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. bobrubino

    B Is imaginary time a fourth dimension?

    In a conversation Dr. Stephen Hawking said that he used Imaginary time as a 4th dimension to show that there was nothing before the big bang. How is it possible for Imaginary Time to act as a fourth dimension when it is still part of the ordinary time dimension?
  2. G

    I Proof of i-th Root of i: Analytic Connection?

    I saw a proof in which they came up with the ith root of i through the typical algebra. $$ i^{1/i} = i^{-i} = e^{i\frac{\pi}{2} \cdot -i} = e^{\frac{\pi}{2}} ~. $$ But it seems the proof is entirely algebraic, so we have no grounds for thinking it works anywhere. The only exception might be an...
  3. Feynstein100

    I Does topology distinguish between real and imaginary dimensions?

    Let's say we have four 3D spaces: (x, y, z) , (x, y, iz) , (x, iy, iz) and (ix, iy, iz), with i being the imaginary unit. Now, let's say we have a donut in each of these spaces. Geometrically, the donuts are different objects, have different equations and different properties (I think) but would...
  4. jedishrfu

    B The Greatest Imaginary Mathematician of the 20th Century

  5. BadgerBadger92

    B Tachyons and imaginary numbers?

    https://www.scientificamerican.com/article/what-is-known-about-tachy/ How do tachyons have mass of a square root of a negative number? I thought this was not mathematically possible?
  6. K

    A Bessel functions of imaginary order

    In Wikipedia article on Bessel functions there is an integral definition of “non-integer order” a (“alpha”). For imaginary order ia I get that Jia* = J-ia, where * is complex conjugate and ia and -ia are subscripts. Then in same article there is a definition of Neumann function, again for...
  7. P

    I Exploring Newton's Third Law in an Imaginary Magnetic Field

    Hi, here's a theoretical problem that I am trying to find a satisfactory answer for. Imagine a coil that is temporarily switched on an off and generates a magnetic field that permeates through space. Now imagine a charged particle passing through this field, at time that the coil is already...
  8. JD_PM

    Find the range of ##k## for which ##\omega## is imaginary

    Out of the given information I see no way of solving neither of the two sections. So I did some reading and it turns out that the given relation is a solution of the following function (I won't prove it here) $$\epsilon (k, \omega) = 1 - \frac 1 2 \left[ \frac{\omega_p^2}{(\omega-kv_0)^2} +...
  9. chwala

    Imaginary Can We Find the Least Time?

    This is the problem; I tried using; ##s##=##ut##+##\frac {1}{2}####at^2## and ended up with, ##240## =## 12t -1.5t^2## clearly we have an imaginary solution here and therefore this may not be correct? My wild guess is that the least time (Without deceleration) should take at least...
  10. PainterGuy

    I Video on imaginary numbers and some queries

    Hi, I was watching the following video. I found some points confusing. Could you please help me to understand the gaps? Thanks, in advance! Question 1: Around 4:22, the video says the following. So for those mathematicians, negative numbers didn't exist. You could subtract, that is find...
  11. R

    Cauchy Riemann complex function real and imaginary parts

    Hi, I have to find the real and imaginary parts and then using Cauchy Riemann calculate ##\frac{df}{dz}## First, ##\frac{df}{dz} = \frac{1}{(1+z)^2}## Then, ##f(z)= \frac{1}{1+z} = \frac{1}{1+ x +iy} => \frac{1+x}{(1+x)^2 +y^2} - \frac{-iy}{(1+x^2) + y^2}## thus, ##\frac{df}{dz} =...
  12. V

    Electric flux through ends of an imaginary cylinder

    When I look at this question, I can see two possible values of electric flux depending on how I take the normal area vector for either ends ##A \text{ and } A^{'}##. What is wrong with my logic below where I am ending up with two possible answers? The book mentions that only ##2E\Delta{S}## is...
  13. karush

    MHB 5.t.11 find x for the imaginary factors

    $\textbf{5.t.11 }$ McKinley HS Find x for $f(x)=0 \quad 5+i\quad 5-i\quad $ $\begin{array}{rl} \textsf{factored} &f(x)=(x-1)[x-(5+i))(x-(5-i)]\\ \textsf{foil} &x^2-x(5+i)-x(5-i)+(5-i)^2\\ \textsf{expand} &x^2-5x-xi-5x+xi+25-2i+i^2 \\ \textsf{simplify}...
  14. M

    I Proving Imaginary Hamiltonian is P-Odd, T-Even & Imaginary

    Hello! The Hamiltonian for nuclear spin independent parity violation in atoms is given by: $$H_{PV} = Q_w\frac{G_F}{\sqrt{8}}\gamma_5\rho(r)$$ Here ##Q_w## is the weak charge of the nucleus (which is a scalar), ##G_F## is the Fermi constant and ##\rho(r)## is the nuclear density. From the papers...
  15. S

    Trying to make a rigorous imaginary world

    My fantasies: 1) There happens a rapid evolution every some thousand years in biological organisms like humans. The sleeping pattern of people changes and by some unknown means people are connected to each other. The number of cycles in the sleep changes and the brain rewires itself to change ...
  16. S

    B Why are imaginary numbers called "imaginary"? If they really exist

    If Imaginary numbers do exist and have real applications, then why do we call imaginary numbers "imaginary numbers"? . They exist. They're used all the time. What makes them "imaginary"?
  17. S

    B What is the Difference between imaginary part and imaginary number?

    I am a high school student and I have a doubt regarding Complex Numbers. when we define a complex number we say that a number of the form z=a+ib ,where a and b are real numbers is called a complex number. a is called the real part and b is called the imaginary part, but I have a doubt here we...
  18. J

    I Imaginary number i in Pauli matrixes

    Is the imaginary number i "necessary" in the pauli matrices simply because of the condition of having 3 mutually orthogonal axi? If space were two dimensional we wouldn't need the i imaginary number?
  19. gonadas91

    A Positive polarity of a retarded Green's function's imaginary part

    Hi all! I am currently working on a problem that involves calculation of Green functions. I am having serious doubts on some part of the calculation, as I am getting a GF that has positive values of its imaginary part. According to Lehman's rule, the retarded GF can be decomposed by the spectrum...
  20. coreanphysicsstudent

    Rational power of the imaginary unit

    but the real answer is −1, (1±i√3)/2 What's wrong with my solution? please help me through.
  21. benorin

    I How do I separate the real and imaginary parts of an infinite product?

    Suppose you have a complex-valued function of a complex variable (namely, ##z=x+iy, \, \, x,y\in \mathbb{R}##) defined as the assumed convergent infinite product $$F(z)=\prod_{k=1}^{\infty}f_{k}(z)$$ Further suppose ##F(x+iy)=u(x,y)+i v(x,y)##, where u and v are real-valued functions. How to...
  22. Bob3141592

    I Imaginary Residuals in Solving the Schrodinger Equation

    I guess the summary says it all, if the question is clear enough. The last time I took physics courses was 45 years ago, and the QM course blew my mind, meaning I was mostly baffled. I could not wrap my head around it, and without a conceptual framework I couldn't remember the details. So I...
  23. S

    Simplify this fraction containing imaginary numbers

    i can get to 3i+1/1-3i but no further. I take it this is the correct way to start
  24. jisbon

    Complex numbers: if (u+v)/(u-v) is purely imaginary, show that mod(u)=mod(v)

    I'm kind of stuck over here at one part, but something is telling me that it might be wrong too :( Do assist, thanks.
  25. Lunct

    B Can virtual particles have an imaginary mass?

    I was talking to a physicist who said to me that virtual particles can have a mass of a constant times by i ,as in the root of -1. I have been thinking about this more and it intrigues me. I have done some research into this and can't find further details. If they have an imaginary mass does...
  26. uzi kiko

    A The relation between imaginary part of permittivity to conductivity

    Hi When I read the following article: http://nopr.niscair.res.in/bitstream/123456789/14183/1/IJPAP%2050%286%29%20405-410.pdf I tried to convert the value of the conductivity for saline in 500 MHz. At the article, they present the conductivity as ε'' = 70. Which ε'' is the imaginary part of...
  27. jk22

    B Should the 'time' axis of a Minkowski diagram be time's imaginary unit?

    Since the metric is euclidean in coordinates ##(ict,x)## it can be drawn in a plane, but if the metric is ##diag(1,-1)##, can both axis still be drawn in a plane ?
  28. F

    B Domain and imaginary numbers

    Consider the function sqrt(x). What is the domain of this function? Is it all real positive numbers? This is what I was taught in high school, but I was also taught that plugging in -1 would give an answer of i. So if the function takes negative inputs, shouldn’t they be part of the domain?
  29. John Greger

    A Imaginary part of the dielectric function

    Hi everyone, I was thinking about the complex part of the dielectric function. To my understanding there's good physical explanation of it. is a superimposed description of dispersion phenomena occurring at multiple frequencies. Say I only have the real part such as the one below, and would...
  30. Tan Thom

    Solving for the Real and Imaginary Number

    Homework Statement Find the 4th Coefficient in a sample of 4 discrete time Fourier Series coefficients in a real time valued periodic sequence. k = 0,1,2,3 a_k = {3, 1-2j, -1, ?} Homework Equations [/B]The Attempt at a Solution Step 1: (1-2j)e^(j*.5pi*n) +a_3 e ^ -(j*.5pi*n) + 3 +...
  31. Tan Thom

    MHB Real and Imaginary Numbers

    Good morning, I am working on a problem where I am finding the 4th Coefficient in a sample of 4 discrete time Fourier Series coefficients. I got the sum but now I have to solve for a_3 which consists of a real and imaginary part. Any assitance on how to solve for the a_3? Thank you. $a_k =...
  32. F

    B Is imaginary "i" a purely aesthetic matter?

    Say that I define a set of pairs called ℂ, such that [a,b] ∈ ℂ iff a ∈ ℝ, b ∈ ℝ, [a,b]+[c,d] = [a+c,b+d] [a,b]*[c,d] = [ac-bd, ad+bc] Then this has exactly the same properties of a+bi, does it not? You can write any equation that uses i exactly the same way with those pairs, so all interesting...
  33. learn.steadfast

    I Hermitian and expectation values.... imaginary?

    I've been studying quantum mechanics, and working problems to get a feel for expectation values and what causes them to be real. I was working the problem of finite 1D wells, when I came across a situation I did not understand. A stationary state solution is made up of a forward and reverse...
  34. rude man

    B Base e to an imaginary exponent seeming contradiction

    <Moderator's note: Moved from a homework forum.> 1. Homework Statement Given 0 < a < 1, i = √(-1), ei2πa = cos 2πa + i sin 2πa but also, ei2πa = (ei2π)a = 1a = 1 How to resolve the apparent contradiction?Homework Equations eab = (ea)b eix = cos x + i sin xThe Attempt at a Solution No clue...
  35. Chris Miller

    B Can the Lorentz Factor Approach 0i as v Exceeds c for FTL Reference Frames?

    As v approaches c, the Lorentz factor approaches infinity. The math and physics is well understood and observed. Is it true that, just mathematically, as v exceeds c the Lorentz factor approaches 0i for imaginary time constriction?
  36. LarryS

    I Imaginary Part of QHO Solutions?

    I have seen a few online lectures on solving the Schrodinger equation for the Quantum Harmonic Oscillator. The various solutions are products of the real-valued Gaussian function and the real-valued Hermite Polynomials. But I have never seen a mathematical expression for the imaginary part of...
  37. A

    Solving Schrodinger's Equation with a weak Imaginary Potential

    Homework Statement A particle of energy E moves in one dimension in a constant imaginary potential -iV where V << E. a) Find the particle's wavefunction \Psi(x,t) approximating to leading non-vanishing order in the small quantity \frac{V}{E} << 1. b) Calculate the probability current density...
  38. T

    MHB Imaginary integral = π/(1+n)

    How to prove this integral, $$\int_{0}^{2\pi}\mathrm dt{\sin t\over \sin t+ i\sqrt{n+\cos^2 t}}={\pi\over 1+n}$$ $n \ne -1$ $i=\sqrt{-1}$
  39. H

    I Intuitive understanding of Euler's identity?

    I'm trying to get a more intuitive understanding of Euler's identity, more specifically, what raising e to the power of i means and why additionally raising by an angle in radians rotates the real value into the imaginary plane. I understand you can derive Euler's formula from the cosx, sinx and...
  40. JTC

    I Using Complex Numbers to find the solutions (simple Q.)

    Say you have an un-damped harmonic oscillator (keep it simple) with a sine or cosine for the forcing function. We can exploit Euler's equation and solve for both possibilities (sine or cosine) at the same time. Then, once done, if the forcing function was cosine, we choose the real part as the...
  41. Manoj Sahu

    Why is the phase of an imaginary signal 90 degrees?

    Like we write "R+jX" where jX is considered to be imaginary term and it said to have 90 degree phase shift.
  42. sams

    I Finding Real and Imaginary Parts of the complex wave number

    In Griffiths fourth edition, page 413, section 9.4.1. Electromagnetic Waves in Conductors, the complex wave number is given according to equation (9.124). Calculating the real and imaginary parts of the complex wave number as in equation (9.125) lead to equations (9.126). I have done the...
  43. S

    A Imaginary position operator

    Hello, some operators seem to "add up" and give real eigenvalues only if they are applied on the imaginary position, ix, rather than the normal position operator, x, in the integral : \begin{equation} \langle Bx, x\rangle \end{equation} when replaced by:\begin{equation} \langle Bix...
  44. Hawksteinman

    Imaginary value for magnetic field?

    [Mod Note: Thread moved from Classical Physics, hence no formatting template] So today we performed an experiment to measure the Earth's magnetic field in three dimensions. For one of the dimensions we got 6.6x10^-5 i We were wondering if an imaginary magnetic field was possible and what it...
  45. T

    B Imaginary Number

    I ran into such problem. Not sure if some one can help. $$\sqrt{-i^2}=\sqrt{-1\times i^2}=\sqrt{-1\times -1}=\sqrt{1}=1$$ I also have $$\sqrt{-i^2}=\sqrt{-1}\times \sqrt{i^2}=\sqrt{-1}\times i=i\times i=-1$$ Can anyone explain to me the inconsistencies?
  46. jdawg

    Laplace Transform with Imaginary Roots

    Homework Statement 4(d2x/dt2) +3x = t*e-3tsin(5t) Homework EquationsThe Attempt at a Solution So I know how to take the Laplace transform and find the function for the Laplace domain: X(s) = 10(s+3)/(((s+3)2+25)2)(4s2+3) + (10s/(4s2+3)) + (2/(4s2+3)) But trying to convert...
  47. M

    B Complex numbers imaginary part

    Hello everyone. Iam reading about complex numbers at the moment ad Iam quite confused. I know how to use them but Iam not getting a real understanding of what they actually are :-( What exactly is the imaginary part of a complex number? I read that it could in example be phase... Thanks in...
  48. L

    I Simplified imaginary unit to the power of imaginary unit

    Where is my simplification wrong?
  49. H

    5 degree equation contains imaginary value?

    Homework Statement Homework Equations polynomial equation $$(a+b)^2 = \sum_{k=0}^{n} \binom nk a^{n-k} b^k $$ The Attempt at a Solution i get $$ \frac {1} {32} + \frac {5} {16}\frac {3^{0.5}i} {2} +\frac {10} {8}\frac {3i^2} {4} + \frac {10} {4}\frac {3^{1.5}i^3} {8} + \frac {5} {2} \frac...
  50. S

    B Is infinity an imaginary number?

    Is infinity a imaginary number?