(1) mathematical terms. A number consisting of real and imaginary parts, such as a+bi. Where a and b are real numbers, I is "imaginary unit" and the square of I is equal to-1. A and b are called the real part and imaginary part of the complex number A+bi, respectively. When b = 0, A+bi = A is a real number; When b≠0, A+Bi is also called imaginary number; When b≠0 and a = 0, bi is called pure imaginary number. Both real and imaginary numbers are subsets of complex numbers. Just as real numbers can be represented on the number axis, complex numbers can be represented on the plane. This representation is usually called "Forrest Gump" in memory of Swiss mathematician J.R. Argand (1768-65438+).

Extending the number set to the real number range, there are still some operations that cannot be performed. For example, the univariate quadratic equation with discriminant less than 0 still has no solution, so the number set is expanded again to reach the range of complex numbers, and the number axis perpendicular to the real number axis is established to represent complex numbers.

A number in the form of z = a+bi (both a and b are arbitrary real numbers) is called a complex number, where A is called the real part, B is called the imaginary part, I is called the imaginary part, and I 2 = I× I =-1.

When the imaginary part is equal to zero, this complex number can be regarded as 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.

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Complex numbers have many applications, such as:

Complex number is very important in quantum mechanics, because its theory is based on infinite Hilbert space in complex number field.

If the time variable in relativity is regarded as an imaginary number, then some space-time metric equations in special and general relativity can be simplified.

Complex numbers used in signal analysis and other fields can conveniently represent periodic signals. The modulus value |z| represents the amplitude of the signal, and the angle arg(z) represents the phase of the sine wave at a given frequency.