1. Solving quadratic equations: quadratic equations with real coefficients do not necessarily have real solutions, but when the imaginary unit I is introduced, all quadratic equations have solutions. This is an important application of imaginary number.
2. Complex operation: The operation of imaginary number and real number constitutes complex operation, which is the basis of algebra and is widely used in physics, engineering, computer science and other fields.
3. Fourier analysis: In signal processing and numerical solution of partial differential equations, Fourier transform and its related technologies depend on complex numbers and imaginary numbers.
4. Quantum mechanics: In quantum mechanics, the wave function is usually expressed as a complex number, in which the real part and imaginary part correspond to the position and momentum of particles respectively.
5. Control system theory: In control system theory, the transfer function is usually expressed as a complex number, in which the real part and imaginary part respectively correspond to the stability and phase margin of the system.
6. Circuit analysis: In circuit analysis, the voltage and current of AC circuit can be expressed by complex numbers, in which the real part corresponds to resistive components and the imaginary part corresponds to capacitive and inductive components.
7. Analytic geometry: In analytic geometry, complex numbers can be used to represent points or vectors in two or three-dimensional space.
8. Modal analysis: In structural engineering, modal analysis is a method to determine the dynamic characteristics of structures, and complex numbers and imaginary numbers are used to describe the vibration modes of structures.
Generally speaking, imaginary numbers are widely used in algebra and are the key tools to understand and solve many practical problems.