Fundamental plane (elliptical galaxies)
In geometry, an imaginary line is a straight line that only contains one real point. It can be proven that this point is the intersection point with the conjugated line.
It is a special case of an imaginary curve.
It can be proven that there exists no equation of the form in which a, b and c are all real coefficients. However there do exist equations of the form , but at least one of the coefficients need be nonreal.
As follows, it can be proven that, if an equation of the form in which a, b and c are all real coefficients, exist, the straight line is a real line, and it shall contain an infinite number of real points.
This property of straight lines in the complex projective plane is a direct consequence of the duality principle in projective geometry.
In the complex plane (Argand Plane), we have a term called "imaginary axis".In Argand plane, y-axis is imaginary axis. All numbers in this axis are in form of 0+ib form.
Argument
An argument is the angle or projection of any complex number in the Argand plane on the real axis (x-axis), denoted Arg(z). The argument can be easily found by following procedure:
If a+ib is any complex number foming angle A on real axis then,
cosA = a/√a^2+b^2 sinA= b/√a^2+b^2 tanA=b/a
arg(z)=A
Properties of argument
- arg(AxB)=arg(A) + arg(B)
- arg(A/B)=arg(A) - arg(B)
- arg(z)=0 if and only if z lies in +ve real axis
- arg(z)=180 if and only if z lies in -ve real axis
- arg(z)=90 if and only if z lies in +ve imaginary axis
- arg(z)=-90 if and only if z lies in -ve imaginary line
- arg(z) lies in (0,90) in first quadrant, in (90,180) in 2nd quadrant, in(-180,-90) in 3rd quadrant, in(-90,0) in 4th quadrant.
Domain of argument = R Range = (-180,180)
Modulus
Modulus of any complex no. a+ib is
mod(z)=√a^2+b^2
In Argand plane, modulus denotes distance between a complex number and the origin (0,0).
Example: mod(z)=2 denotes locus of all complex numbers z lying in circle of radius 2 at centre (0,0)