Write your introduction by relating it to a real life context of complex numbers, that is, follow one of the areas below and use it to explain why we need complex numbers and their operations.

Complex Numbers are useful in representing a phenomenon that has two parts varying at the same time, for example an alternating current. Also, radio waves, sound waves and microwaves have to travel through different media to get to their final destination. There are many instances where, for example, engineers, doctors, scientists, vehicle designers and others who use electromagnetic signals need to know how strong a signal is when it reaches its destination. The two parts in this context are: the rotation of the signal and its strength. The following are examples of this phenomenon:
• A microphone signal passing through an amplifier
• A mobile phone signal travelling from the mast to a phone a couple of miles away
• A sound wave passing through the bones in the ear
• An ultrasound signal reflected from a foetus in the womb
• The song of a whale passing through miles of ocean water
Complex Numbers are also used in:
• The prediction of eclipses
• Computer game design
• Computer generated images in the film industry
• The resonance of structures (bridges, etc.)
• Analysing the flow of air around the wings of a plane in aircraft design

PART 2
Use your own examples to explain De Moivre's Theorem and explain how to find the nth roots of a complex number you have made up, where neither the real nor the imaginary part are 0