Operation formula of imaginary number I Four operation formulas of imaginary number I
(a+bi) (c+di)=(a c)+(b d)i
(a+bi)(c+di)=(ac-bd)+(ad+bc)i
(a+bi)/(c+di)=(ac+bd)/(c？ +d？ )+(bc-ad)i/(c？ +d？ )
r 1(isina+cosa)R2(isin b+cosb)= r 1r 2[cos(a+b)+isin(a+b)]
r 1(isina+cosa)/R2(isin b+cosb)= r 1/R2[cos(a-b)+isin(a-b)]
r(isina+cosa)n =(isina+cosna)
Formulas of trigonometric functions of imaginary number I
sin(a+bi)= sin(a)cos(bi)+sin(bi)cos(a)= sin(a)cosh(b)+isinh(b)cos(a)
cos(a-bi)= cos(a)cos(bi)+sin(bi)sin(a)= cos(a)cosh(b)+isinh(b)sin(a)
tan(a+bi)=sin(a+bi)/cos(a+bi)
cot(a+bi)=cos(a+bi)/sin(a+bi)
sec(a+bi)= 1/cos(a+bi)
csc(a+bi)= 1/sin(a+bi)
The practical significance of imaginary number I The value of anything can be expressed as: a+bi, not just a real number.
We can draw an imaginary system in a plane rectangular coordinate system. If the horizontal axis represents all real numbers, then the vertical axis can represent imaginary numbers. Every point on the whole plane corresponds to a complex number, which is called a complex plane. The horizontal and vertical axes have also been renamed as real and imaginary axes. At this time, the coordinate of point p is P(a, bi). Multiply the coordinates by I, that is, the point rotates 90 degrees counterclockwise around the center of the circle.
Students or scholars who are dissatisfied with the interpretation of the above images can refer to the following topics and explanations:
If the reciprocal of a number is equal to its reciprocal (or its reciprocal is itself), what is the form of this number?
According to this requirement, the following equation can be given: -x=( 1/x).
It is not difficult to know that the solution of this equation x = I (imaginary unit)
Therefore, if there is an algebraic expression t'=ti, we will understand I as the conversion unit from the unit of t to the unit of t', then t'=ti will be understood as
-t'= 1/t, that is, t' =-1/t.
This expression has little significance in geometric space, but if it is understood in time by special relativity, it can be explained that if the relative speed of motion can be greater than the speed of light C, then the virtual value generated by the relative time interval is essentially the negative reciprocal of the real value. That is, the so-called time interval value of going back to the past can be calculated from this.
Imaginary number has become the core tool for designing microchips and digital compression algorithms, and it is the theoretical basis of quantum mechanics that triggered the electronics revolution.
Imaginary number is an abstract concept, which is used to represent the factors that can not constitute an abstract concept in things.