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CHAPTER 8 : COMPLEX NUMBERS

8.1 Geometry of Complex Numbers

The set of real numbers seems to fulfil the need to algebra. But at times in algebra too, a state is reached when a set of real numbers falls short. For example, it is impossible to solve
the quadratic equation x2+ 16 = 0 in terms of real roots. For, x2= - 16 leads to x = , where the square root of -16 can not be extracted in terms of any real number. In other words this means, there is no real number whose square is - 16. At the most we simplify
as,

At this stage we introduce a new kind of number , which is different from the real numbers and hence called an "imaginary number", usually denoted by 'i'. It is customary to designate 'i' by 'j' in most of the work in electrical engineering . This number permits the

solution of the above equation and those similar to it. This number i = such that i2 = -1.

The symbol 'i' now enables us to write (i) as = 4 = 4i. The credit of introducing 'i' goes to an eminent French mathematician Euler (1707 - 1783).

Power of 'i' : More often, while effecting easy operations with complex number, we come across with various powers of 'i'. The following illustrations would serve a guidance in such cases

[Thus in = ± 1 or ± i ]

This requires the knowledge of indices and a bit careful working.

[next page]

Index

8.1 Geometry of complex numbers
8.2 De - Moivres's theorem
8.3 Roots of complex numbers
8.4 Cirsular functions of complex angles & hyperbolic function
Supplementary Problems

Chapter 9

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