As we all know about the topic of complex numbers, we are familiar with the term iota (i), where i = √(-1). A question arises that is there any value possible for ii.
So, its simple answer is yes, there is a value for ii. The below solution is mentioned for it.
We have to find the value of ii. So, Let y = ii
Taking ln on both sides,
ln(y)= i ln(i) ----- ( i ) [ ln (ab) = b*ln(a) ]
Now, for solving ln(i), we have to understand the following concept :
In the polar representation of complex numbers, we write z = reiθ, where –
z = a + ib, r = |a2 + b2| θ = tan-1(b/a), So, taking log on both sides of the equation z = reiθ ln(z) = ln(r) + iθ [ln(ea) = a, and ln(a*b) = ln(a) + ln(b)] Putting the value of z, r and θ in the above equation ln(a+ib) = ln(|a2 + b2|) + i*tan-1(b/a)
So, writing ln(i) = ln(0 + 1i), and applying the above formula
ln(0+1i) = ln(|02 + 12|) + i*tan-1(1/0) ln(i) = ln1 + i*∏/2 [ tan-1(1/0) = tan-1(∞) = ∏/2 ] ln(i) = i*∏/2 [ ln1 = 0 ]
Now putting the value of ln(i) in equation ( i )
ln(y) = i * ( i*∏/2 ) ln(y) = i2 * ∏/2 ln(y) = -1 * ∏/2 [i2 = -1] ln(y) = -∏/2 y = e -∏/2 [ln(a) = b ⇒ a = eb]
As we have assumed y = ii.
ii = e -∏/2
If we calculate the value for e -∏/2 with the help of a calculator, we get its approximate value as 0.20788.
ii = 0.20788