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Mathematics and Statistics · Ch 3 — Complex Numbers

Powers of i

2

Powers of i

Because i2=−1i^{2}=-1, the powers of ii repeat in a cycle of four, which lets us simplify any integer power of ii quickly.

The four-step cycle

i1=i,i2=−1,i3=i2⋅i=−i,i4=(i2)2=(−1)2=1.i^{1}=i,\qquad i^{2}=-1,\qquad i^{3}=i^{2}\cdot i=-i,\qquad i^{4}=\left(i^{2}\right)^{2}=(-1)^{2}=1.

After i4=1i^{4}=1 the pattern starts over: i5=i4⋅i=ii^{5}=i^{4}\cdot i=i, i6=−1i^{6}=-1, and so on.

Tip

How to evaluate ini^{n} fast

Divide the exponent nn by 44 and keep only the remainder rr (where 0≤r≤30\le r\le 3). Then in=iri^{n}=i^{r}, using i0=1, i1=i, i2=−1, i3=−ii^{0}=1,\ i^{1}=i,\ i^{2}=-1,\ i^{3}=-i.

For example i35i^{35}: 35=4×8+335=4\times8+3, remainder 33, so i35=i3=−ii^{35}=i^{3}=-i.

Negative powers

For negative exponents, i−1=1i=1i×ii=ii2=i−1=−ii^{-1}=\dfrac{1}{i}=\dfrac{1}{i}\times\dfrac{i}{i}=\dfrac{i}{i^{2}}=\dfrac{i}{-1}=-i. More generally i−n=1ini^{-n}=\dfrac{1}{i^{n}}, which can then be simplified with the same cycle. …

Definition 1Cyclic powers of $i$

i1=i, i2=−1, i3=−i, i4=1i^{1}=i,\ i^{2}=-1,\ i^{3}=-i,\ i^{4}=1, repeating with period 44. So $i^{n}=i …

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