Question

How do you divide $\dfrac{-1+5i}{-8-7i}$?

Answer 1

Hint: A complex number is of the form $p+iq$ , where p, q are real numbers. The division of complex numbers is performed by multiplying and dividing the conjugate of the denominator. In our question, we multiply and divide the conjugate of $-8-7i$ and simplify the expression to get the required result.

Complete step-by-step answer:
A complex number has two parts, a real part, and an imaginary part.
Example:
For a complex number $2+3i$ ,
Real part: 2
Imaginary Part: 3
The conjugate of the given complex number is obtained by changing the sign of its imaginary part.
Example:
The conjugate of a complex number $2+3i$ is given as $2-3i$ that is obtained by changing the sign of the imaginary part.
We are two complex numbers and need to divide them. We can solve the given question by multiplying and dividing the conjugate of the denominator.
In our question,
Numerator: $-1+5i$
Denominator: $-8-7i$
Conjugate of the denominator: $-8+7i$
We need to divide $\dfrac{-1+5i}{-8-7i}$
Multiplying and dividing the conjugate of the denominator, we get,
$\Rightarrow \dfrac{-1+5i}{-8-7i}=\dfrac{-1+5i}{-8-7i}\times \dfrac{-8+7i}{-8+7i}$
Multiplying the conjugate of denominator with each term on the numerator and denominator,
$\Rightarrow \dfrac{-1+5i}{-8-7i}=\dfrac{-1\times \left( -8+7i \right)+5i\times \left( -8+7i \right)}{\left( -8-7i \right)\times \left( -8+7i \right)}$
Simplifying the above expression on the right-hand side,
$\Rightarrow \dfrac{-1+5i}{-8-7i}=\dfrac{8+7i-40i+35{{i}^{2}}}{\left( -8-7i \right)\times \left( -8+7i \right)}$
We know that $\left( p+iq \right)\left( p-iq \right)={{p}^{2}}+{{q}^{2}}$
Substituting the same, we get,
$\Rightarrow \dfrac{-1+5i}{-8-7i}=\dfrac{8+7i-40i+35{{i}^{2}}}{{{8}^{2}}+{{7}^{2}}}$
Substituting the value of ${{i}^{2}}=-1$ ,
$\Rightarrow \dfrac{-1+5i}{-8-7i}=\dfrac{8+7i-40i+35\left( -1 \right)}{64+49}$
Let us evaluate further.
$\Rightarrow \dfrac{-1+5i}{-8-7i}=\dfrac{8+7i-40i-35}{64+49}$
$\Rightarrow \dfrac{-1+5i}{-8-7i}=\dfrac{-27-47i}{113}$
Writing the right-hand side in the form of $p+iq$ , we get,
$\therefore \dfrac{-1+5i}{-8-7i}=-\dfrac{27}{113}-i\dfrac{47}{113}$

Note: The complex number $z=x+iy$ can be represented on the plane as the coordinates, $\left( x,y \right)$Given that $x$ is the real part of the complex number and $y$ is the imaginary part of the complex number. Complex numbers are used to perform geometric operations.