The four operational formulas of 1 and imaginary number I are: (a+bi) (c+di) = (a c)+(b d) i.
2. formulas of trigonometric functions of imaginary number I: csc(a+bi) 1/sin(a+bi).
3. Properties of imaginary number I: i 1=i, i2=- 1, i3 =-i. ..
In mathematics, the imaginary number is a+b×i number, where A and B are real numbers, and b≠0, I? =- 1。 The word imaginary number was founded by Descartes, a famous mathematician in the17th century, because the concept at that time thought it was a non-existent real number.
Later, it was found that the real part A of the imaginary number a+b×i can correspond to the horizontal axis and the imaginary part B can correspond to the vertical axis on the plane, so that the imaginary number a+b×i can correspond to the points (a, b) on the plane.
The imaginary number bi can be added to the real number A to form a complex number in the form of a+b×i, where the real numbers A and B are called the real part and imaginary part of the complex number respectively. Some authors use the term pure imaginary number to represent the so-called imaginary number, which refers to any complex number whose imaginary part is not zero.
Application of imaginary number
1, Mathematics and Physics: In mathematics and physics, imaginary numbers are used to describe various phenomena, such as wave phenomena, quantum mechanics, AC signals in circuits, etc. Imaginary number can represent the phase difference of waves and can also be used to describe the wave function in quantum mechanics. In electrotechnics, the imaginary number can be used to analyze the phase difference of alternating current.
2. Signal processing: In the field of signal processing, imaginary and complex numbers are widely used in spectrum analysis and filter design. By representing the signal as a complex number, it is easier to process the signal and extract useful information.
3. Engineering: In electrical engineering, imaginary numbers are used to describe AC circuits and signal processing. In control theory, complex frequency is particularly useful for designing and analyzing control systems. In mechanical engineering, imaginary number is also used to describe vibration and elasticity theory.
4. Economics and Finance: Complex numbers are used to describe compound interest and option pricing in economics and finance.