# Imaginary line (mathematics)

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 $ax+by+cz=0$ in which a, b and c are all real coefficients. However there do exist equations of the form $ax+by+cz=0$, but at least one of the coefficients need be nonreal.

As follows, it can be proven that, if an equation of the form $ax+by+cz=0$ 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

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)