Earlier in the book the distinction between analytical and graphical methods of solution of problems was discussed. In that case the problems concerned the addition or subtraction of vector or phasor quantities and the two methods consisted of graphical addition or subtraction of the quantities waveforms to produce a resultant waveform or the analytical use of trigonometrical identities, a method which did not require graphs. The analytical method was, however, confined only to additions or subtractions of the form a sin B ± b cos 8, that is, the vector or phasor quantities were sinusoidal and cosinusoidal. Complex numbers give us an analytical method of vector or phasor addition, subtraction, multiplication or division without the use of graphs and often even without the use of any form of diagram. Without the use of complex numbers we are restricted to the use of vector or phasor diagrams and solution either by the application of trigonometry to the diagram or by drawing to scale. The topic of complex numbers as an aid to calculation is by far one of the most useful topics of mathematics we can study.
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