A number in the form of z = A+bi(A and B are both real numbers) is called a complex number, where A is called the real part, B is called the imaginary part, and I is called the imaginary part. When the imaginary part of z is equal to zero, z is often called a real number; When the imaginary part of z is not equal to zero and the real part is equal to zero, z is often called pure imaginary number. Complex number field is an algebraic closure of real number field, that is, any polynomial with complex coefficients always has roots in complex number field.

Complex number is a generalization of real number, which makes any polynomial equation have roots. There is an imaginary unit I in the complex number, which is the square root of-1, that is, the square of I is equal to-1. Any complex number can be expressed as a+bi, where a and b are both real numbers, which are called the "real part" and "imaginary part" of the complex number respectively. The discovery of complex numbers stems from the expression of the roots of cubic equations. Mathematically, the word "complex" means that the number field in question is complex, such as complex matrix and complex variable function.

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Application of Complex Numbers-System Analysis

In system analysis, Laplace transform is often used to transform the system from time domain to frequency domain. Therefore, the poles and zeros of the system can be analyzed on the complex plane. Root locus method, Nyquistplot method and Nicholsplot method for analyzing system stability are all carried out on the complex plane. Whether the poles and zeros of the system are in the left half plane or the right half plane, the root locus method is very important. If the system pole

Located in the right half plane, the causal system is unstable; If they all lie in the left half plane, the causal system is stable; On the imaginary axis, the system is critically stable. If all zeros and poles of the system are in the left half plane, it is the minimum phase system. A system is all-pass if its poles and zeros are symmetrical about the imaginary axis.

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