11Some more doubts
2) Every real number is purely real and viceversa. But an imaginary number may not be purely imaginary and a purely imaginary number may not be an imaginary number? WHY? I am really confused :(
1) Why and how any root of an imaginary number is an imaginary number?
I would be grateful if someone give me a deep knowledge about it. :)
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11 Answers
71We can get this result from EUler's Formula:
by substituting x = π/2, giving
Now we have to find the root. So take the Square root on both sides.
which again, through application of Euler's formula to x = π/4, gives
Thus Root of an Imaginary no are Imaginary!
11113) A complex number z is said to be an imaginary number if imaginary part of z is non zero and it is said to be purely imaginary if its real part is zero.(imaginary part may be zero or non zero)
Can I say that "A complex number z is said to be a real number if real part of z is non zero and it is said to be purely real if its imaginary part is zero.(real part may be zero or non zero)
114)While writing a polar form of a complex number if we write principal value of the argument, it is not said to be in polar form. Why?
For example, let z = -1 - i
arg of z = 5Ï€/4
principal value of argument = (-3Ï€/4)
But while writing polar form we will write √2[cos (5π/4) + i sin(5π/4)] and not
√2[cos(3π/4) - i sin (3π/4)].{its written in the book that √2[cos(3π/4) - i sin (3π/4)] is not the polar form}
Please someone tell why is it so?
Thanks is advance :)
11Well for the 2nd and 3rd , Hodge Conjecture posted answer in mathlink and the same I am posting here,
Here's what I think
2)An imaginary number may not be purely imaginary.: Set the real part to a nonzero number
A purely imaginary number may not be an imaginary number. Set the real and imaginary parts to zero. It is purely imaginary but real at the same time (I think that's what it means, but it doesn't seem to make much sense).
3)For your last one, a real number always has an imaginary part of zero.
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=366706
71This might help:
Polar form of a complex number z = a + ib is z = r(cosΘ + isinΘ)
where angle Θ is known as argument(or amplitude) of z, written as arg(z) or amp(z) and is determined by
tanΘ = (b/a) or Θ = tan-1(b/a).
The unique value of Θ such that -π < Θ ≤ π for which a = rcosΘ and b = rsinΘ, is known as principle value of argument.
The general value of the amplitude is (2nπ + Θ), where n is an integer and Θ is the principal value of arg(z).
NOTE: Θ = Theta and π = pie
So Probably ! I think you have understood the difference between argument and Principal value of argument and where they are used where they aren't.
11Thanks :)
Its mean that we have to always use general value of argument while writing the polar form.