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Imaginary unit

During exponentiation with the imaginary unit as base, you get a development of

for the even powers           i 0 = 1,     i 2 = –1,     i 4 = 1,     i 6 = –1,     etc.
for the odd powers            i 1 =  i,     i 3 =  –i,     i 5 = i,      i 7 =  –i,     etc.

 


Explanation

For the powers follows, clockwise

i 0 = 1 ,       i 1 = ,       i 2 = –1 ,     i 3 = –i
i 4 = 1 ,       i 5 = i ,       i 6 = –1 ,     i 7 = –i

or counter-clockwise

i 0  = 1 ,     i –1 = –i ,     i –2 = –1 ,    i –3 =  i
i –4 = 1 ,    i –5 = –i ,     i –6 = –1 ,     i –7 =  i

This is often used for series developments.

 


Example 1

In the complex plane can you draw a unit circle. All points on this circle comply with Euler's formula

e = cos φ + i sin φ

and these are therefore the points indicated in the Argand diagram.

Substitution of φ in the formula gives the values

 


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