Further Complex Numbers (CIE A Level Maths: Pure 3)

Express the following complex numbers in exponential form:

Given the point on an Argand diagram, where  is a complex number, describe the geometrical transformations that will map  to each of the following points:

 (where  is a non-zero complex number)

Let  be a square root of the complex number .

By first expressing  in modulus-argument form, show that

Use the geometry of complex numbers to explain why

Use your answers to parts (a) and (b) to find the two square roots of the complex number  ,  giving your answers in modulus-argument form.

Express the square roots in the form    where  and  are real numbers.

By considering the position of  in an Argand diagram and using geometrical reasoning, find the two possibilities for  in exponential form.

Express the following complex numbers in exponential form:

Given the point on an Argand diagram, where  is a complex number, describe the geometrical transformations that will map  to each of the following points:

(where  is a non-zero complex number)

Let  be a square root of the complex number .

Show that

Use the geometry of complex numbers to explain why

for any value of , where  is a real number.

Use your answers to parts (a) and (b) to find the two square roots of the complex number . Give your answers both in modulus-argument form and in the form where  and  are real numbers.

Use geometrical reasoning to find the two possibilities for , giving your answers in exponential form.

By considering the exponential and modulus-argument forms of a complex number, prove Euler’s identity

Express the following complex numbers in exponential form:

Given the points 1 and  on an Argand diagram, where is a complex number, explain how to find each of the following points by geometrical construction.  In each case provide a sketch to illustrate your answer.

Let  be a general complex number, where  and .

Use the geometry of complex numbers to explain why

for any value of .

Hence use the properties of complex numbers to determine the two distinct square roots of , giving your answers in exponential form in terms of  and .

Use geometrical reasoning to find the two possibilities for , giving your answers in exponential form.

Note: You may assume throughout this question that  behaves exactly the same as any other constant for purposes of algebraic manipulation, differentiation and integration.

For a complex number ,  where  ,  show that

Utilising your knowledge of differential equations, explain briefly why the result of part (a) supports the validity of Euler’s relation