Find the complex number that lies at the midpoint between 𝑧 and 𝑖𝑧 on the given complex plane.
We recall that any complex number 𝑧 can be written in the form 𝑎 plus 𝑏𝑖, where 𝑎 is the real part or component and 𝑏 is the imaginary part. On the Argand diagram shown, we see that the horizontal axis corresponds to the real component and the vertical axis the imaginary component. The point 𝑧 has coordinates six, two. This means it corresponds to the complex number six plus two 𝑖.
To calculate 𝑖𝑧, we need to multiply six plus two 𝑖 by 𝑖. Distributing our parentheses gives us six 𝑖 plus two 𝑖 squared. From our knowledge of complex numbers, we know that 𝑖 squared is equal to negative one. This means that 𝑖𝑧 is equal to six 𝑖 plus two multiplied by negative one. This in turn simplifies to negative two plus six 𝑖.
We now have two complex numbers, and we need to find their midpoint. We could plot 𝑖𝑧 on the Argand diagram. It would have the coordinates negative two, six, as the real part is negative two and the imaginary part is six. The midpoint of two coordinates is half the sum of their corresponding components. Clearing some space, we need to add 𝑧 and 𝑖𝑧 and then divide our answer by two. This is equal to six plus two 𝑖 plus negative two plus six 𝑖 all divided by two.
Grouping the real terms on the numerator gives us four, as six plus negative two is equal to four. Two 𝑖 plus six 𝑖 is equal to eight 𝑖. This gives us four plus eight 𝑖 divided by two. As a half of four is two and a half of eight 𝑖 is four 𝑖, this simplifies to two plus four 𝑖. The midpoint of 𝑧, six plus two 𝑖, and 𝑖𝑧, negative two plus six 𝑖, is two plus four 𝑖. This complex number corresponds to the coordinate two, four on the Argand diagram.