What is Imaginary numbers: Definition and 78 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. 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
  2. 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! =)
  3. 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
  4. 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...
  5. 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...
  6. 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...
  7. 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...
  8. 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?
  9. 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...
  10. 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...
  11. 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?
  12. P

    Solving Wave Equation / Imaginary Numbers

    Homework Statement Consider the simplified wave function: \psi (x,t) = Ae^{i(\omega t - kx)} Assume that \omega and \nu are complex quantities and that k is real: \omega = \alpha + i\beta \nu = u + i\omega Use the fact that k^2 = \frac{\omega^2}{\nu^2} to obtain expressions for \alpha and...
  13. B

    What is the absolute value of imaginary numbers, why not supernatural numbers?

    what is the absolute value of imaginary numbers, why not "queer" numbers? the square root of -1 is "i". the absolute value of an interger is itself, and of a negative number, it is a positive interger. |-5| = 5 |5| = 5 what is |5i| = ? |-5i| = ? why not invent a queer number...
  14. N

    Factoring Imaginary Numbers in a 2x1 Matrix: Solving the Physics Problem

    This is a physics problem but I am having trouble factoring this matrix. Basically, there shouldn't be anything left inside the matrix except 0's, 1's, or i's (any of which can be negative). This seems like such an easy problem but I cannot find something that works. Any ideas? \frac {1}...
  15. R

    Imaginary numbers entertwined with quadratics

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  16. J

    Solving 2 inequalities with imaginary numbers?

    I have 2 equations, imaginary ones, and 2 unknowns...trying to solve for them..but the answer i got, works with one, but not the other: i*Z1 - i*Z2 = -2 - i Z1 + 3i*Z2 = 4 + 7i where i is the imaginary number, and Z1 and Z2 are the 2 unknowns the answer i got: Z1 : 1.33333 +...
  17. Q

    Imaginary numbers past calculus level

    My teacher in the charter school I go to wanted to be a mathematition. He said calculus was no problem for him and he got past vector calculus, although he can't remember because it was so long ago. He said he got stuck on imaginary numbers past the caluclus level and this made him quit is...
  18. A

    Capacitor/Inductor Imaginary Numbers

    Dear All, Why do we introduce complex numbers when talking about the voltage behaviour through capacitors and inductors. Any help would be appreciated, Thanks
  19. P

    What are Imaginary Numbers used for in mathematics?

    I have a very simple question. What are Imaginary Numbers (i.e. \sqrt[4]{-16}=2\mbox{i}) used for in mathematics besides negetive roots with an even index? Thank you in advance... ---- Life is a Problem... SOLVE IT!
  20. D

    Why were Imaginary numbers invented

    Why was i invented?
  21. E

    Epsilon Pi's ideas on imaginary numbers

    It is known that it was Descartes the one that gave the symbol i the connotation of imaginary; in electrical engineering there is the concept of apparent power(MVA) S = P + i Q where P(MW)=generation or consumed power and Q(MVAR) = reactive power and they both can be measured, so they...
  22. E

    Hermitian Operators and Imaginary Numbers

    So I understand what a hermitian operator is and how if A and B are hermitian operators, then the product of AB is not necessarily Hermitian since *Note here + is dagger (AB)+=B+A+=BA I also recognize that (AB-BA) is not Hermitian since (AB-BA)+=B+A+-A+B+ In addition, I know that...
  23. N

    What is the purpose of imaginary numbers and how do they work?

    Sorry if this isn't the right forum, I didn't know so I just went to general. Could someone explain how this i (imaginary numbers) thing works? I know i is supposed to be a number which is the sqrt of a negative number, which isn't supposed to exist, but what's its use? And yeah...really any...
  24. A

    Solving Imaginary Numbers: What is x^^2 = x^x = -4?

    I've been doing a lot of thinking about imaginary numbers lately. My first question was "What is sqr(i)?". I thought it was unsolvable until I punched it into my trusty (and often right) caclulator and found out it was (sqr(.5)i + sqr(.5))^2 So obvious now. Of course. Anyways, a...
  25. Z

    Finding Imaginary Numbers with Cosine of 3

    Question: Use the formula cos(x + iy) = cos(x)cos(iy) - sin(x)sin(iy) to find two imaginary numbers whose cosine is 3 My workings so far: cos(x)cos(iy) - sin(x)sin(iy) = 3 Therefore: cos(x)cosh(y) - i(sin(x)sinh(y)) = 3 cos(x)cosh(y) = 3 sin(x)sinh(y) = 0 sin(x) = 0 x = 0...
  26. E

    Solve 3-7i/2+3i: Imaginary Numbers

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  27. A

    Solving an Algebra Problem: Imaginary Numbers

    I have an Algebra problem giving me trouble. I am supposed to determine without graphing, if the following equation is symetrical to the x or y-axis or to the origin. The equation is y^2 = -5/x^2. The problem is, I think that this is an imaginary number. Am I right about that?
  28. J

    What are imaginary numbers and how and why are they used in physics?

    What are imaginary numbers and how and why are they used in physics? Please could you try and make your answers as simple as possible and bear in mind that I have not even finished my GCSE course in maths yet.
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