This question is from Nov 2002, paper 1:
i) The complex number x+iy is such that
(x+iy)2 = i. Find the possible values of the real
numbers x and y.
ii) HENCE find the possible values of w such
that w2 = −i.
The first part is straightforward. They want the square roots of
i, and the answers are ±( 1/√ 2 + i/√ 2). The second part says HENCE find the square
roots of −i. So I
asked my tuition students to try this, and one of them answered in
one line:
w = ±( 1/√ 2 − i/√ 2).
Her "explanation" was that the square roots of a
conjugate should just be the conjugates of the original square
roots.
My question is whether this is
an acceptable "explanation" at A-levels? I mean, it is quite true that the square
roots of a conjugate are just the conjugates of the original square
roots. But is this a standard theorem (at A-levels) that a student
can quote without proof?
I asked my student how she
knew that the square
roots of a conjugate are just the conjugates of the original square
roots? - and she just shrugged her shoulders and said it was
obvious!
So is it obvious? Can a student just appeal to this fact without
proof in the A-level exams? Because I was expecting a longer answer
for part (ii). Something like this:
w2 = −i
⇒ w2 = i2 × i
⇒ w2 = i2 × (1/√ 2 + i/√
2)2 , using the result in part (i)
⇒ w = ± i × (1/√ 2 + i/√ 2))
⇒ w = ±(1/√ 2 − i/√ 2).
But she just answered the question in one line:
w2 = −i
⇒ w = ±(1/√ 2 − i/√ 2) , by conjugating the
answers from part (i).
I told her I would check whether this answer is acceptable. Can
any experienced teacher advise?