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Errors in algebra
Published in Breach Mark, Essential Maths for Engineering and Construction, 2017
This indicates that the squares of all real numbers are positive. Therefore, the square root of a negative but real number cannot be a real number. But not all numbers are real; there exists a branch of mathematics that deals with complex numbers which have real and imaginary parts. A complex number is of the form of a + ib where a and b are real numbers and i=-1. Thus, a is the real part and ib is the imaginary part of the complex number a + ib. So i2=(-1)2=-1..
From the Top Down: The Physical Layer–AM OFDM
Published in David P. Maxson, The IBOC Handbook, 2007
The complex conjugate is a sign reversal of the imaginary component. Complex conjugation is often represented by an asterisk (*) or a caret (^). The IBOC Reference Documents use the asterisk, which is used here. In this text the conjugation symbol * is distinguished from the convolution symbol * by its smaller size and superscript position. Stripped of its subtleties, the basic function for a complex signal is Equation 8.1. Negating the phase component creates the complex conjugate, as in the lower right quarter of Figure 8.3. The negation of the phase has no effect on the real component. Equation 8.2 is complex conjugation, showing how mathematically it can be reduced to negating the imaginary component, j sin θ. Figure 8.3 shows how this results in rotating the constellation point in the opposite direction from the real axis. If Equation 8.1 represents the modulated AM USB, Equation 8.2 describes the AM LSB.
Mechanical Nanosensors
Published in Vinod Kumar Khanna, Nanosensors, 2021
where A1, A2, A3, and A4 are complex constants which are determined from the boundary condition. The relationship between the trigonometric functions and the complex exponential function, Euler’s formula for the complex exponential states that, for any real number θ, eiθ = cos θ + i sin θ, where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine). Using the Euler equations and the comparable equations for sinh and cosh [sinh x = (ex−e−x)/2 and cosh x = (ex + e−x)/2], Equation 4.23 can be transformed to an equation with real constants B1, B2, B3, B4, as Uyx,ω=B1sinαxω+B2cosαxω+B3sinhαxω+B4coshαxω
Comparative analysis of quaternion modulation system with OFDM systems
Published in International Journal of Electronics Letters, 2021
Anam Zahra, Qasim Umar Khan, Shahzad Amin Sheikh
A complex number is defined by, where are real numbers and is an imaginary number such that. Complex numbers are two-dimensional vectors space over the real numbers. In addition to, quaternions are constructed by adding two new imaginary units and with one real part. A quaternion is an extension of the complex number system (Catoni, Bordoni, Cannata, & Zampetti, 1997). In 1843, Irish mathematician William Rowan Hamilton was the first person who described quaternions and practically applied them in three-dimensional mechanics (Farouki, Al-Kandari, & Sakkalis, 2002). The mathematical notation of quaternions represents three-dimensional rotations of objects. A quaternion can be written as a sum of one real part and three imaginary parts.
Multidual Sensitivity Method in Ray-Tracing Transport Simulations
Published in Nuclear Science and Engineering, 2021
Matthew R. Balcer, Harry Millwater, Jeffrey A. Favorite
where the imaginary number is defined as . Taking the imaginary part, denoted by , of Eq. (3) and dividing both sides by gives an approximation to the first partial derivative: