Complex numbers

· i = √-1

· a+bi is a complex number, any multiple of i is an imaginary number.

· The real part (a) is plotted as an x-coordinate, the imaginary part (bi) is plotted as a y-coordinate.

· |z| is the modulus of a complex number (the distance between the point and the origin).

· Use Pythagoras’ theorem to get the modulus.

· z = a + bi, √z = c + di, where 2cd = b and c² - d² = a.

Arguments and argand diagrams

· Argument (arg(z))= smallest angle (radians) between the modulus and the +ve real-axis.

· In the top right quadrant ~ tan =real/imaginary.

· In the bottom right quadrant ~ tan =real/imaginary (size of angle is +ve but angle is below axis).

· In the top left quadrant ~ arg(z) = π - ||.

· In the bottom left quadrant ~ arg(z) = -(π - ||).

· Summary = Angle above axis has +ve argument, below has -ve argument.

· - π < arg(z) ≤ π

· For |z| = r, where r > 0, draw a circle with radius r and the origin as the centre.

· For |z – c| = r, where r > 0, the circle is centred on c with radius r.

· For |z – a| = |z – b|, the complex numbers that solve for Z are shown by a perpendicular bisector of the line joining a and b.

· For arg(z – c) = α, where - π…