Video Transcript
Put one plus two 𝑖 divided by nine plus seven 𝑖 in the form 𝑥 plus 𝑦𝑖.
In this question, we need to divide two complex numbers and write our answer in the form 𝑥 plus 𝑦𝑖 where 𝑥 is the real part and 𝑦 is the imaginary part of our complex number. We recall that in order to divide two complex numbers, we multiply the numerator and denominator by the complex conjugate of the denominator. We do this as this results in the denominator being purely real. If the complex number 𝑧 is equal to 𝑥 plus 𝑦𝑖, then the conjugate 𝑧 bar is equal 𝑥 minus 𝑦𝑖.
In this question, we need to multiply one plus two 𝑖 over nine plus seven 𝑖 by nine minus seven 𝑖 over nine minus seven 𝑖. We will do this by distributing the parentheses or expanding the brackets on the numerator and denominator separately. One way of doing this is using the FOIL method. Multiplying the first terms of our numerator gives us nine. The outer terms have a product of negative seven 𝑖. The inner terms have a product of 18𝑖. And multiplying the last terms gives us negative 14𝑖 squared. Repeating this with the denominator gives us 81 minus 63𝑖 plus 63𝑖 minus 49𝑖 squared. The middle terms cancel to give a zero.
We also recall that 𝑖 squared is equal to negative one, so we can substitute this into the last term on the numerator and the denominator. Negative 14 multiplied by negative one is equal to 14, and adding this to nine gives us 23. Negative seven 𝑖 plus 18𝑖 is equal to 11𝑖. The numerator becomes 23 plus 11𝑖. On the denominator, we have 81 minus 49 multiplied by negative one. This simplifies to 130. As we are asked to give our answer in the form 𝑥 plus 𝑦𝑖, we need to separate the real and imaginary parts. This gives us 23 over 130 plus 11 over 130𝑖. 𝑥 is equal to 23 over 130, and 𝑦 is equal to 11 over 130.
