Complex numbers

• 1. $i =$$\sqrt-1$
  2. $i^2 =$$-1$
  3. $i^3 =$$-i$
  4. $i^4 =$$1$
• The conjugate of a complex number has an imaginary part of the negative of the imaginary part of the first complex number - it is represented by z*
• To find the square roots of a complex number:

  1. Define the given expression as $z^2$
  2. Define $z=$$a+$$bi$
  3. Square the algebraic form of the equation and simplify
  4. Compare the real and imaginary parts of the two equations
  5. Solve the equations simultaneously
  6. Write both roots in complex form
• To remove a complex denominator, multiply by the conjugate of the denominator
• A complex number multiplied by its conjugate is always real
• An Argand diagram has the real component on the x-axis and the imaginary component on the y-axis
• A complex conjugate represented on an Argand diagram is a reflection of the original complex number in the real axis
• The modulus is the distance of a point from the origin. It is represented by r or |z|
• The argument is the angle anticlockwise relative to the real axis. If the point is below the real axis, then the argument is negative. The argument is measured in radians
• The equation |z-a| = r represents a circle with a radius r and centre a
• The equation |z-a| = |z-b| represents the perpendicular bisector of the line joining points a and b
• The equation $arg(z-a) =$$\theta$ represents a half line that starts at point a and has an argument of $\theta$
• |zw|=|z||w|

  arg(zw) = arg(z) + arg(w)

  |z/w| = |z|/|w|

  arg(z/w) = arg(z) - arg(w)