Complex numbers: Lecture 6

We learnt 3 important things in today's lecture.
1. Geometrical interpretation of multiplication/division of complex numbers
2. Loci in the form of |z-a|=r
3. Loci in the form of |z-a|=|z-b|

1. Geometrical interpretation of multiplication/division of complex numbers

• Multiplying a complex number z by i is equivalent of rotating z anti-clockwise about the origin by pi/2
• Multiplying a complex number z by -1 is equivalent of rotating z anti-clockwise about the origin by pi
• In general, multiplying a complex number z by another complex number with modulus k and argument x is equivalent of rotating z anti-clockwise about the origin by x and scale z by a factor of k.

2. Loci in the form of |z-a|=r

• Locus of |z-a|=r means the collection of all points that satisfy the equation, i.e. all the points whose distance to complex number a is r.
• Points with fixed distance to a given point a forms a circle with center at a and radius r.
• If the equation is changed to an inequality |z-a|>r or |z-a|< r, it means all the points whose distance to a is greater or smaller than r, which is the region outside or the interior region of the circle.

3. Loci in the form of |z-a|=|z-b|

• Locus of |z-a|=|z-b| means the collection of all points that satisfy the equation, i.e. all the points whose distances to a and b are equal.
• Points with equal distances to two fixed points a and b forms the perpendicular bisector of the line ab.
• If the equation is changed to an inequality |z-a|<|z-b|,it means all the points whose distance to a is smaller than the distance to b, which is the region on one side of the perpendicular bisector that is closer to a.
• If the equation is changed to an inequality |z-a|>|z-b|, it means all the points whose distance to a is greater than the distance to b, which is the region on one side of the perpendicular bisector that is closer to b.

Remark: when you draw the loci of an inequality, take note of whether the equal sign is included. If the circle or the perpendicular bisector itself are not included, you have to use dotted line.

After today's lecture, you can do the whole tutorial 10b and Q1,2, 4, 6, 8 and 9 in tutorial 10c.