1, the definition and properties of imaginary number
Imaginary number is a special number, corresponding to real number, which can be expressed as a+bi, where A and B are real numbers respectively, and I is imaginary unit. The square of imaginary unit I is equal to-1, which is the basic property of imaginary number. Imaginary numbers can be added, subtracted, multiplied and divided to meet the operation rules similar to real numbers.
2. The geometric meaning of imaginary number and complex plane.
Imaginary numbers can be represented by complex planes. The complex plane is a two-dimensional coordinate system composed of real axis and imaginary axis. The real axis represents the real part and the imaginary axis represents the imaginary part. On the complex plane, the imaginary number extends along the imaginary axis and the real number extends along the real axis. Through the complex plane, we can intuitively understand the relationship between imaginary number and real number.
3. Application of imaginary number in physics.
Imaginary numbers are widely used in physics. In circuit theory, imaginary numbers are often used to represent the phase angle and frequency response of voltage and current. In quantum mechanics, imaginary numbers are also used to describe wave functions. Imaginary numbers also have important applications in vibration, fluctuation and signal processing.
4. The relationship between Euler formula and imaginary number.
Euler formula is an important formula in mathematics, which relates trigonometric function, imaginary index and complex index. Euler's formula is e (IX) = cos (x)+I * sin (x), where e is the base of natural logarithm and I is the imaginary unit. This formula connects imaginary number with trigonometric function, which enriches the expression form and application scope of imaginary number.
Expand knowledge:
The * * * yoke of imaginary number: the * * yoke of imaginary number a+bi is a-bi, and the real part of the * * * yoke of complex number is the same but the imaginary part is opposite in sign. The modulus and amplitude angle of imaginary number: the modulus of imaginary number a+bi is √(a? +b？ ), the amplitude is arctan(b/a). Real part and imaginary part of complex number: the real part of complex number a+bi is A and the imaginary part is B.
To sum up, the imaginary number is the number obtained by multiplying the real number by the imaginary unit I, and the relationship between the imaginary number and the real number can be expressed intuitively through the complex plane and geometric figures. Imaginary numbers are widely used in mathematics and physics, such as circuit theory and quantum mechanics. Euler formula relates imaginary number, complex exponent and trigonometric function, which enriches the expression form and application scope of imaginary number.