// Numbas version: exam_results_page_options {"name": "Complex Numbers; Division 01 (Rect')", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Complex Numbers; Division 01 (Rect')", "tags": [], "metadata": {"description": "

Division (Rationalising) complex numbers in rectangular (Cartesian) Form.

Thanks are due to Christian for his Gap Fills in Fractions code and method

", "licence": "All rights reserved"}, "statement": "

Let $z$ and $w$ be any two complex numbers

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$   z=a+bi   $          and          $   w=c+di   $ 

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Division is carried out by rationalising.

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Multiply both \"top and bottom\" of  the fraction by the complex conjugate of the bottom (the denominator).

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$\\Large \\frac{z}{ w} = \\frac{(a + bi)}{(c + di)}$

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$\\Large \\frac{z}{ w} = \\frac{(a + bi)}{(c + di)}\\times\\frac{(c - di)}{(c - di)}$

", "advice": "

We are asked to carry out the division of two complex numbers by rationalising.

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If:

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$\\large z_1=\\var{z1}$     and      $\\large z_2=\\var{z2}$

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Then:

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\\[\\frac{z_1}{z_2}=\\frac{\\var{z1}}{\\var{z2}}\\]

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We deal with this by multiplying both top and bottom by the complex conjugate of the denominator  $z_2^*=\\var{z2C}$

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\\[\\frac{z_1}{z_2}=\\frac{\\var{z1}}{\\var{z2}} \\times       \\frac{\\var{z2C}}{\\var{z2C}}                                \\]

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We can do this as all we are really multiplying by is $1$.

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Now we multiply across the top, then across the bottom - just as we would do with ordinary fractions:

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\\[\\frac{z_1}{z_2}=\\frac{\\simplify{{z1}{z2C}}}{\\simplify{{z2}{z2C}}}\\]

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All that remains is to express this as a proper complex number, with separate real and imaginary parts:

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\\[ \\frac{z_1}{z_2}= \\frac{\\var{Rans1}}{\\var{Ans2}}                           +          \\frac{\\var{Ians1}}{\\var{Ans2}}    i \\]

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If:

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$\\large z_1=\\var{z1}$     and      $\\large z_2=\\var{z2}$

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Evaluate: $\\large \\frac{z_1}{z_2}$ 

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Lay out the two \"fractions\":

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$  \\Large \\frac{z_1}{z_2}=$ [[0]][[1]]$ \\large  \\times$[[2]][[3]]
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Multiply the two complex numbers on top, then those on the bottom:

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$  \\Large \\frac{z_1}{z_2}=$ [[4]][[5]]
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 We can now write the answer as a single complex number with real and imaginary parts (they will most likely be fractions).

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$  \\Large \\frac{z_1}{z_2}=$ [[6]] $+$ [[7]] $i$

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Now enter the complete answer as one entry, not forgetting to use fractions where needed:

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$  \\Large \\frac{z_1}{z_2}=$ [[8]]

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\n

\n


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