What is Imaginary number: Definition and 37 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.

View More On Wikipedia.org
  1. BurpHa

    Find all possible solutions of x^3 + 2 = 0

    I actually know one way to solve, ##x ^ 3 + 2 = 0## ##x ^ 3 + \left (\sqrt[3] 2\right) ^ 3 = 0## ##\left(x + \sqrt[3] 2\right) \left(x ^ 2 - x\sqrt[3] 2 + \left(\sqrt[3] 2 \right)^2\right) =0## ##x + \sqrt[3] 2 = 0, x = -\sqrt[3] 2## ##x ^ 2 - x\sqrt[3] 2 + \left(\sqrt[3] 2\right)^2 = 0, x =...
  2. 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...
  3. 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?
  4. 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 +...
  5. S

    How to write the complex exponential in terms of sine/cosine?

    I apologize in advance if any formatting is weird; this is my first time posting. If I am breaking any rules with the formatting or if I am not providing enough detail or if I am in the wrong sub-forum, please let me know. 1. Homework Statement Using Euler's formula : ejx = cos(x) + jsin(x)...
  6. T

    B Understanding the Inconsistencies of Imaginary Numbers

    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?
  7. 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...
  8. Cocoleia

    Working with phasors (Circuits, such as complex power)

    Homework Statement I am going over examples in my textbook and I came across this: I don't understand how they converted 18.265 at angle of 39.9 to 14.02+j11.71 Homework Equations I know how to convert from the imaginary numbers into the angle form, usually I use: Is there another equation...
  9. S

    B Is infinity an imaginary number?

    Is infinity a imaginary number?
  10. AaronQ

    B Predicting outputs of f(x)=(1+i)^x

    I got bored a while back and deiced to create a table of the integer inputs of f(x)=(1+i)^x and I noticed quiet a few patterns which I am trying to catalog here, although most of my work so far deal with Natural inputs, all patterns continue into the negative, see here, I was wondering if anyone...
  11. NatFex

    I Proving De Moivre's Theorem for Negative Numbers?

    Or basically anything that isn't a positive integer. So I can prove quite easily by induction that for any integer n>0, De Moivre's Theorem (below) holds. If ##\DeclareMathOperator\cis{cis} z = r\cis\theta, z^n= r^n\cis(n\theta)## My proof below: However I struggle to do this with...
  12. 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 =...
  13. 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
  14. Byeonggon Lee

    B 1 = -1, which part of this proof is wrong?

    Of course 1 isn't same as -1. This proof must be wrong but I can't find which part of this proof is wrong. Could you help me with this problem? (1)$$1 = \sqrt{1}$$ (2)$$= \sqrt{(-1)(-1)}$$ (3)$$= \sqrt{(-1)} \cdot i$$ (4)$$= i \cdot i$$ $$=-1$$
  15. 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...
  16. T

    How old is i? (Imaginary number)

    A couple of things I've read or heard in class suggest it's been around for a very long time. I got the impression it's from antiquity, but am I wrong?
  17. 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
  18. 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.
  19. 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...
  20. 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...
  21. 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...
  22. 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...
  23. M

    Can speed be a imaginary number validity of work energy theoram in 1D

    Consider mass m_{1}and m_{2}with position vector (from an inertial frame) \overrightarrow{x_{1}} and \overrightarrow{x_{2}} respectively and distance between them be x_{0}. m_{1}\frac{d^{2}}{dt^{2}}\overrightarrow{x_{1}}=\overrightarrow{F} \Rightarrow...
  24. C

    Imaginary number -i raised to negative power

    Homework Statement I came across this expression in homework and for the life of me I can't figure out how this evaluates to 0: 1 - ( -i )^-4 = 1 - 1 = 0 I know that i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. I'm just not sure how to treat the negative on the i. Do I just treat i as if it...
  25. R

    How do you determine the cube root of (-1+i) in complex numbers?

    I'm doing some practice problems for my mechanics exam tomorrow (good ol' SHM) and I can't solve this for the life of me: Determine: (-1+i)^(1/3) Any help would be greatly appreciated.
  26. 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...
  27. 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...
  28. 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...
  29. R

    Square root of an imaginary number

    Where on the imaginary number axis do i graph sqrt(3i)? At sqrt3?
  30. 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...
  31. 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...
  32. 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]...
  33. T

    Wick rotation and imaginary number

    Can you use Wick rotation to turn any real variable to an imaginary one, not necessary time, such that your integration converges, and then, return back to the real? I'm not really sure how to use Wick rotation.
  34. G01

    Whats wrong with this imaginary number problem?

    I saw this thing where someone proved that the imaginary number, i, the sqrt(-1) was equal to 1. here it is: i= sqrt(-1) i^2 = [sqrt(-1)]^2 i^2 = sqrt(-1) * sqrt(-1) i^2 = sqrt(-1*-1) i^2 = sqrt(1) i^2 = 1 so i = 1 I know there's something wrong here but i can't...
  35. J

    Imaginary Number to Indicate Division by Zero?

    Forgive me if I'm being ignorant, but this recently occurred to me. We all know division by zero is undefinfed, but \sqrt {-1} used to be undefined too, until i was created. Has anyone ever proposed an imaginary number to indicate the result after division by zero?
  36. A

    How do you test the convergence of i^n (imaginary number) or (-1)^n?

    I was thinking alternating series test, but can't that only be used for testing for convergence, not for testing divergence?
  37. V

    Derivative of an imaginary number

    I was just wondering if anyone knows the rule when taking the derivative of an imaginary number(i). For example: d(ix)/dx=? Thanks:)