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Square roots of a conjugate (H2 maths)

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  • Mad Hat's Avatar
    91 posts since May '10
    • 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?

  • Moderator
    wee_ws's Avatar
    1,502 posts since Mar '08
    • Hi,

        Hence part question is allocated 2 marks, so her answer is acceptable. It will be good if she quotes the reason. Thanks.

      Cheers,
      Wen Shih

    • Hi,

        Students should strive to clarify their working steps as far as possible.

        In the H2 Maths syllabus, it is stated clearly the following:

        Assessment objective 2 - reason and communicate mathematically through writing mathematical explanation, arguments and proofs, and inferences.

        Use of graphic calculator - Where unsupported answers from GC are not allowed, candidates are required to present the mathematical steps using mathematical notations and not calculator commands. Incorrect answers without working will receive no marks.

        Thanks.

      Cheers,
      Wen Shih
      wenshih.wordpress.com

    • Hi,

        My project supervisor used to joke that when he sees a proof in a mathematics research paper that says "the proof of this result is obvious" it is usually not the case :P

      Cheers,
      Wen Shih
      wenshih.wordpress.com

  • Mad Hat's Avatar
    91 posts since May '10
    • Eh Wen Shih, thanks for your input. Ha yes, I didn't find it "obvious" that the square roots of a conjugate are just the conjugates of the original square roots. I didn't even know it was true. But when I checked, it was quite true :-) So I will let my student know that her answer is good. Learn things from them all the time!

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