First, complex numbers can be used to solve algebraic equations. By introducing complex numbers, the algebraic equations in real number domain can be extended to complex number domain, and more solutions can be obtained. For example, for the quadratic equation AX 2+BX+C = 0, when the discriminant B 2-4ac is less than 0, the equation has no real root. However, if we introduce complex numbers, we can get two complex roots.
Secondly, complex numbers also have important applications in geometry. Complex numbers can be used to represent points and vectors on a plane, thus solving some problems related to geometry. For example, by using the multiplication of complex numbers, the points on one unit circle can be mapped to another unit circle, thus realizing the rotation transformation.
In addition, complex numbers can also be used to solve problems such as signal processing and circuit analysis. In signal processing, complex numbers can be used to represent the amplitude and phase information of signals. By representing the signal as a complex number, we can filter, modulate and demodulate the signal conveniently. In circuit analysis, the amplitude and phase difference of AC current and voltage can be expressed by complex numbers. By using complex circuit models, we can simplify the circuit analysis process.
In a word, as a mathematical tool, complex numbers are widely used in algebra, geometry, signal processing, circuit analysis and other fields. It provides us with a powerful mathematical language, which can solve many practical problems.