Complex numbers

Complex Numbers in Cartesian FormSign up

understand the idea of a complex number; recall real/imaginary part, modulus, argument and conjugate (Re z, Im z, |z|, arg z, z*); equality iff real and imaginary parts equal · addition, subtraction, multiplication and division of complex numbers in Cartesian form x+iy with full working · for a polynomial equation with real coefficients, non-real roots occur in conjugate pairs (e.g. solving a cubic/quartic given one complex root) · find the two square roots of a complex number in exact Cartesian form (e.g. square roots of 5+12i)

The Argand Diagram and Polar (Modulus-Argument) FormSign up

represent complex numbers on an Argand diagram, plotting z, its conjugate and sums/differences · multiplication and division in polar form r(cos theta + i sin theta) / r e^(i theta), including |z1 z2|=|z1||z2|, arg(z1 z2)=arg z1+arg z2 and the division results · geometrical effects of conjugating and of adding, subtracting, multiplying and dividing complex numbers

Loci in the Argand DiagramSign up

illustrate simple equations and inequalities involving complex numbers by loci in an Argand diagram, e.g. |z-a|<k (disc/circle), |z-a|=|z-b| (perpendicular bisector), arg(z-a)=alpha (half-line)