to use: on the polar complex plane, one multiplies
the radius of one, adds so many degrees - rads -
to the angle measure. But what does the expression
e^i*pi mean. Not always helpful is that the historical
progression of concepts - and proofs - can be confusing.
One thing to keep in mind is the particuliarity of i, (-1)^.5.
9^.5 = 3
9^-.5 = 1/3
9^i = 9^((-1)^.5)
The only way to escape i is by squaring it!!
Below, seeing how the series noted e can be viewed in terms
of sin and cos. Historically, the Taylor series came within Euler's lifetime
in 1710 but is based on calculus. Taylor adjudicated Newton at the
Royal Society!?
Having made it there, one more step gets one to e^(i*pi). The takeaway
on all this: the expression does have meaning but it is about e!! (The
Donald Trump of mathematical constants!) So the uses
of cos and sin might be unexpected, counter-intuitive for many. It
is algebraic...
* * *
* * *
Google gave me the above numbers for 'evaluate 9^i'.
One starts by solving for e^(i θ)= 9^i
The angle is some 54 degrees...
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