Complex numbers

hey i know the basics about complex numbers

like: 5*i^7 = 5*i^3 = 5 * -i = -5i = (- pi/2, 5)

but now :

how would i represent :

-> 1 ^ i = ? = ( ? , ? ) or would it involve another mathematical dimention and be more of a (? , ? , ?) ?

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and now, how can i draw a cube of lenght = i

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i mean , at my school , we told me how to use i , but not how to understand it
we dont even really know why we have the graph with real numbers and pure imaginary numbers as axis ?

1^i = 1. 1^x = 1 for all complex numbers x.

The complex numbers are closed, so you'll never need another unit j, say, to solve equations involving complex numbers, such as 1^i=x.

Lengths are positive real numbers. You can't have something with an imaginary length.

Putting complex numbers on a graph with real part and imaginary parts as axes is just a representation. It doesn't much mean anything. You can do it differently if you want (magnitude and angle axes, for example). It's just useful to think of them as points in the plane to help your intuition.

ok since 1^x= 1 , x belonging to all complex numbers

then 2 ^ i = ?

mathwonk
Homework Helper
2020 Award
if a is a positive real number, tghen a^z can be defined as e^(ln(a)z) where ln(a) is the positive real natural log of a.
if a is a more complicated complex number, there is no such nice unique choice of a natural log of a, so a^z has more than one meaning.

i know iof no way to make sense of a complex length, so a cube of side lnegth i amkes no sense to me. what does it mean to you? maybe you cn think of something interesting.

mathwonk
Homework Helper
2020 Award
so your second exmple 2^i equals e^(i.ln(2)), which is approximated as closekly as desired by the series for e^z.
so the first two terms are 1 + i.ln(2).

thx for the a^i

and for the cube, maybe a cube with imaginary borders, sides , ect = an imaginary cube : )

or maybe a cube with no lengh in our 3 main dimentions (we cannot see it), but with an existance in another dimention : )

HallsofIvy
Homework Helper
thx for the a^i

and for the cube, maybe a cube with imaginary borders, sides , ect = an imaginary cube : )

or maybe a cube with no lengh in our 3 main dimentions (we cannot see it), but with an existance in another dimention : )
How did you get off complex numbers to geometry? I know of know way of defining "a cube with imaginary borders, sides, etc." I have no idea what you could mean by an imaginary length.

i didnt mean into geometry, but in existence
i admit, the idea of the cube was a bad idea, but complex numbers surely must be found somewhere in nature (or somewhere in space) ? i mean is there somewhere in space, or more precisely earth, where we see sqroot(-1) ?

i didnt mean into geometry, but in existence
i admit, the idea of the cube was a bad idea, but complex numbers surely must be found somewhere in nature (or somewhere in space) ? i mean is there somewhere in space, or more precisely earth, where we see sqroot(-1) ?

Complex numbers can be applied to models dealing with alternating current. There are probably more.

can you tell me in what exactly with alternative current we find complex numbers ?

HallsofIvy