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. putongren

    Quantum Cosmology - Imaginary Time

    I have trouble understanding the concept of imaginary time. As I understand it (by reading other online sources), imaginary time was developed by Stephen Hawking and Hartle in their theory of quantum cosmology. Quantum cosmology is applying quantum mechanical principles to cosmology as a...
  2. Y

    Tips for Solving Imaginary Numbers Quiz Problems

    How does one go about solving the first and second problems of this quiz? http://www.wiziq.com/online-tests/3503-algebra-iit-jee
  3. J

    So the imaginary component is just -ωCVV*.

    For the following, I was wondering if someone could help me determine the imaginary part of the complex number: Im<GVV* - j\omega CVV*>, where V* is the complex conjugate of V. I was thinking the imaginary component would be -\omega CVV*. But I am reluctant on the conclusion, because I am not...
  4. J

    Engineering Circuit elements (Real or Imaginary)

    Statement: I was wondering if someone could provide more background knowledge on why certain elements in a circuit (e.g. capacitors) are associated with imaginary components, whereas other elements (e.g. resistors) are only associated with the real components. In particular, the equation below...
  5. J

    Solving cubic roots (with imaginary number)

    Homework Statement (0.1 - 0.3j)^(1/3) = a + bj, where j is the imaginary number or more specifically sqrt(-1). Does anyone know how to solve for a and b? Homework Equations I've looked at cubic function equations, along with some polar equations. However, the latter requires some angle...
  6. A

    Imaginary into real but not vice versa?

    Its kinda weird (atleast to me) that we can convert imaginary numbers into real, there is just a simple procedure/function called multiplication. You multiply two no. and you get a real ,though negative. But not to worry we can still multiply it with 2 i's or say i^2. To make it positive. But...
  7. N

    What is time? Is it an imaginary thing?

    What is time? Is it an imaginary thing? Why do time slow down when it is closer to a black hole?
  8. G

    Is the rotation angle in Minkowski's diagram real or imaginary?

    In Figure_1(b) I have depicted a simplified version of Minkowski's diagram, where \beta = \frac{v}{c}= \tanh \psi= - i \tan i \psi= \frac{( e^{\psi} - e^{-\psi})}{( e^{\psi} + e^{-\psi})} , the rotation of (x',t')-axes being defined as imaginary (considering x and t real, and c=1)...
  9. N

    Solving a Complex Math Problem: Real and Imaginary Solutions

    Homework Statement Apparantley the following problem can't be solved with a graphics calculator and we have to state why this is, and give all the possible solutions. A ladder AB leans against the side of a building. Where the wall meets the ground is a cubic packing case 1m x 1m x 1m. The...
  10. J

    Eigenvalues - real and imaginary

    Am I understanding this right? Let's say I have a 15x15 matrix called Z. Then the matrix of eigenvalues calculated from Z, called D, can have two forms - either diagonal or block diagonal. If the matrix D comes out with values only on the diagonal, then there are only real values. But...
  11. T

    Phasors and Imaginary and Real help

    Ok so i just want to clear up something about phasors and just circuit input functions in general. So based on the imaginary and real graph, where imaginary is like the y-axis and the real is the x axis, when let's say a circuit falls on the imaginary axis is it considered a stable circuit? Do...
  12. P

    Identity for exponential function with imaginary arguments

    I have seen the following identity used. Exp[iw/2]-Exp[iw/2]=Exp[iw]-1 I can't find this in any book and I can't prove it myself. The left side equals 2isin(w/2) The right side equals cos(w)+isin(w)-1 On the face of it, that seems to make the identity absurd How can one go about proving...
  13. G

    Does divisibility apply to imaginary numbers?

    For example, is 5i "divisible" by 5? Or does divisibility only apply to integers? On that note, is 5pi divisible by 5? Is 5/6 not divisible by 5? Thanks in advance! =)
  14. matt_crouch

    What are Real and Imaginary Numbers and How Do They Relate to Each Other?

    Im doing A-level maths at the moment an its nowhere in the sylabus I am just generally interested :D can anyone giv me like a really simplish explanation :D cheers
  15. S

    Solving for all values of x including imaginary numbers

    Homework Statement Solve for all values of x both real and imaginary 1) x^3=-8 2)(3/x+1)+(x/x-1)=(x-5/(x+1)(x-1)) 3)2x^3-8x^2+5x=0 4)x^4-7x^2+12=0 5)x - sqroot ( 6 - 5x ) = 0 6)(absvalue(3x)) + 6 = 10 The Attempt at a Solution I tried all of them but came out with wonky answers or I...
  16. R

    Imaginary Zeros of Zeta Function

    I was doing some work with the zeta function and have a question. I am aware that the Riemann Hypothesis claims that all of the critical zeros of the analytically continued zeta function have a real part Re(z)=1/2. My question is, does the concept apply only to the complex zeros, or the...
  17. T

    Story behind imaginary numbers

    Hey Guys, I am looking for a book that talks about the history of imaginary numbers and how people came to need it. I'm just having trouble imagining how people could have accepted imaginary numbers before negative numbers (at least in the western world). Thank you, Eric P.S. If...
  18. H

    Sketching EM Waves with imaginary amplitudes

    Homework Statement Sketch in 3D, the following waves. (both E fields are in x direction E_1 = 3*exp(-j*8*Pi*z) E_2 = 4j*exp(-j*8*Pi*z) (where j=sqrt(-1)The Attempt at a Solution I know how to sketch E_1, but my question is how to treat the imaginary amplitude, 4j in E_2. here is my stab at...
  19. E

    Purely imaginary bound state wave functions?

    Hello all. I’m researching rotational motion with a nearly harmonic potential using the basis of the particle on a ring eigenstates e(n*i*theta) defined from theta=0 to theta=2*pi. The total systems wave functions (eigenfunctions of the full Hamiltonian (KE+PE)) are then linear combinations of...
  20. J

    Efficiency of an imaginary ideal-gas engine cycle

    Homework Statement The figure represents a simplified PV diagram of the Joule ideal-gas cycle. All processes are quasi-static, and C_P is constant. Prove that the thermal efficiency of an engine performing this cycle is \eta = 1 - \gamma \frac{\frac{V_{1}}{V_{2}} - 1}{\frac{P_{3}}{P_{2}}...
  21. haushofer

    Exploring Imaginary Shifts in QFT: A Necessary Tool for Convergence?

    Hi, I'm studying QFT (quantum field theory) again, with the help of Peskin&Schroeder. Altough the book is much better to read for a second time, there are some things that keep me wondering. It's about shifting physical quantities by a small imaginary constant in order to be sure that...
  22. M

    Rafael Bombelli & Imaginary Numbers: The Pros & Cons

    Rafael Bombelli first used them and at the time they were thought to be useless. with other discplines (notably physics) finding a use for 'imaginary' numbers, why wasn't it's use curtailed? afterall if I was paid to fill a room with a 1000 people and could only manage 500. would i be within my...
  23. Evo

    Please welcome another imaginary friend

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  24. A

    Understanding Complex & Imaginary Numbers

    Hi, Just reading through a good physics book about games and came across some stuff on complex numbers. The book states that the following statements are true. i*j*k=-1 I have a problem with this one. Surely doesn't this mean the same as: -1*sqrt(-1)=-1 or -k=-1 How is minus...
  25. C

    Complex imaginary Number Problem

    Hi, i have this question which is related to complex number and i have just no idea how i should solve it. Some guide and help please. Given that z = x + yi and w = (z+8i)/(z-6) , z \neq 6. If w is totally imaginary, show that x^2 + y^2 + 2x - 48 = 0 I've tried a lot of way comparing them...
  26. A

    Real World Applications of Imaginary Numbers

    Something has been puzzling me...what is an imaginary number in real life? I know that engineers sometimes use it but how do they apply it to real world situations? How is it anything but a mathematical constant?
  27. rohanprabhu

    Imaginary Friends Comic | NFC Comic #205

    http://nfccomic.com/comics/205.jpg
  28. R

    Can expectation value of observables be imaginary?

    I am quite new to Quantum Mechanics and I am studying it from the book by Griffiths, as kind of a self-study..no instructor and all... For a gaussian wavefunction \Psi=Aexp(-x^{2}), I calculated <p^{2}> and found it to be equal to ah^{2}/(1-2aiht/m) (By h I mean h-bar..not so good at...
  29. G

    Write as sum of real and imaginary part

    Hi, I have to write the 1/(a+ib) as a sum of a real part and an imaginary part. I was thinking of using the complex conjugate (a-ib) and multiply with this (a-ib) / (a^2 + b^2) I am a little bit lost on what to do next - any help appreciated thanks in advance Best
  30. F

    Imaginary numbers outside of math

    In Algebra we are learning and using imaginary numbers. Someone asked if imaginary numbers are ever used outside of math, and our teacher said he talked to an electrical engineer who used imaginary numbers all the time. Our teacher didn't know how or why they were used in electrical...
  31. S

    Questions about imaginary number and root of 4

    I am thinking that if the imaginary number is bigger than the other number, is it right to say that: i> 5 ? 7i> 3i ? Does i has magnitude? if Z_1=4+5i then Z_2=1-3i whether Z_1>Z_2 or Z_2>Z_1 is true? If we say Z_a is bigger than Z_b, does that means the absolute value of these complex...
  32. E

    Mathematica Mathematica solution with an imaginary part

    Can someone please explain why the result of the following has an imaginary part when I put it into mathematica 6.0? Sum[Sin[n 6]/2^n, {n, 1, Infinity}]
  33. I

    Lost: Imaginary term in weak lagrangian

    Homework Statement I'm working on the last section of the last chapter (11.9 The Higgs Mechanism) of Griffith's book on Elementary Particles. In Problem 11.25 I'm asked to derive the lagrangian that he gets juts after gauging [Eq. 11.131]. That's the one with a Goldstone boson and a strange...
  34. M

    Real & Imaginary Parts of Logarithms

    Homework Statement Find the real & imaginary parts of log(1+i)log(i) ? Homework Equations The Attempt at a Solution
  35. G

    MathCAD - separation of real and imaginary parts of a complex variable

    Hi guys, just wondering whether you could help. I've got a complex number in terms of a lot of variables, and need to separate it into its real and imaginary parts. How do I do that? I spent past hour trying to look for tutorials, unfortunately none tells you how to do it...
  36. R

    Square root of an imaginary number

    Where on the imaginary number axis do i graph sqrt(3i)? At sqrt3?
  37. D

    A new set of numbers as a z-Axis to imaginary and real numbers?

    hello, first: excuse me if the question is stupid but I am still at school. my question: some days ago i came across imaginary numbers. You know what I mean - the imaginary number i^2=-1 and the imaginary numberline is not on the the same line as the other numbers. The imaginary numberline is...
  38. R

    Is 0 a real and imaginary number?

    this is a question of mine because of an edit to the Wikipedia article Imaginary number . the funny thing is that i couldn't find, in three of my old textbooks a clear definition of "imaginary number". (they were pretty good at defining "imaginary part", etc.) i understand that that...
  39. A

    Curious about imaginary solution

    It's clear that f(x)=e^x has no solution for f(x)\le0. So can there be imaginary solutions for values not in the range of the exponential function? Just as an example, can this have imaginary solutions, and how would one go about solving it: e^x=-e?
  40. J

    How do you test convergence for an infinite series with imaginary coefficients?

    If you have an infinite series some or all of whose coefficients involve imaginary terms how do you check for convergence?
  41. H

    Can These Points Form a Square in Geometric Constructions?

    let ABDE and BCGH be squares lying outside the traingle ABC. The centres of these sqaures are P & Q respecitvely, and the midpoints of the line segments AC and DH are R & S respectively. Show that the points P,Q,R,S are vertices of a square? any ideas on how to do this please? the only hint i...
  42. J

    Simplifying a solution such that the imaginary number goes away

    Homework Statement Given a general solution to a system of differential equations: Y(t)= C1(1;2i)e^(2it)+C2(1;-2i)e^(-2it) side note: i is sqrt of -1, and the (1;2i) is a 2 by 1 matrix. The idea is to simplify the solution such that the imaginary components go away. Homework...
  43. G

    Understanding the Significance of Imaginary Frequencies in Chemical Reactions

    Hi, In the transition state of a chemical reaction defined as one imaginary eigenvalue of the hessian matrix - the size of my frequency is 1000i what can one say about the size of the imaginary frequency - is it related to energy in some way ? I have not been able to find any documents...
  44. M

    Finding Imaginary Roots for X2 –3X +C

    please see my question i can't dfind its imaginary roots .the equ is X2 –3X +C,here 2 is the power of X and Cis constant we have to show that there exixts no reak number C for which the givev equation has two distinct rootss in [-1,1] i solve this by quadic formula but i got its real...
  45. S

    Exploring Imaginary Dimensions & Time Motion

    Apparently, when one calculates a length in flat space-time, one must add the lengths in the three spatial dimensions, and subtract... square root(-1)ct, my first question is... is there some sort of physical significance to a dimension being imaginary or is this simply a mathematical trick so...
  46. P

    Imaginary Action Friends: Ann. Phys. & Textbook for Tunneling

    Friends, I am reading that the standard way to discuss tunneling is via imaginary action. The reference cited is J.S. Langer, Ann. Phys. 1967. (1) What is the full name of Ann. Phys. so I can look up this paper? (2) What textbook can I acquire to learn this technique?
  47. L

    Express Imaginary number in standard form.

    Homework Statement Need to see if what I am doing is right, here is the problem: z= 4i/(-1+i), express z^20 in standard form. so, z= 4i/(-1+i) = 2-2i z^20 = 2*sqrt(2) x cis[20(-pi/4)] = 2*sqrt(2) x cis(-5pi) = 2*sqrt(2) x [cos(-5pi) + i sin(-5pi)] = 2*sqrt(2) x [-1 + 0i]...
  48. M

    Imaginary numbers are inherently unobservable

    Maybe this should be in the philosophy of science/math forum, but i thought it fitted here. Why is it that variables taking imaginary values are inherently unobservable, whereas real numbered variables correspond to observables like position/momentum? As far as I can see there is no a priori...
  49. T

    Polar Form of Imaginary Number w=8i - Undefined Tan(theta)

    w=8i I need to put this in polar form but how can i do this since this would be w=8(cos(theta)+isin(theta)) I can't find the angles because tan(theta)=8/0 which of course is undefined. Is there something that I am doing wrong?
  50. A

    Calculating the Imaginary Part of (1-2i)^2-i

    How can i find The imaginary part of (1-2i)^2-i. This is what i have done so far: (1-2i)^2 (1-2i)^-i =(1-4i+4)(1-2i)^-i
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