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Find square root

  1. Nov 24, 2004 #1
    does anyone know how to find the square root of 5+12i ?
     
  2. jcsd
  3. Nov 24, 2004 #2

    Hurkyl

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    Try asking yourself the same question written differently:

    Can you find a number z such that z*z = 5 + 12i?
     
  4. Nov 24, 2004 #3
    That sounds easier but im not sure how to figure that out.

    This was a multiple choice question so the answer is 2+i
    I dont know how they got that!!
     
    Last edited: Nov 24, 2004
  5. Nov 24, 2004 #4
    Actually... no, that's wrong. The answer is 3 + 2i.
     
  6. Nov 24, 2004 #5

    Hurkyl

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    If it's multiple choice, then it's even easier.

    You say one of the choices was 2+i. Well, what is (2+i)*(2+i)? Is it 5+12i?

    Do you know of a way to write down an arbitrary complex number, z, in terms of two (real) variables? If so, then you can compute what z^2 is...
     
  7. Nov 24, 2004 #6
    can u plz tell me how you got that answer?
     
  8. Nov 24, 2004 #7
    remember i^2 = -1 .
     
  9. Nov 24, 2004 #8
    Well, you know that z is a complex number in the form a+bi. Therefore, you can say this:

    [tex](a+bi)^2=5+12i[/tex]
    [tex]a^2+2abi-b^2=5+12i[/tex]

    Now, since a and b are real, you know a^2-b^2 = 5, and 2abi = 12i. Therefore, you can solve the system of equations to find a and b, thus finding z.
     
  10. Nov 24, 2004 #9
    Unfortunately, it was only a matter of guessing (and reverse factoring).

    I simply used: (ai + b)^2 = 12i + 5

    Expanding the equation: (ai)^2 + (ab)i + b^2 = 12i + 5

    Knowing that i^2 = -1, this equation can be simplified further:

    -(a^2) + 2(ab)i + b^2 = 12i + 5

    so... therefore:

    a*b must equal 12 and 2(b^2 - a^2) must equal 5

    Make a system of equations and solve:

    2ab = 12
    b^2 - a^2 = 5

    Solving, we find that a = 2 and b = 3

    Input them into the original expanded equation (-(a^2) + 2(ab)i + b^2 = 12i + 5)

    So:

    -(2^2) + 2(2*3)i + 3^2 = 12i + 5

    -4 + 12i + 9

    12i + 5

    There's an easier method to this, probably... sorry I can't really help you out.

    (EDIT: This came 3 min. after nolachrymose made a post... I'm really slow at this.)
     
    Last edited: Nov 24, 2004
  11. Nov 24, 2004 #10
    Phreak everything makes sense except for where did u get a=2 b=3? what did u solve and how? Im so dumb sorry lol I really dont like math.

    Thanks Nolachrymose I would have never figured this out without u and Phreak thanks guys, :smile:
     
    Last edited: Nov 24, 2004
  12. Nov 24, 2004 #11

    Gokul43201

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    You have ab = 6, so b = 6/a
    And b^2 - a^2 = 5, so 36/a^2 - a^2 = 5. This is just a quadratic in a^2. Solve it to find a^2, and hence a and b.
     
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