If i 2 = - 1 and 1 + i 2 n = 1 - i 2 m = 1 , ∀ n , m ∈ N , then the minimum value of n + m is equal to

Question

If i 2 = - 1 and 1 + i 2 n = 1 - i 2 m = 1 , ∀ n , m ∈ N , then the minimum value of n + m is equal to,

MARKS App · Accepted Answer

Here, 1 + i 2 = e i π 4 and 1 - i 2 = e - i π 4 Now, e i π 4 n = e - i π 4 m = 1 ⇒ n π 4 = - m π 4 = 2 k π , ∀ k ∈ Z Hence, n = 8 k and m = - 8 k So n m i n = m m i n = 8 i.e. n + m = 16

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