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While the fundamental signal used in electrical engineering is the sinusoid, it can be expressed mathematically in terms of an even more fundamental signal: the complex exponential . Representing sinusoids in terms of complex exponentials is not a mathematical oddity. Fluency with complex numbers and rational functions of complex variables is a critical skill all engineers master.Understanding information and power system designs and developing new systems all hinge on using complex numbers. In short, they are critical to modern electrical engineering, a realization made over a century ago.
The notion of the square root of originated with the quadratic formula: the solution of certain quadratic equations mathematically exists only if the so-called imaginary quantity could be defined. Euler first used for the imaginary unit but that notation did not take hold untilroughly Ampère's time. Ampère used the symbol to denote current (intensité de current).It wasn't until the twentieth century that the importance of complex numbers to circuit theory became evident. By then, using for current was entrenched and electrical engineers chose for writing complex numbers.
An imaginary number has the form . A complex number , , consists of the ordered pair ( , ), is the real component and is the imaginary component (the is suppressed because the imaginary component of the pair is always in the second position). The imaginary number equals ( , ). Note that and are real-valued numbers.
[link] shows that we can locate a complex number in what we call the complex plane . Here, , the real part, is the -coordinate and , the imaginary part, is the -coordinate. From analytic geometry, we know that locations in the plane can be expressed as the sum of vectors, with the vectors corresponding to the and directions. Consequently, a complex number can be expressed as the (vector) sum where indicates the -coordinate. This representation is known as the Cartesian form of . An imaginary number can't be numerically added to a real number;rather, this notation for a complex number represents vector addition, but it provides a convenient notation when we perform arithmetic manipulations.
Some obvious terminology. The real part of the complex number , written as , equals . We consider the real part as a function that works by selecting that componentof a complex number not multiplied by . The imaginary part of , , equals : that part of a complex number that is multiplied by . Again, both the real and imaginary parts of a complex number are real-valued.
The complex conjugate of , written as , has the same real part as but an imaginary part of the opposite sign.
Using Cartesian notation, the following properties easily follow.
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