Given that 𝑧 is negative eight 𝑖,
determine the square roots of 𝑧 without first converting to trigonometric form.
Since we want to find the square
root of the complex number 𝑧 is negative eight 𝑖, let’s give our square root some
form and call them 𝑤 equal to 𝑥 plus 𝑖𝑦 so that 𝑥 is the real part and 𝑦 is
the coefficient of the complex part of our square root. So then, if 𝑤 is the square root
of 𝑧, then 𝑤 squared is 𝑧, which is negative eight 𝑖. And since 𝑤 is 𝑥 plus 𝑖𝑦, then
𝑥 plus 𝑖𝑦 squared must be negative eight 𝑖. If we expand our left-hand side, we
have 𝑥 squared plus two 𝑖𝑥𝑦 minus 𝑦 squared is negative eight 𝑖. And comparing real and imaginary
parts on the left-hand side, we have 𝑥 squared minus 𝑦 squared is equal to zero
and two 𝑥𝑦 is negative eight.
Our second equation implies 𝑥𝑦 is
negative eight over two; that is negative four. And this tells us, for one thing,
that the signs of 𝑥 and 𝑦 are not the same. So, if 𝑥 is positive, 𝑦 must be
negative and vice versa. Now, recalling that for a complex
number 𝑧, which is 𝑎 plus 𝑖𝑏, the modulus of 𝑧 is the square root of 𝑎 squared
plus 𝑏 squared. This then means that the modulus of
𝑧 squared is 𝑎 squared plus 𝑏 squared. And applying this to our square
root 𝑤, where 𝑤 is 𝑥 plus 𝑖𝑦, we have the modulus of 𝑤 squared is 𝑥 squared
plus 𝑦 squared. And this must be equal to the
modulus of our original complex number 𝑧 is negative eight 𝑖. That is the square root of zero
squared plus negative eight squared, which is eight, that is, that 𝑥 squared plus
𝑦 squared is equal to eight.
And we now have a set of two
equations we can solve for 𝑥 and 𝑦. That’s 𝑥 squared minus 𝑦 squared
is zero and 𝑥 squared plus 𝑦 squared is eight. Now, just tidying up and making
some room, if we call our first equation equation one, that’s 𝑥 squared minus 𝑦
squared is zero. And equation two is 𝑥 squared plus
𝑦 squared is eight. Adding equations one and two gives
us two 𝑥 squared is equal to eight. That is, 𝑥 squared is equal to
four so that 𝑥 is equal to positive or negative two.
Our equation one tells us that 𝑥
squared is equal to 𝑦 squared so that if 𝑥 is positive or negative two, then the
corresponding 𝑦-values are negative or positive two. Remembering that our square roots
were 𝑤 which is equal to 𝑥 plus 𝑖𝑦 and using the values of 𝑥 and 𝑦 that we
just found, then the square roots of 𝑧 is negative eight 𝑖 are two minus two 𝑖
and negative two plus two 𝑖.