Advanced Math/A question in complex number
I have a question in complex number:
What should be the value of (-a * -b)^0.5
This question seems to give two answers:
(-a * -b)^0.5 = ( (-1)*(-1)*a*b)^0.5 = (a*b)^0.5
(-a * -b)^0.5 = ( (-1)*(-1)*a*b)^0.5 = ((-1)*(-1))^0.5 * (a*b)^0.5 = (i^2) * (a*b)^0.5 = -(a*b)^0.5
where, square root of (-1) is taken as i, and i^2 = -1
Many people would go with second method. But why? What is mathematically wrong with first method?
This is a classic apparent paradox. The inconsistency comes down to the fact that
(1) √(ab) ≠ √a･√b when both a, b < 0.
Probably the best way to prove that this doesn't hold is to make exactly the same calculation you did, which assumes the relation does hold, and show a contradiction, which you did. Your calculation is equivalent to assuming a,b >0, for which the relation would hold.
Lets take a look at the case a < 0 and b > 0, then
√(ab) = √[(-1)|a||b|] since absolute value of b = |b| = b for b >0.
= √(-1)√(|a||b|) = i√|a|√|b
and everything is OK (no inconsistency) and we haven't had to use the above relation (1) where both terms under the radical are negative: note also that everything under the radical is real.
Now let a,b <0 and
√(ab) = √[ (-1)|a|･(-1)|b| ] √[ (-1)(-1)|a||b| ] = √[ (-1)(-1) ] ･√( |a||b| );
up to now we haven't used (1) where both terms are negative, and if we want to keep everything under the radical real, we would substitute (-1)･(-1) = 1 and get √(|a||b|) with no inconsistency.
This just shows you have to be careful with imaginary numbers. For instance, you could write
(a) -1 = e^(iπ)･e^i(0･π) = e^(iπ) or
(b) -1 = e^(iπ)･e(i2π) = e^(i3π).
Then if we take the square root
(a) (-1)^1/2 = [e^(iπ)]^1/2 = e^(π/2) = i or
(b) (-1)^1/2 = [e(i3π)] = e^(i3π/2) = -i.
Here, it looks like there is an inconsistency, but there isn't; what really happened is that we used the PRINCIPAL VALUE (PV) of 1 ( = e^(iπ0) ) in (a) and a multiple of the PV in (b). Both are technically correct, but to use (b) you would have to be explicit about your use of the non-PV and do extra bookeeping to stay clear of inconsistencies down the road.