Loci involving Complex Numbers

1. Loci involving Complex Numbers

2. Modulus • For real numbers, |x| gives the distance of the number x from zero on the number line • For complex numbers, |z| gives the distance of the number z from the origin in an Argand diagram • The locus of points representing the complex number z, such that |z| = 2 means all points 2 units from the origin

3. Modulus • |z - a| gives the distance of z from a • |z - a| = r gives a circle and |z – a| = |z – b| gives a perpendicular bisector • For more complicated questions, may be easier to use |z| = √(x2 + y2) • |z + 4| = 3|z| • ↔ (x + 4)2 + y2 = 9(x2 + y2) • ↔ 8x2 – 8x – 16 + 8y2 = 0 • ↔ (x – ½ )2 + y2 = 9/4

4. Modulus Resources • Flash: Investigation of Loci • Excel: Spreadsheet Investigating Loci

5. Argument • If z and w are complex numbers represented by points Z and W in the Argand diagram, z-w can be represented by the translation from W to Z • Like position vectors in C4, WZ = z – w • So arg (z – (a + bi)) = θ gives the set of all possible translations (vectors) from a +bi in the direction given by θ