Turn the plane.
Three Further Maths instruments: move a complex number and watch its square and conjugate follow, feed a 2x2 matrix a shape to distort, and grow a polar curve petal by petal.
Argand diagram · z, z̄, z²
FIG. 01 · PURE B · COMPLEX NUMBERSOr drag the accented point on the diagram. The dashed gold point is the conjugate; the pale point is z².
cart
mod-arg
expo
z²
Exam technique. Multiplying multiplies moduli and adds arguments — squaring therefore squares the modulus and doubles the argument, which is why z² swings round twice as fast as you drag. The n roots of any complex number always sit on one circle of radius |z|1/n, evenly spaced 2π/n apart, so once you have found one root you can write the rest down. Keep arg z in (−π, π] unless told otherwise.
Matrices · the determinant is an area
FIG. 02 · PURE F · MATRICESM
Load a standard one
Exam technique. The columns of M are exactly the images of i and j — read a transformation straight off the matrix rather than solving anything. |det M| is the area scale factor; a negative determinant means the orientation has flipped, so the F comes back mirror-image; and det M = 0 squashes the whole plane onto a line, which is precisely when M⁻¹ does not exist.
Polar curves · r = a + b cos nθ
FIG. 03 · PURE H · POLAR COORDINATESExam technique. Area in polars is ½∫r² dθ, never ∫y dx — and the limits are the angles where the curve starts and stops sweeping the region you want, which is usually where r = 0. AQA works with r ≥ 0, so switch that toggle off only to see what the negative-r branch would draw. Tangents parallel to the initial line come from dy/dθ = 0, where y = r sin θ.