If (1 + i/1 - i)m = 1, then find the least positive integral value of m.

Question

Shiksha Nation · Accepted Answer

(1 + i/1 - i)m = 1

⇒ ((1 + i)(1 + i)/(1 - i)(1 + i))m

= (1 + 2i + i2/1 - i2)m

= (1 + 2i - 1/1 + 1)m

= (2i/2)m = (i)m

So, im = 1

Now, i1 = i, i2 = -1, i3 = -i, i4 = 1

∴ m must be a multiple of 4

Therefore, the least positive integral value of m = 4

If (^{1 + i}/_{1 - i})^m = 1, then find the least positive integral value of m.