# How do you add, subtract, multiply, and divide the complex numbers z = −10 + 1 i and w = 7 i?

Feb 19, 2017

$z + w = - 10 + 8 i$

$z - w = - 10 - 6 i$

$\setminus \setminus \setminus \setminus \setminus z w = - 7 - 70 i$

$\setminus \setminus \setminus \setminus \setminus \frac{z}{w} = \frac{1}{7} + \frac{10}{7} i$

#### Explanation:

We have:

$z = - 10 + i$, and $w = 7 i$

To add and subtract complex numbers we simply add subtract the real and imaginary parts separately, thus:

$z + w = \left(- 10 + i\right) + \left(7 i\right)$
$\text{ } = \left(- 10\right) + \left(1 + 7\right) i$
$\text{ } = - 10 + 8 i$

And:

$z - w = \left(- 10 + i\right) - \left(7 i\right)$
$\text{ } = \left(- 10\right) + \left(1 - 7\right) i$
$\text{ } = - 10 - 6 i$

To multiply complex numbers we multiply every combination in one term with every combination of the other term, and use ${i}^{2} = - 1$, so

$z w = \left(- 10 + i\right) \left(7 i\right)$
$\text{ } = \left(- 10\right) \left(7 i\right) + \left(i\right) \left(7 i\right)$
$\text{ } = - 70 i + 7 {i}^{2}$
$\text{ } = - 70 i + 7 \left(- 1\right)$
$\text{ } = - 7 - 70 i$

And for division we generally remove the complex denominator by multiplying the numerator and denominator by the complex conjugate of the denominator (as the product of a complex number with its conjugate is always real). As the denominator in this example is purely imaginary we can multiply the numerator and denominator by $i$ to make it real.

$\frac{z}{w} = \frac{- 10 + i}{7 i}$
$\text{ } = \frac{- 10 + i}{7 i} \cdot \frac{i}{i}$
$\text{ } = \frac{- 10 i + {i}^{2}}{7 {i}^{2}}$
$\text{ } = \frac{- 10 i - 1}{- 7}$
$\text{ } = - \frac{1 + 10 i}{- 7}$
$\text{ } = \frac{1}{7} + \frac{10}{7} i$