Roots of a complex number

Use the Euler formula and the De Moivre theorem to evaluate powers of complex numbers, to determine / th roots of a complex number, and to identify real and imaginary parts of functions of a complex variable... 

It follows that one square root of a complex number (where = ) is given by ... 

The square root of a complex number is a number that will yield the first number when multiplied by itself. Just as with real numbers, there are two square roots of a complex number. If z = re, one of the square roots is given by... 

There are three cube roots of a complex number. These can be found by looking for the numbers that when cubed yield and. These... 

When the dielectric function itself is complex, we must take the square root of a complex number or n = Ve where the tildes represent complex quantities. We can write... 

When a system has poles that are widely different in value, it is difficult to plot them all on a root locus plot using conventional rectangular coordinates in the s plane. U is sometimes more convenient to make the root locus plots in the log s plane. Instead of using the conventional axis Re s and Im s, an ordinate of the arg s and an abscissa of the log s arc used, since the natural logarithm of a complex number is defined ... 

We denote the set of complex numbers by C. Readers should be familiar with complex numbers and how to add and multiply them, as described in many standard calculus texts. We use i to denote the square root of —1 and an asterisk to denote complex conjugation if x and y are real numbers, then (x + iy) = X — iy. Later in the text, we will use the asterisk to denote the conjugate transpose of a matrix with complex entries. This is perfectly consistent if one thinks of a complex number x + iy as a one-by-one complex matrix ( x + iy ). See also Exercise 1.6. The absolute value of a complex number, also known as the modulus, is denoted... 

Neither of equations (iii) or (iv) are solutions to equation (7.46). However, if n was such that n2 was negative, then both functions would be solutions to the equation. This would require us to define the square root of a negative number, which is at odds with our understanding of what constitutes a real number. In Chapter 2, Volume 2, we extend the concept of the number to include so-called imaginary and complex numbers, which embrace the idea that the square root of a negative number can be defined. 

Operations on complex numbers all start with IM, and use text strings to squeeze the two components of a complex number into one cell. In order to use the results of complex number operations, you must therefore first extract its real and imaginary components, using IMREALO and IMAGINARY () Instead of i you can use j to denote the square root of minus one (which you must then specify as such), but you cannot use the corresponding capitals, I or /. 

Complex numbers owe their origin to the quest for the square root of a negative number. Thus the so-called imaginary number i = is a fundamental element of complex numbers, written as z = X + iy, in which x is the real part and y is the imaginary part. Although real numbers quantify physical quantities, complex numbers provide very convenient representations of many physical phenomena. In quantum mechanics, the wave function is a complex function. Two-dimensional, incompressible, irrotational flows are represented by a complex flow potential, w = 9 h- t /, with 9, the velocity potential, as the real part, and /, the stream function, as the imaginary part. 

As we have seen, if is a root of a polynomial equation, then z is also a root. Recall that for real numbers, absolute value refers to the magnitude of a number, independent of its sign. Thus, 3.14 = — 3.14 = 3.14. We can also write - 3.14 = - 3.14. The absolute value of a complex number z, also called its magnitude or modulus, is likewise written as z - It is defined by... 

Modulus ma-j3-l3s -li [NL, fr. L, small measure] (1753) n, pi. (1) A modulus is a measure of a mechanical property of a material, most frequently a stiffness property. (2) The absolute value of a complex number or quantity, equal to the square root of the sum of the squares of the real and imaginary parts. (3) Modulus at 300% n The tensile stress required to elongate a specimen to three times its original length (200% elongation) divided by 2. Although other elongations are used, 300% is the one most often employed for rubbers and flexible plastics. 

A complex number consists of two parts a real and a so-called imaginary part, c = a + ib. The imaginary part always contains the quantity i, which represents the square root of -1, i = /—1- The real and imaginary parts of c are often denoted by a = R(c) and b = 1(c). All the common rules of ordinary arithmetic apply to complex numbers, which in addition allow extraction of the square root of any negative number. If... 

Most of the graphs we deal with are trees- that is, they do not have any self tracing loops. Choose a vertex which will be the root . In the example we consider here we shall take vertex 5 as the root (see figure l.(c)). A wave which propagates from the root along a certain branch is reflected, and this reflection can be expressed by a reflection from the vertex which is next to the root. Once we know the reflection coefficient, which because of unitarity is a complex number with unit modulus, we can construct the SB(k) matrix. In the present cases, when the valency of the root is 3, we get... 

If the roots and S2 are both real numbers, Bq. (6.116) shows that Uq and a I are certainly both real. If the roots and. S are complex, the coefficients and di must still be real and must also satisfy Eq. (6.116). Complex conjugates are the only complex numbers that give real numbers when they are multiplied together and when added together. To illustrate this, let z be a complex number z = X + iy. Let z be the complex conjugate of z z = Jt — iy. Now... 

The set of roots of a polynomial with real coefficients can include conjugate pairs of complex numbers. Thus eigenvalues can be complex, appearing in conjugate pairs. When this happens the corresponding column eigenvectors also form a conjugate pair, as do the rows. 

In the answer to Worked Problem 2.1, we obtained the required roots of the quadratic equation in the form of a sum of a real number (-1) and an imaginary number (2i or -2i). Such numbers are termed complex numbers, and have the general form ... 

The fundamental theorem of algebra states that a polynomial of degree n has n number of roots, although some of the roots may be complex numbers. If a root of a polynomial of degree n is known to be, say r, then a polynomial of degree n - 1 is given by... 

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