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Mathematics · Ch 10 — Complex Numbers

Powers of i

10.2.7

Powers of i

Powers of i

We already know −1=i, i2=−1, i3=−i, i4=1\sqrt{-1}=i,\ i^2=-1,\ i^3=-i,\ i^4=1. Consider ini^n for a positive integer n>4n>4. Divide nn by 44 to get a quotient mm and remainder rr: n=4m+rn=4m+r, where 0≤r<40\le r<4. Then

in=i4m+r=i4m⋅ir=(i4)m⋅ir=1m⋅ir=iri^n=i^{4m+r}=i^{4m}\cdot i^r=(i^4)^m\cdot i^r=1^m\cdot i^r=i^r

So only the remainder on dividing the exponent by 44 matters — any power of ii collapses to one of just four values, 1,i,−1,−i1,i,-1,-i, arranged in a repeating cycle (see the accompanying Fig. 1.1).

Examples.

  • i50=(i4)12⋅i2=i2=−1i^{50}=(i^4)^{12}\cdot i^2=i^2=-1 (since 50=4(12)+250=4(12)+2).
  • i318=(i4)79⋅i2=−1i^{318}=(i^4)^{79}\cdot i^2=-1 (since 318=4(79)+2318=4(79)+2).
  • i999=(i4)249⋅i3=−ii^{999}=(i^4)^{249}\cdot i^3=-i (since 999=4(249)+3999=4(249)+3).

General remark, for any integer n∈Zn\in\mathbb{Z} (this covers negative exponents too, using i−1=1/i=−ii^{-1}=1/i=-i and continuing the same cycle backwards):

i4n=1,i4n+1=i,i4n+2=−1,i4n+3=−ii^{4n}=1,\qquad i^{4n+1}=i,\qquad i^{4n+2}=-1,\qquad i^{4n+3}=-i …

Figure Fig.1.1Fig. 1.1 — the four-value cycle of powers of i

What this figure shows. A small cycle diagram lays out the four possible values that any integer power of ii can take — 11, ii, −1-1, −i-i — arranged so that repeatedly multiplying by ii walks around the cycle one step at a time and returns to the start after four steps. It visually backs the remainder rule: to evaluate ini^n for any integer nn (positive or negative), divide nn by 44, read off the remainder r∈{0,1,2,3}r\in\{0,1,2,3\}, and the answer is whichever of the four cycle values sit …

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