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. F

    CR Equations: Real & Imaginary Parts Satisfy Cont. & Diff.

    Complex differentiable <--> real and imaginary parts satisfy C-R eqns and are cont. Say we have a complex function f(z) we can break this into real and imaginary parts: f(z)=u(x,y)+iv(x,y)In my book I am told the following:(1) f complex differentiable at z0 in ℂ --> the Cauchy Reimann...
  2. A

    Misbehaving Imaginary Fractions

    Homework Statement Why is \sqrt{\frac{1}{-1}} \neq \sqrt{\frac{-1}{1}} when quite obviously \frac{1}{-1} = \frac{-1}{1} Homework Equations N/A The Attempt at a Solution By the above inequality, I mean when one calculates \sqrt{\frac{1}{-1}} as \frac{\sqrt{1}}{\sqrt{-1}}, and...
  3. A

    MHB Separate imaginary from real

    Is it possible to separate imaginary part from the real part in this question $\sin ^{-1} ( e^{i\theta}) $ I tired to find u such that $\sin u = e^{i\theta} $ $ \sin u = \cos \theta + i sin \theta $ $ \sin (x + iy) = \cos \theta + i \sin \theta $ $ \sin x \cos iy + \sin iy \cos x =...
  4. C

    Imaginary eigenvalues of gamma matrices

    Hi! I'm reading David Tong's notes on QFT and I'm now reading on the chapter on the dirac equation http://www.damtp.cam.ac.uk/user/tong/qft/four.pdf and I stumbled across a statement where he claims that (\gamma^0)^2 = 1 \ \ \Rightarrow \text{real eigenvalues} while (\gamma^i)^2 = -1 \...
  5. H

    Power spectrum for real, imaginary and complex functions.

    Homework Statement What can we say about the evenness and oddness of the power spectrum (|F(s)|^{2}) if the input fuction is purely real, purely imaginary or complex? I know that a real function will give an even power spectrum. But I can't prove it! Homework Equations F(s) =...
  6. E

    Question regarding imaginary numbers and euler's formula

    So, I was thinking about Euler's formula, and I noticed something interesting. Based on the fact that e^\frac{i\pi}{2} = 1 , it seems as though \frac{i\pi}{2} = 0. However, this doesn't make any sense. Not only can I not see how this expression could possibly equal 0, but that would imply that...
  7. K

    Can Lagrange's Undetermined Multiplier Have Imaginary Points?

    Hey guys, new here and this is my first post. Wondering if anyone could help me. So I've encountered a problem on Lagrange's undetermined multiplier. Usually i have no problem with these, but this one caught me off a little. g(x,y) = x^2 + y^2 - 4xy - 6 = 0 Find the points closest to the...
  8. G

    Using imaginary time to unify QM and GR

    I find this passage from A Brief History of Time a bit hard to believe. When he talks about using imaginary time for the purposes of calculation, is it the same like in the Schro eq which uses an imaginary number? How plausible is the following passage? Is using imaginary time a common practice...
  9. T

    Hi, I m trying to find out, what is imaginary unit/number. i^2

    Hi,I m trying to find out,what is imaginary unit/number.How to imagine it.I was reading many articles and analogies,but I still have some questions to get better idea what is it.So I m going to write some points,that are mystic for me. My question is maybe more philosophic than mathematic...
  10. phosgene

    Inequalities between a real number and an imaginary number

    Homework Statement I'm just having a problem with a step that's part of a larger problem. Specifically, if I have: \sqrt{2}i\leq\sqrt{2} I'm not sure what this actually means. If I ignore the i, each side is the same distance from the origin if I imagine both points on a graph...
  11. Z

    Understanding Symbolic Math in MATLAB: Real and Imaginary Parts

    I have such MATLAB problem: I create variables R1 RF R2 and w so: syms RF R1 R2 w then I write expression: 3*R1*w*(RF + 200)/((R2*w*29*i + 3)*(3*R1*w - 2*i)) which gives: (3*R1*w*(RF + 200))/((3*R1*w - 2*sqrt(-1))*(R2*w*29*sqrt(-1) + 3)) why sqrt(-1) and not i? furthermore? if I want real part...
  12. DryRun

    Solving Complex Equation: Real & Imaginary Parts of z=x+iy

    Homework Statement Given that the real and imaginary parts of the complex number z=x+iy satisfy the equation (2-i)x-(1+3i)y=7. Find x and y. The attempt at a solution I know it's quite simple. Just equate the real and imaginary parts, but i checked and redid it again, but the answer still...
  13. K

    Complex Integration - Poles on the Imaginary axis

    Homework Statement evaluate the integral: I_1 =\int_0^\infty \frac{dx}{x^2 + 1} by integrating around a semicircle in the upper half of the complex plane. Homework Equations The Attempt at a Solution first i exchange the real vaiable x with a complex variable z & factorize...
  14. T

    How does phase affect the Nyquist Diagram - imaginary axis, how should it look?

    Please consider http://gyazo.com/e5c5b4f7808a63e7e664440259ac3058 I agree with all notes made on that slide, but I don't actually get how they constructed the diagram from that? I understand that they line represents frequency so going to 0 to infinity means the line would travel from -0.5 to...
  15. H

    Imaginary numbers and the real part of the Schrodinger Equation

    At the moment I am studying the Schrodinger equation using this resource. In a 1D solution (sec 3.1 in the paper) they show that a wave function can be expressed as \Psi(x,t)=\sqrt{2}e^{-iE_{n_x}t}\sin (n_x\pi x) where n_x is the quantum number. And they show the real part of the solution in...
  16. C

    The principle argument when no imaginary part

    Homework Statement Its not really a problem I was just wondering if the argument for any negative real number but no imaginary part was always = to pi? ie -1, -2,-3, -0.65... is the arg(z)=pi for all these cases if so I am guessing for positive real numbers with no imaginary part then...
  17. P

    Splitting Infinite Series into Real and Imaginary Parts

    I need a quick reminder that this is (hopefully) true: Let \sum a_n be an infinite series of complex terms which converges but not absolutely. Then can we still break it up into its real and imaginary parts? \sum a_n = \sum x_n + i\sum y_n
  18. F

    Taylor expansion - imaginary coefficients?

    Homework Statement Find the first two non-zero terms in the Taylor expansion of \frac{x}{\sqrt{x^2-a^2}} where a is a real constantHomework Equations f(x)=f(x_0)+f^{\prime}(x_0)(x-x_0)+\frac{f^{\prime\prime}(x_0)}{2!}(x-x_0)^2+...+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n The Attempt at a Solution If...
  19. V

    Derivation of imaginary exponential function

    I would like to find derivations of exp(-ik0r) respect to k in order to calculate exp(-ik1r) by using Taylor expansion: exp(-ik1r) = (exp(-ik0r))(0) +(k1 -k0)(exp(-ik0r))(1)/1! + (k1 -k0)2(exp(-ik0r))(2)/2! + ... This expansion converges when the value of r is relative low (0.3 - 1.2)...
  20. A

    Why imaginary co-ordinates and complex numbers?

    Most of advance/modern physics has i(imaginary components like E and P are represented so ) in it..How does these imaginary co-ordinates or axes fit into application of physics which explains real world phenomenon..Hope my question sounds reasonable.?.THANK YOU IN ADAVANCE
  21. N

    Complex Analysis - Values of Real and Imaginary parts

    Homework Statement Simplify in terms of real and imaginary parts of x and y and sketch them. 1) Re \frac{z}{z-1} = 0 2) I am \frac{1}{z} ≥ 1 The Attempt at a Solution 1) \frac{x + iy}{x + iy -1} = 0 Am I allowed to just vanish the imaginary components here and have \frac{x}{x...
  22. N

    Complex Analysis - Value of imaginary part.

    Homework Statement Suppose both c and (1 + ic)^{5} are real (c \neq 0). Show that c = ± \sqrt{5 ± 2\sqrt{5}} Now use another method to show that either c = ± tan 36◦ or c = ± tan 72◦ The Attempt at a Solution I expanded it out, but I'm not entirely too sure how to solve this for...
  23. C

    What is the meaning of the imaginary part of the plane wave function

    The plane wave function sometimes could be represented as: U(\mathbf{r} ,t ) = A_{0} e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t + \phi)} and we could separate the expression above into: U(\mathbf{r} ,t = \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \phi) + i \sin(\mathbf{k}...
  24. M

    Simplifying with Imaginary Numbers

    Homework Statement I'm trying to see if what I have before the e match up with the correct answer. the correct answer is (2+.5i)e^(1+3i)x + (2-.5i)e^(1-3i)x The Attempt at a Solution This is what I have so far. I don't know how I would simplify anymore. Please help.
  25. D

    E to an imaginary power, equivalent expressions, inequal outcomes?

    I'm completely stumped. So is my high-school calculus teacher, but he hasn't done imaginary powers for forty-five years. Hopefully somebody can explain this... To clarify, I understand the reasoning between the following equation: e^{i x}=cos(x)+i sin(x) Now, I need to put some things...
  26. atyy

    Imaginary Time in Spin Foams: How is it Utilized in Quantum Gravity Theories?

    In QM and QFT, imaginary time is used to make the oscillatory path integral converge, and also to handle terms that are not semibounded in Minkowski spacetime. In CDT, imaginary time is also used after the path integral is restricted to "causal" configurations. How is the oscillatory...
  27. H

    Question about imaginary numbers

    Homework Statement For a real number x, √√(-x) equals : a) +x b) -x c) complex d) pure imaginary Homework Equations √-1 = i The Attempt at a Solution Here is what i did: If x is a positive real number then the answer comes out to be x^0.25 * √i (now what is square...
  28. M

    Complex / imaginary numbers trigonometric graphs

    Hi, I have been representing complex numbers in graphical form in school recently. My teacher was telling me about a graph which shows all 4 quadrants and basically shows you what each quadrant is in terms of pi. Hopefully you understand what I mean, I have been looking on the internet for this...
  29. D

    Splitting an exponential complex number into real and imaginary parts

    e-z2 where z is a complex number a+ib
  30. G

    Derivative of a complex function in terms of real and imaginary parts.

    Hi, I wonder if anyone knows when (maybe always?) it is true that, where z=x+iy \text{ and } f : \mathbb{C} \to \mathbb{C} \text{ is expressed as } f=u+iv, \text{ that } f'(z)=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}? I'm pretty sure that this is true for f=exp. I...
  31. R

    Imaginary numbers concept help

    I've been learning about imaginary numbers and while I understand the concept of them I have tried a few examples with them and I don't get some of the answers. why can you not take xi = 90i and multiply it by i x*i*i = 90*i*i x*-1 = 90*-1 -x=-90 x=90 Thanks AL
  32. N

    Are imaginary numbers really necessary?

    You guys are probably sick of people who know little math posting here, but there's something that's been bugging me. I've bought The Feynman Lectures on Physics and have been reading through it slowly, and I'm up to the part where he talks about probability amplitudes of the electrons/photons...
  33. M

    Real or Imaginary? Solving x^6 + 1

    My friends teacher posted a sample problem saying find the real roots of x^6 + 1. Is this a trick question? All roots for this function are imaginary right?
  34. D

    Resistor Heat Dissipation (with imaginary numbers)

    Friends: I am wondering about heat dissipation when you have imaginary numbers. Lets say a current I = (3 + 4j) Amps is going through an impedance Z = (2 + 3j) Ohms. What is the amount of heat dissipated by the impedance? I think that you take the magnitude of the current, |I| = 5 Amps, and...
  35. R

    Why does the graph of y = ln(x) have an imaginary part?

    I was messing around with wolframalpha and tried to make this graph. http://www.wolframalpha.com/input/?i=plot+y+%3D+log+x Now I understand why the blue real part exists and has that shape but I don't understand why it has an imaginary part? Thanks AL
  36. B

    Show that the real and imaginary parts of the wavenumber, k, are given by

    Homework Statement Show that the real and imaginary parts of the wavenumber, k, are given by k(real)=[sqrt(epsilon(real))]omega/c and k(imaginary)=[epsilon(imaginary) *omega/(2c sqrt(espilon(real))) The Attempt at a Solution k^2= mu epsilon omega^2 (1+(i g/epsilon*omega)) k^2...
  37. L

    Imaginary numbers what to do with them

    I have attached an image showing the three possible solutions (as determined by Mathematica) when solving for the peak velocity(Vs) in a trapezoidal move where the following are already known: distance(d),total time(t),units of acceleration(Ma),units of deceleration(Ma),initial velocity(Vi)...
  38. F

    Imaginary Center of Mass? What is that?

    Suppose I give you a curve f(x) = \sin^2 (x) + ln(x) And suppose I tell you to rotate this curve about the x axis, we get disks. Now, I ask you, what is the center of mass of this object? Now immediately, you could say that \bar{y} = 0 because it is symmetric about the x-axis. I don't...
  39. J

    Simplification with exponentials and imaginary numbers

    Hi, I'm using euler's identity : exp(i∏) = exp (-i∏) = -1 to simplify the equation after integrating it. [PLAIN]http://img443.imageshack.us/img443/5504/captureikm.jpg Note: the equation to be integrated is exp(0.5it) + exp(-0.5it) and they have simplified it, it was actually a cos(0.5t)...
  40. J

    Imaginary Ideal Gas Cycle Proof

    Homework Statement Given an imaginary ideal-gas cycle. Assuming constant heat capacities, show that the thermal efficiency is η = 1 - γ[((V1/V2)-1)/((P3/P2)-1)] Since i can't show you the cycle we are shown that l Qh l = which is absolute value of the heat at high temperature =...
  41. A

    Imaginary fraction questions. lost.

    im studying for my circuits midterm and the proff has handouts with questions and answers but not detailed answers. i can't figure out how he went from a fraction to an answer. (-j2)(2+j2)/-j2+2+j2 the answer on the paper is 2-j2 i do not know what I am allowed to do with the 2 next to...
  42. L

    Imaginary numbers multiply and divide

    I don't really understand how to multiply and divide when numbers are in a+bi form
  43. T

    Medical How logisticlly are imaginary images formed in the brain?

    Is it a collage of interacting nuerons and the cohesive accumulation of information? Or something else, don't tell me its dependent on quantum proporties?
  44. vibhuav

    Physical meaning of imaginary numbers

    Can someone give a physical meaning for imaginary numbers? The imaginary numbers, in my opinion, are truly imaginary. What do they even represent? Irrational numbers are, well, preposterous but I can accept them. √2, π and φ have some tangible meaning, but √(-1)? What does it mean? A solution...
  45. N

    What causes Bragg planes in materials?

    I am having trouble understanding what exactly Bragg planes are physically. I understand how they behave, in that they act like mirrors and reflect matter waves, but what exactly is the wave bouncing off of? for instance I can guarantee any physics textbook always has a picture like...
  46. G

    MATLAB mistaken Inertia I for imaginary number what do i do?

    MATLAB tends to mistake I or i for imaginary number when ever I try to use it as a variable, for example when do with a pendulum with a mass attached to it then 2 pi f = sqrt( (m g L)/I ) were I is the inertia not imaginary number but when I try to get MATLAB to solve this equation for L it...
  47. G

    Simplifying Rational Imaginary Functions

    Why does (5 i)/(3 (1-i)) = -5/6+(5 i)/6
  48. L

    On the influence of our imaginary on string theory's beauty

    Dear physicists, Please forgive my naive and general question but I have a something in mind that I would like to answer... String theorists say something like: "Although string theory cannot be tested, the mathematical beauty coming from this theory is such that it is very convincing and...
  49. K

    How Can I Remove Imaginary Parts from a Complex Number Expression?

    Hi all, I have this expression containing complex numbers and I wanted the expression to be displayed with real parts only. How can i do this? For instance, the original expression is, eqn = (16.0001+3.16141*10^-21 i)-(0.00860351-1.16927*10^-18 i) Ao[1]+(0.00537811-4.47536*10^-19 i)...
  50. I

    What does it mean for a wave to have an imaginary part?

    This post is in General Math because it is focuses on the complex plane and justifications for using it. I do not understand what it means for a wave to have an imaginary part. I can understand expressing a wave as e^(iθ) and then extracting the information you want since complex...
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