2 interactive labs · AS & A Level Mathematics 9709
Argand Diagram: Operations on z
3.9 Complex numbers · Drag z₁ and z₂ on an Argand diagram, predict, then see z*, z₁ ± z₂, z₁z₂ and z₁ ÷ z₂ worked in Cartesian and polar form.
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About this lab
Drag z₁ and z₂ on an Argand diagram, predict, then see z*, z₁ ± z₂, z₁z₂ and z₁ ÷ z₂ worked in Cartesian and polar form.
You will be able to
Calculate Re z, Im z, |z|, arg z and z* for a complex number, giving arg z in radians in −π < θ ≤ π (or 0 ≤ θ < 2π when asked), and describe z* as the reflection of z in the real axis.
Carry out addition, subtraction, multiplication and division of complex numbers in Cartesian form x + iy with full working, dividing by multiplying top and bottom by the conjugate of the denominator.
Use |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg z₁ + arg z₂ (and the corresponding results for division), and describe multiplication by z₂ as a rotation through arg z₂ with an enlargement by |z₂|.
Explain why |z₁ + z₂| is not |z₁| + |z₂| in general, using the parallelogram on an Argand diagram.
Loci in the Argand Diagram
3.9 Complex numbers · Set |z − a| ◇ k, |z − a| ◇ |z − b| or arg(z − a) ◇ α, predict whether z satisfies it, then see the locus or region and combine two conditions.
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About this lab
Set |z − a| ◇ k, |z − a| ◇ |z − b| or arg(z − a) ◇ α, predict whether z satisfies it, then see the locus or region and combine two conditions.
You will be able to
Describe the locus |z − a| = k as the circle with centre a and radius k, and |z − a| < k and |z − a| > k as the inside and outside, with the boundary dashed or solid according to the inequality.
Describe |z − a| = |z − b| as the perpendicular bisector of the line joining a and b, and |z − a| < |z − b| as the side containing a.
Describe arg(z − a) = α as a half-line from a at angle α to the positive real direction, with a itself excluded, and use −π < arg ≤ π for inequalities on arg.
Determine which conditions a given z satisfies, and shade the region where two conditions both hold, showing where their boundaries meet.
