Question

How to use the clock rule for imaginary numbers? My teacher taught it a while ago but I forgot it. Its like if i^67 and you take out the lowest multiple of 4. You get either 0, 1/4, 1/2 or 3/4. 0=1, 1/4=i, 1/2=-1, 3/4=-i. Can someone please re-explain this to me?

Answer 1

shoot i dont remember a clock rule but...
\[i^{2} = -1\]
\[i^{3} = i*i^{2} = -i\]
\[i^{4} = i^{2}*i^{2} = 1\]
then i^5 = i*i^4 = i and pattern continues
rule: divide by 4, then look at remainder
67/4 = 16 remainder 3 so it matches with i^3
\[i^{67} = i^{3} = -i\]

Answer 2

The clock rule is basically take out a multiple of 4 until the lowest remainder. I just remembered!