?

    Since an electron generated a negative charge around itself and can push other electrons around itself, waves can travel through electrons. These are electromagnetic waves. But quantum theory proposes that the pushes between electrons happen in discrete packets. Electromagnetic packets called...
  43. T

    Gaussian integral w/ imaginary coeff. in the exponential

    So I've seen this type of integral solved. Specifically, if we have ∫e-i(Ax2 + Bx)dx then apparently you can perform this integral in the same way you would a gaussian integral, completing the square etc. I noticed on wikipedia it says doing this is valid when "A" has a positive imaginary part...
  44. G

    Feynman rules - where do the imaginary numbers come from?

    I'm trying to learn how to derive Feynman rules (what else to do during xmas, lol). The book I'm using is QFT 2nd ed by Mandl&Shaw. On p 428 they're trying to show how to derive a Feynman rule for W W^\dagger Z^2 interaction term g^2 \cos^2\theta_W\left[W_\alpha W_\beta^\dagger Z^\alpha Z^\beta...
  45. David Carroll

    Why is this way of computing imaginary arctangents wrong?

    I was doing some things in my head the other day (these moments usually don't come out so well :rolleyes:). And I "came up" with the following way to compute the arctangent with imaginary arguments. Consider the identity x2 = - 2, where x = ± i√2. Now rearrange this identity to x2 + 1 = -1...
  46. S

    Does (-2)^(⅔) have an imaginary component?

    Some calculators say (-2)2/3 is equal to ##-\frac{1}{2^\frac{1}{3}}+i\frac{3^\frac{1}{2}}{2^\frac{1}{3}}## while others say its equal to ##4^{\frac{1}{3}}## i.e. ##|-\frac{1}{2^\frac{1}{3}}+i\frac{3^\frac{1}{2}}{2^\frac{1}{3}}|##. I think I am right to imply from above that (-2)2/3 does have an...
  47. E

    Imaginary Roots and Vieta: 3a < 2+4c

    Homework Statement If both roots of the equation ax^2 + x + c - a = 0 are imaginary and c > -1, then: Ans: 3a < 2+4c Homework Equations Discriminant < 0 for img roots Vieta The Attempt at a Solution 1-4(a)(c-a)<0 4ac > 4a^2 + 1 Minimum value of 4a^2 + 1 is 1 so 4ac>1 I can't think of...
  48. T

    Exploring Real & Imaginary Parts of Delbruck Scattering

    My question here involves Delbruck Scattering specifically but my curiosity is more general. Delbruck Scattering is the scattering of a photon off of the Coulomb field of a nucleus via the creation and annihilation of real and virtual electron-positron pairs. The process can occur at energies...
  49. D

    MHB Plotting Complex Functions: Does it Look Like a Riemann Surface?

    I plotted the real and imaginary parts of a complex function \(z^{1/3}\). The two plots are similar to the Riemann surface is that correct?
  50. R

    Nonlinear Dispersion Relation with Imaginary Part

    Homework Statement I've determined the dispersion relation for a particular traveling wave and have found that it contains both a real and an imaginary part. So, I let k=\alpha+i\beta and solved for \alpha and \beta I found that there are \pm signs in the solutions for both \alpha and \beta...
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