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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.

 

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Explanation

For the powers follows, 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

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 power series developments.

 

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Example 1

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

e^iφ = 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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