Argand Diagrams

• The argument is the angle between the positive real axis and the line from the origin to the point on the plane representing the complex number.
• The modulus of a complex number represents its distance from the origin, while the argument represents its direction relative to the positive real axis.
• Complex numbers can be represented as points on an Argand diagram, with their coordinates being the real part and imaginary part respectively.
• The complex number z = x + iy can be represented by vector (x y) on an argand diagram
• For z = x + iy, the modulus is given by the square root of x^2 + y^2
• For z = x + iy, the argument, θ, satisfies tanθ = y/x
• Let α be the positive acute angle made with the real axis by the line joining the origin and z:

  • if z lies in the first quadrant then arg(z) = α
  • if z lies in the second quadrant then arg(z) = π - α
  • if z lies in the third quadrant then arg(z) = -(π - α)
  • if z lies in the fourth quadrant then arg(z) = -α
• For complex number |z| = r and arg(z)= θ, the modulus-argument form of z is z = r(cosθ + isinθ)
• For two complex numbers z1 = x + iy and z2 = x + iy, |z1 - z2| represents the distance between the points z1 and z2 on an Argand diagram
• Given that z1 = x + iy, the locus of points z on an Argand diagram such that |z - z1| = r or |z - (x1 + iy)| = r is a circle with centre (x1, y1) and radius r
• Given that z1 = x + iy and z2 = x + iy, the locus of the points z on an Argand diagram such that |z - z1| = |z - z2| is the perpendicular bisector of the line segment joining z1 and z2
• Given that z1 = x + iy, the locus of points z on an Argand diagram such that arg(z - z1) = θ is a half-line from, but not including, the fixed point z1 making an angle θ with a line from the fixed point z1 parallel to the real axis