WHY STUDY COMPLEX NUMBER : PART 1

 Why should study imaginary numbers?

Most students including mine, (who are studying BA201), will ask the same question to me in the beginning of lecture of this topic. Answering this kind of question is so simple. For the sake of our life, we study them for the same reason we study other numbers including whole numbers, prime numbers, integer numbers etc. (Refer to the video@youtube).

 

However, if this question to be asked to students, their answers
1.  To torture students
2. Part of my mathematics engineering syllabus

 It is so useful for modelling certain situations.For most human tasks, real numbers (or even rational numbers) offer an adequate description of data. For instance, fractions such as ⅔ and ⅛ would be meaningless to a person counting stones, but essential to a person comparing the sizes of different collections of stones. In the same way, negative numbers such as –3 and –5 would be meaningless when measuring the mass of an object, but essential when keeping track of monetary debits and credits. Similarly, imaginary numbers have essential concrete applications in a variety of scientific and related areas such as signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis.

In electrical engineering, for example, the voltage produced by a battery is characterized by one real number called potential (e.g. +12 volts or –12 volts), but the alternating current (AC) voltage in a home requires two parameters — potential and an angle called phase. The AC voltage is, therefore, said to have two dimensions. A two dimensional quantity can be represented mathematically as either a vector or as a complex number (known in the engineering context as phasor). In the vector representation, the rectangular coordinates are typically referred to simply as x and y. But in the complex number representation, the same components are referred to as real and imaginary. When the complex number is purely imaginary, such as a real part of 0 and an imaginary part of 120, it means the voltage has a potential of 120 volts and a phase of 90°, which is, physically speaking, very much a real voltage.