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Express the following in the form $a+bi$.

Input $a$ and $b$ as fractions or integers and not as decimals.

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Division of two complex numbers can be performed by mutiplying both the numerator and denominator by the conjugate of the denominator.
Suppose that \\[ z = \\frac{a+bi}{c+di},\\;\\; c+di \\neq 0\\] then we have:
\\[\\begin{eqnarray*}\n \n z&=&\\frac{a+bi}{c+di}\\\\\n \n &=&\\frac{(a+bi)(c-di)}{(c+di)(c-di)}\\\\\n \n &=&\\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}\\\\\n \n &=&\\frac{ac+bd}{c^2+d^2}+\\frac{bc-ad}{c^2+d^2}i\n \n \\end{eqnarray*}\n \n \\]
Although this is a formula for the inverse, the best way to find these complex numbers is to remember to multiply top and bottom by the conjugate of the denominator.
(a)
\\[\\begin{eqnarray*}\\simplify[std]{{c1}/{z1}} &=&\\simplify[std]{({c1}*{conj(z1)})/({z1}*{conj(z1)})}\\\\\n \n &=&\\simplify[std]{{c1*conj(z1)}/{abs(z1)^2}}\\\\\n \n &=& \\simplify[std]{{c1*re(z1)}/{abs(z1)^2}-{c1*im(z1)}/{abs(z1)^2}*i}\n \n \\end{eqnarray*} \\]
(b)
\\[\\begin{eqnarray*}\\simplify[std]{{c2}/{z2}} &=&\\simplify[std]{({c2}*{conj(z2)})/({z2}*{conj(z2)})}\\\\\n \n &=&\\simplify[std]{{c2*conj(z2)}/{abs(z2)^2}}\\\\\n \n &=& \\simplify[std]{{c2*re(z2)}/{abs(z2)^2}-{c2*im(z2)}/{abs(z2)^2}*i}\n \n \\end{eqnarray*} \\]
(c)
\\[\\begin{eqnarray*}\\simplify[std]{{z1}/{z3}} &=&\\simplify[std]{({z1}*{conj(z3)})/({z3}*{conj(z3)})}\\\\\n \n &=&\\simplify[std]{{z1*conj(z3)}/{abs(z3)^2}}\\\\\n \n &=& \\simplify[std]{{re(z1*conj(z3))}/{abs(z3)^2}+{im(z1*conj(z3))}/{abs(z3)^2}*i}\n \n \\end{eqnarray*} \\]
(d)
\\[\\begin{eqnarray*}\\simplify[std]{{z3}/{z2}} &=&\\simplify[std]{({z3}*{conj(z2)})/({z2}*{conj(z2)})}\\\\\n \n &=&\\simplify[std]{{z3*conj(z2)}/{abs(z2)^2}}\\\\\n \n &=& \\simplify[std]{{re(z3*conj(z2))}/{abs(z2)^2}+{im(z3*conj(z2))}/{abs(z2)^2}*i}\n \n \\end{eqnarray*} \\]

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$\\displaystyle \\simplify[std]{{c1}/{z1}}\\;=\\;$[[0]].

Do not include brackets in your answer.

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Make sure that you input the real and imaginary parts as fractions and not as decimals. Do not include brackets in your answer.

$\\displaystyle \\simplify[std]{{c2}/{z2}}\\;=\\;$[[0]].

Do not include brackets in your answer.

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Make sure that you input the real and imaginary parts as fractions and not as decimals. Do not include brackets in your answer.

$\\displaystyle \\simplify[std]{{z1}/{z3}}\\;=\\;$[[0]].

Do not include brackets in your answer.

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Make sure that you input the real and imaginary parts as fractions and not as decimals. Also do not include brackets in your answer.

$\\displaystyle \\simplify[std]{{z3}/{z2}}\\;=\\;$[[0]].

Do not include brackets in your answer.

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Make sure that you input the real and imaginary parts as fractions and not as decimals. Do not include brackets in your answer.

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15/07/2015:

Added tags.

4/07/2012:

Added tags

Question a - sometimes the complex number is generated as a/(b+i*c) but sometimes the complex number is displayed as a decimal, i.e. 0.0975609756+0.1219512195i if this happens then the question is invalid.

16/07/2012:

The above issue has been resolved,

Also forbid brackets in the answers as otherwise can repeat the question and be marked as correct.

0.0975609756+0.1219512195i

", "description": "

Inverse and division of complex numbers. Four parts.

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