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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 $-1$ originated with the quadratic formula: the solution of certain quadratic equations mathematically exists only if the so-called imaginary quantity $\sqrt{-1}$ could be defined. Euler first used $i$ for the imaginary unit but that notation did not take hold untilroughly Ampère's time. Ampère used the symbol $i$ 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 $i$ for current was entrenched and electrical engineers chose $i$ for writing complex numbers.
An imaginary number has the form $ib=\sqrt{-b^{2}}$ . A complex number , $z$ , consists of the ordered pair ( $a$ , $b$ ), $a$ is the real component and $b$ is the imaginary component (the $i$ is suppressed because the imaginary component of the pair is always in the second position). The imaginary number $ib$ equals ( $0$ , $b$ ). Note that $a$ and $b$ are real-valued numbers.
[link] shows that we can locate a complex number in what we call the complex plane . Here, $a$ , the real part, is the $x$ -coordinate and $b$ , the imaginary part, is the $y$ -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 $x$ and $y$ directions. Consequently, a complex number $z$ can be expressed as the (vector) sum $z=a+ib$ where $i$ indicates the $y$ -coordinate. This representation is known as the Cartesian form of $z$ . 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 $z=a+ib$ , written as $\Re (z)$ , equals $a$ . We consider the real part as a function that works by selecting that componentof a complex number not multiplied by $i$ . The imaginary part of $z$ , $\Im (z)$ , equals $b$ : that part of a complex number that is multiplied by $i$ . Again, both the real and imaginary parts of a complex number are real-valued.
The complex conjugate of $z$ , written as $\overline{z}$ , has the same real part as $z$ but an imaginary part of the opposite sign.
Using Cartesian notation, the following properties easily follow.
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