In mathematics, a complex number is a number consisting of real and imaginary parts. It is an extension of real number set, which can be used to represent the fluctuation and oscillation phenomena in vector rotation, circuit analysis, signal processing and other fields.
Complex numbers are usually expressed in the form of a+bi, where a is the real part, b is the imaginary part, and I is the imaginary unit, which satisfies I 2 =- 1. Both the real part and the imaginary part can be real numbers, and the imaginary part is represented by I, which together is a complex number.
Complex numbers have many properties and operation rules. Basic arithmetic operations such as addition, subtraction, multiplication and division can be performed. For example, two complex numbers are added, the real part and imaginary part are added, and the imaginary part and imaginary part are added; When two complex numbers are multiplied, the real part is multiplied by the real part and the imaginary part is multiplied by the imaginary part.
Complex numbers also have important properties such as modulus, yoke and amplitude angle. The modulus of a complex number indicates the distance from the complex number to the origin, which can be calculated by Pythagorean theorem. The * * * yoke of the complex number indicates the sign of changing the imaginary part, that is, the * * * yoke of a+bi is a-bi. The amplitude angle of complex number indicates the angle between complex number and positive real axis, which can be calculated by inverse trigonometric function.
Complex numbers are widely used in solving equations, Fourier transform, vector rotation, complex plane and so on. An equation that has no solution in the real number range, such as x 2+1= 0, can find the solution in the complex number field. Fourier transform can transform time domain signals into frequency domain signals, so as to analyze the frequency spectrum of the signals.
You can also use complex numbers to represent the rotation of vectors, and the rotation operation is realized by complex multiplication. In addition, in the complex plane, complex numbers can be expressed by the coordinates of points, which is convenient for intuitive understanding and calculation.
Data expansion
There are several kinds of numbers, and Arabic numerals are the most common one. Arabic numerals were not invented by Arabs, but by Indians. In fact, it should be listed as Hindi, but it spread to Arabia first and then to the world, so it is called "Arabic numerals".
Numbers are written symbols used to represent numbers. Different counting systems can use the same number.
In a word, complex number is an important mathematical concept in mathematics, which expands the set of real numbers and has rich properties and applications. Through the operation and properties of complex numbers, we can better understand and describe all kinds of fluctuations and oscillations in nature and science.