The imaginary unit i has the defining property i2=−1, and its higher powers cycle through just four values: i1=i, i2=−1, i3=i⋅i2=−i, and i4=i2⋅i2=1, after which the pattern repeats — i5=i, i6=−1, and so on. This four-value cycle is the key to evaluating in for any integer n, however large: divide n by 4 to get n=4m+r with remainder 0≤r<4, and then in=i4m⋅ir=(i4)m⋅ir=1m⋅ir=ir — the whole power collapses to whichever of 1,i,−1,−i corresponds to the remainder. In compact form: i4k=1, i4k+1=i, i4k+2=−1, i4k+3=−i for any integer k (this works for negative exponents too, using i−1=i1=−i and continuing the same four-cycle backwards). This single reduction rule is what makes it possible to simplify expressions like i592 or sums such as i30+i40+i50+i60 instantly, without ever multiplying out a long chain of i's — only the remainder on division by 4 matters.