Complex Numbers And Circular Functions

When the coefficients of a second-order linear differential equation, y"(t) + aly'(t) + a2y - b, are such that a] < 4a2, the characteristic-root formula (15.5) would call for taking the square root of a negative number. Since the square of any positive or negative real number is invariably positive, whereas the square of zero is zero, only a nonnegative real number can ever yield a real-valued square root. Thus, if we confine our attention to the real number system, as we have so far, no characteristic roots are available for this case (Case 3). This fact motivates us to consider numbers outside of the real-number system.

Imaginary and Complex Numbers

Conceptually, it is possible to define a number i = 1 , which when squared will equal — 1. Because i is the square root of a negative number, it is obviously not real-valued; it is therefore referred to as an imaginary number. With it at our disposal, we may write a host of other imaginary numbers, such as -f- 9 = /9 v^T = 3/ and /= /2 i.

Imaginary axis

Real axis h

Figure 15.2

Extending its application a step further, we may construct yet another type of number—one that contains a real part as well as an imaginary part, such as (8 + /) and (3 + 5/). Known as complex numbers, these can be represented generally in the form (h + vi), where h and v are two real numbers.* Of course, in case v = 0, the complex number will reduce to a real number, whereas if h = 0, it will become an imaginary number. Thus the set of all real numbers (call it R) constitutes a subset of the set of all complex numbers (call it C). Similarly, the set of all imaginary numbers (call it I) also constitutes a subset of C. That is, R c C, and I c C. Furthermore, since the terms real and imaginary are mutually exclusive, the sets R and I must be disjoint; that is R n I = 0.

A complex number (h + vi) can be represented graphically in what is called an Argand diagram, as illustrated in Fig. 15.2. By plotting h horizontally on the real axis and v vertically on the imaginary axis, the number (h + vi) can be specified by the point (h, v), which we have alternatively labeled C. The values of h and v are algebraically signed, of course, so that if h < 0, the point C will be to the left of the point of origin; similarly, a negative v will mean a location below the horizontal axis.

Given the values of h and v, we can also calculate the length of the line OC by applying Pythagoras' theorem, which states that the square of the hypotenuse of a right-angled triangle is the sum of the squares of the other two sides. Denoting the length of OC by R (for radius vector), we have

* We employ the symbols h (for horizontal) and v (for vertical) in the general complex-number notation, because we shall presently plot the values of h and v, respectively, on the horizontal and vertical axes of a two-dimensional diagram.

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