in engineering i feel that every standerd equcation have a complex no. or involve complex plane. so what is real mean:
a example..... we all of know that there is no mean of negetive no. but in math they exist and if they not in use the math is incomplete so if there is real then must be imagenery to fulfill the requirement. but there physical mean nothing.
now a mathmetical example:
Equation 1: x2 - 1 = 0.Equation 1 has two solutions, x = -1 and x = 1. We know that solving an equation in x is equivalent to finding the x-intercepts of a graph; and, the graph of y = x2 - 1 crosses the x-axis at (-1,0) and (1,0).
Equation 2: x2 + 1 = 0
Equation 2 has no solutions, and we can see this by looking at the graph of y = x2 + 1.
Since the graph has no x-intercepts, the equation has no solutions.but according to maths rule the second ordered equcation have two solution. then what is the solution it is imagenery solution "i".
now it hasbeen proved a quantity which have no real solution have imagenery solution and vice varsa.
so z=a+ib a is real and b also real but a called real quantity and second part called imagenery. and if real=0 then totally imagenery and second=0 then real.
another fact a real quantity make by airthmatic operation with i. so imagenery quantity makes a plane including pure real which is real axix and pure imagenery which is y.that is called complex plane.
in ractangular cordinates:-
in polar form(r,@):-
r radious from origen and @ its angle with axis. so according to trignometric here the sin and cosine will introduce
now by projection rule there is r(cos@+isin@) here r wil be sqrt of cos@ and sin@ that will be 1 means a circle with unit radius.
now we know solution of power of i.
ok then take the solution of cos@ and sin@ with taylor's equcation means expention of sin@ and cos@. and the some of that with i gives a combind form like e(iz) where z makes complex plane means =real+i(real)
so this is theoritical proof ot eulers equcation.
a example..... we all of know that there is no mean of negetive no. but in math they exist and if they not in use the math is incomplete so if there is real then must be imagenery to fulfill the requirement. but there physical mean nothing.
now a mathmetical example:
Equation 1: x2 - 1 = 0.Equation 1 has two solutions, x = -1 and x = 1. We know that solving an equation in x is equivalent to finding the x-intercepts of a graph; and, the graph of y = x2 - 1 crosses the x-axis at (-1,0) and (1,0).
Equation 2: x2 + 1 = 0
Equation 2 has no solutions, and we can see this by looking at the graph of y = x2 + 1.
Since the graph has no x-intercepts, the equation has no solutions.but according to maths rule the second ordered equcation have two solution. then what is the solution it is imagenery solution "i".
now it hasbeen proved a quantity which have no real solution have imagenery solution and vice varsa.
so z=a+ib a is real and b also real but a called real quantity and second part called imagenery. and if real=0 then totally imagenery and second=0 then real.
another fact a real quantity make by airthmatic operation with i. so imagenery quantity makes a plane including pure real which is real axix and pure imagenery which is y.that is called complex plane.
in ractangular cordinates:-
r radious from origen and @ its angle with axis. so according to trignometric here the sin and cosine will introduce
now by projection rule there is r(cos@+isin@) here r wil be sqrt of cos@ and sin@ that will be 1 means a circle with unit radius.
now we know solution of power of i.
ok then take the solution of cos@ and sin@ with taylor's equcation means expention of sin@ and cos@. and the some of that with i gives a combind form like e(iz) where z makes complex plane means =real+i(real)
so this is theoritical proof ot eulers equcation.
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