Simplify two minus two 𝑖 divided by three minus 𝑖.
In order to divide two complex numbers, we multiply the numerator and denominator by the conjugate of the denominator. This is because when we multiply a complex number by its conjugate, the answer is a purely real number.
We recall that for any complex number 𝑧, which is equal to 𝑥 plus 𝑦𝑖, then the conjugate, written 𝑧 bar, is equal to 𝑥 minus 𝑦𝑖. In this question, the conjugate of three minus 𝑖 is three plus 𝑖. We need to multiply the fraction two minus two 𝑖 over three minus 𝑖 by the fraction three plus 𝑖 over three plus 𝑖. We can now expand the parentheses on the numerator and denominator using the FOIL method.
On the numerator, the first terms have a product of six. Multiplying the outer terms gives us two 𝑖. The inner terms gives us negative six 𝑖. And multiplying the last terms gives us negative two 𝑖 squared. On the denominator, we get nine plus three 𝑖 minus three 𝑖 minus 𝑖 squared. Plus three 𝑖 minus three 𝑖 is equal to zero. We recall from our knowledge of imaginary and complex numbers that 𝑖 squared is equal to negative one. The numerator therefore simplifies to eight minus four 𝑖, as six plus two is equal to eight and two 𝑖 minus six 𝑖 is negative four 𝑖. On the denominator, we have nine plus one.
Our expression is simplified to eight minus four 𝑖 divided by 10. All three of our integers are even, so we can divide each term by two. The expression simplifies to four minus two 𝑖 divided by or over five. For any simplification of this type, we can leave our answer as shown or we can split the real and imaginary parts. Four minus two 𝑖 over five is equivalent to four-fifths minus two-fifths 𝑖.
The real part of our complex number is four-fifths, and the imaginary part is negative two-fifths. These are the values of 𝑥 and 𝑦 in the complex number 𝑧 is equal to 𝑥 plus 𝑦𝑖, respectively.