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Math Help - write the expression in standard form

  1. #1
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    write the expression in standard form

    Type your answer in the form a + bi

    4/(5+i)
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  2. #2
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    Re: write the expression in standard form

    Are you familiar with the conjugate of a complex number? The conjugate of a+bi is a-bi. Multiply top and bottom of your expression by the conjugate of the denominator. Multiplying any complex number by its conjugate will give a real number, so you are guaranteed to get a real number in the denominator.
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    Re: write the expression in standard form

    Quote Originally Posted by asilvester635 View Post
    Type your answer in the form a + bi
    4/(5+i)
    Here is a very useful fact: z\ne 0,~~\frac{1}{z} = \frac{{\overline z }}{{{{\left| z \right|}^2}}}
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    Re: write the expression in standard form

    If all else fails, here's how it's done:

    Spoiler:
    4/(5 + i) x (5 - i)/(5 - i) -------- (Multiplying top and bottom by conjugate)
    = 4(5 - i)/((5 + i)(5 - i))
    = (20 - 4i)/(25 - i2) -------- (We know that i2 is equal to -1, therefore 25 - (-1) = 26)
    = (20 - 4i)/26
    = (10 - 2i)/13 -------- (Dividing throughout by 2)
    = (10/13) - (2/13)i -------- (Rearranging it into standard form, i.e., a + bi)


    Apologies for the formatting.
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  5. #5
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    Re: write the expression in standard form

    Hi Plato,
    I don't understand why this is a useful fact.
    Could you please explain ?
    (I don't know if this is relevant but the original post posed no problem for me)
    Thanks.
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  6. #6
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    Re: write the expression in standard form

    Quote Originally Posted by Melody2 View Post
    I don't understand why this is a useful fact.
    (I don't know if this is relevant but the original post posed no problem for me)
    It gives a one step solution:
    \frac{4}{5+i}=\frac{20}{26}-\frac{4}{26}i
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    Re: write the expression in standard form

    I'm sorry but I still don't understand what you have done.
    Maybe I don't understand the notation.
    I really want to understand.
    Thanks
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  8. #8
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    Re: write the expression in standard form

    Quote Originally Posted by Melody2 View Post
    I'm sorry but I still don't understand what you have done.
    Maybe I don't understand the notation.
    I really want to understand.
    \frac{1}{a+bi}=\frac{\overline{a+bi}}{a^2+b^2}= \frac {a}{a^2+b^2}-\frac{b}{a^2+b^2}i

    \overline{z} is almost the universal notation for the conjugate of z.
    Last edited by Plato; November 7th 2013 at 04:09 AM.
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  9. #9
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    Re: write the expression in standard form

    Thanks Plato,

    It was  {{\left| z \right|}^2} that threw me.

    So if z=5+6i then  {{\left| z \right|}^2} =  {5^2+6^2}

    I assume that this is also standard notation? Or is there some logic to it that I am missing?



    This is the first time I have tried to use Latex. I copied yours. I am just a little pleased with myself.

    Thanks.
    Last edited by Melody2; November 7th 2013 at 04:42 AM.
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  10. #10
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    Re: write the expression in standard form

    Quote Originally Posted by Melody2 View Post
    It was  {{\left| z \right|}^2} that threw me.
    So if z=5+6i then  {{\left| z \right|}^2} =  {5^2+6^2}
    I assume that this is also standard notation? Or is there some logic to it that I am missing?
    |a+bi|=\sqrt{a^2+b^2} therefore |a+bi|^2=a^2+b^2=(a+bi)(a-bi)=(a+bi)\overline{(a+bi)}.
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