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Loci involving Complex Numbers

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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

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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| = (x 2 + y 2 ) |z + 4| = 3|z| (x + 4) 2 + y 2 = 9(x 2 + y 2 ) 8x 2 – 8x – 16 + 8y 2 = 0 (x – ½ ) 2 + y 2 = 9/4

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Modulus Resources Flash: Investigation of Loci Excel: Spreadsheet Investigating Loci

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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

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STROUD Worked examples and exercises are in the text PROGRAMME 2 COMPLEX NUMBERS 2.

STROUD Worked examples and exercises are in the text PROGRAMME 2 COMPLEX NUMBERS 2.

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