Imaginary number was first put forward by mathematician Euler in18th century, which was used to solve some problems that real numbers could not solve. The appearance of imaginary number makes complex number a complete algebraic structure, which provides convenience for solving many mathematical problems.
Imaginary numbers have some special properties. First of all, imaginary and real numbers can be added, subtracted, multiplied and divided, but the result is still complex. Secondly, the imaginary number can be expressed as a point or vector in a rectangular coordinate system, where the real part represents the abscissa and the imaginary part represents the ordinate. In addition, the imaginary number can also be expressed in polar coordinates, where r represents the module length and θ represents the radiation angle.
Imaginary numbers are widely used in mathematics, physics and engineering. For example, in circuit analysis, imaginary numbers can be used to represent the frequency and phase of AC signals; In quantum mechanics, imaginary number is used to describe the wave function of particles; In signal processing, imaginary numbers are used to represent signals with complex values. Imaginary number is an important mathematical concept, which plays an important role in both mathematical theory and practical application.
Application of imaginary number:
1. Analysis of AC circuit: In AC circuit, voltage and current change with time and can be expressed by complex numbers. The real part represents the magnitude of voltage or current, and the imaginary part represents the phase difference of voltage or current. Through imaginary number operation, the parameters such as power and impedance in AC circuit can be calculated conveniently.
2. Signal processing: In signal processing, signals can usually be expressed as complex values. For example, an audio signal can be converted into a complex form of a frequency domain signal by Fourier transform. Through imaginary number operation, filtering and modulation can be carried out.
3. Analysis of control system: In the control system, the transfer function can be expressed by complex numbers. Where the real part represents the frequency response of the system and the imaginary part represents the phase response of the system. The stability and step response of the system can be analyzed by imaginary number operation.
4. Quantum mechanics: In quantum mechanics, the state of particles can be expressed by wave functions. Wave function is a complex function, in which the real part represents the probability density of particle appearance and the imaginary part represents the phase of particle. Through imaginary number operation, the energy and wave function propagation of particles can be calculated.
5. Image processing: In image processing, an image can be represented as a matrix of pixel values. By multiplying the pixel value by a complex number and performing imaginary operation, the rotation, translation and scaling of the image can be realized.