complex numbers - polar form

[lieinking]

Sorry I don't know latex if that's what you use on here. I know of the identity

theta = a

re^ia identical to re^i(a+2npi) where n is any integer

now given an equation

z^4 = 16i. How do I get from there to 16e^ipi/2

???

 

[Deleted member 4993]

Sorry I don't know latex if that's what you use on here. I know of the identity

theta = a

re^ia identical to re^i(a+2npi) where n is any integer

now given an equation

z^4 = 16i. How do I get from there to 16e^ipi/2

???

a + i * b = r * e^iΘ

then

r = √(a² + b²) and

tan(Θ) = b/a

then, using above

i = ?? (here a = 0 and b = 1)

What are your thoughts?

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[Ishuda]

Sorry I don't know latex if that's what you use on here. I know of the identity

theta = a

re^ia identical to re^i(a+2npi) where n is any integer

now given an equation

z^4 = 16i. How do I get from there to 16e^ipi/2

???

As Subhotosh Khan says,
tan(Θ) = b/a
where
z = a + i b
but, I believe more accurately, it is actually the 'proper' angle accounting for the signs of a and b and putting \(\displaystyle \theta\) in the proper quadrent. That is
\(\displaystyle \theta\, =\, atan2(b,\, a)\)
as described in
https://en.wikipedia.org/wiki/Atan2

 

[pka]

As Subhotosh Khan says,
tan(Θ) = b/a
where
z = a + i b
but, I believe more accurately, it is actually the 'proper' angle accounting for the signs of a and b and putting \(\displaystyle \theta\) in the proper quadrent. That is
\(\displaystyle \theta\, =\, atan2(b,\, a)\)
as described in
https://en.wikipedia.org/wiki/Atan2

Here is the way the 'proper' argument function is defined in programs on complex variables:
\(\displaystyle \text{Arg}(x + yi) = \left\{ {\begin{array}{{rl}} {\arctan \left( {\frac{y}{x}} \right),}&{x > 0} \\ {\arctan \left( {\frac{y}{x}} \right) + \pi ,}&{x < 0\;\& \;y > 0} \\ {\arctan \left( {\frac{y}{x}} \right) - \pi ,}&{x < 0\;\& \;y < 0} \end{array}} \right. \)