How do you divide $\frac{6 - i}{7 - 2 i}$ $(6-i) / (7-2i)$ ?

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How do you divide $\frac{6 - i}{7 - 2 i}$ $(6-i) / (7-2i)$ ?

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Answer 1 · 2016-04-08T10:14:28

$\frac{44}{53} + \frac{5}{53} i$ $44/53 + 5/53 i$

Explanation:

To divide this fraction we require to rationalise the denominator.

We do this by multiplying numerator/denominator by the $complex conjugate of the denominator$ $color(blue)" complex conjugate " " of the denominator "$

If $a \pm bi is a complex number then$ $color(blue)" a ± bi " " is a complex number then "$

$a \mp bi is it's conjugate$ $color(red)" a ∓ bi " " is it's conjugate "$

Note that the 'real part' remains unchanged , while the sign of the 'imaginary part' changes.

Also (a+ bi)(a - bi) = $a^{2} - b^{2} a real number$ $a^2 - b^2 " a real number "$

and $i^{2} = {(\sqrt{- 1})}^{2} = - 1$ $i^2 = (sqrt(-1))^2 = -1$

Now the conjugate of 7 - 2i is 7 + 2i

multiplying numerator / denominator by (7 + 2i)

$\Rightarrow \frac{(6 - i) (7 + 2 i)}{(7 - 2 i) (7 + 2 i)} = \frac{42 + 5 i + 2}{49 + 4} = \frac{44}{53} + \frac{5}{53} i$ $rArr ((6 - i)(7 + 2i))/((7 - 2i)(7 + 2i)) = (42 + 5i +2)/(49 + 4)= 44/53 + 5/53 i$

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