Question

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

Answer 1

`z+w = -10 + 8i`

`z-w = -10 -6i`

`\ \ \ \ \ zw = -7-70i`

`\ \ \ \ \ z/w = 1/7+10/7i`

Explanation:

We have:

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

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

`z+w = (-10+i) + (7i)`
`" "= (-10) + (1+7)i`
`" "= -10 + 8i`

And:

`z-w = (-10+i) - (7i)`
`" "= (-10) + (1-7)i`
`" "= -10 -6i`

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

`zw = (-10+i)(7i)`
`" "= (-10)(7i)+(i)(7i)`
`" "= -70i+7i^2`
`" "= -70i+7(-1)`
`" "= -7-70i`

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.

`z/w = (-10+i)/(7i)`
`" " = (-10+i)/(7i) *i/i`
`" " = (-10i+i^2)/(7i^2)`
`" " = (-10i-1)/(-7)`
`" " = -(1+10i)/(-7)`
`" " = 1/7+10/7i`