8.2 Argand Diagrams

The modulus of a complex number  is the distance of the point  from 0 (i.e. from the origin) in an Argand diagram.

For a complex number  given in  form, where  and  are real numbers, the modulus of  may be calculated by using the formula

Calculate the modulus of each of the following, giving your answers as exact values:

The argument of a complex number  is the angle in an Argand diagram measured (in radians!) from the positive real axis to the position vector of . The positive direction is defined to be anticlockwise from the positive real axis (so a negative argument means a clockwise measurement). 

The following diagram summarises the method for finding the principal argument of a complex number, i.e. the argument that lies in the interval  :

Write down the principal arguments of

Find the principal arguments of the following complex numbers, giving your answers as exact values:

Find the principal arguments of the following complex numbers, giving your answers correct to 3 significant figures:

The complex number  is such that  ,  where    is a real number constant.

Show that  .

Given that ,  find the two possible values of .

For each of the values of  found in part (b), find the principal argument of the corresponding version of  .  Give your answers correct to 3 significant figures.

Given additionally that   lies in the fourth quadrant of the Argand diagram, find the precise value of  .

It is possible to express a complex number  in modulus-argument form,  i.e. in the form

where  and  .

Express the complex number   in modulus-argument form.  The value for should be in radians correct to 3 significant figures, and should be given in the interval .

A complex number has modulus 14 and argument .  Using the fact that

express  in the form  ,  where  and   are real numbers given as exact values.

By considering the definition of modulus-argument form above, explain why the following numbers are not in modulus-argument form:

When multiplying and dividing two complex numbers  and  ,  the moduli and arguments of the numbers are connected by the following sets of relationships:

When the numbers are given in modulus-argument form as  and  , where ,  these relationships mean that

Use those results to work out the following, giving your answers in modulus-argument form:

If the complex number    is represented by a fixed point in an Argand diagram, then the value of the modulus    gives the distance in the Argand diagram from    to any other complex number  .

It follows from this that the locus (i.e., set of points) in an Argand diagram for which an equation of the form

is true (where   is a real number constant), is a circle of radius  with its centre at  .

On an Argand diagram, sketch the locus for which the equation

is true.

Use your sketch from part (a) to write down a complex number   that satisfies each of the following inequalities:

If the complex number    is represented by a fixed point in an Argand diagram, then the value of the modulus    gives the distance in the Argand diagram from    to any other complex number  .

It follows from this that the locus (i.e., set of points) in an Argand diagram for which an equation of the form

is true, is the perpendicular bisector of the line segment connecting the points  and  .

On an Argand diagram, sketch the locus for which the equation

is true.

Use your sketch from part (a) to write down a complex number   that satisfies each of the following inequalities:

If the complex number    is represented by a fixed point in an Argand diagram, then the value of    gives the angle measured (in radians!) from the half-line that starts at  and goes right in the same direction as the positive real axis, to the line segment connecting  and :

As with the argument of a complex number, the positive direction is defined to be anticlockwise from the line parallel to the positive real axis.

It follows from this that the locus (i.e., set of points) in an Argand diagram for which an equation of the form

is true (where  is a real number constant), is the half-line starting at    and going off at a direction that makes an angle of    radians measured anticlockwise from the line going to the right from    parallel to the positive real axis.  Note that    is not included in the locus.  If the value of    is negative, it means that the angle is measured clockwise instead.

On an Argand diagram, sketch the locus for which the equation

is true.

Use your sketch from part (a) to write down a complex number   that satisfies each of the following inequalities:

(i)                (ii)