Using the Argand diagram shown, find the value of 𝑧 sub one plus 𝑧 sub two.
We begin by recalling that any complex number 𝑧 can be written in the form 𝑥 plus 𝑦𝑖, where 𝑥 is the real part of our complex number and 𝑦 is the imaginary part. On an Argand diagram, the 𝑥-coordinate is the real part and the 𝑦-coordinate is the imaginary part. The point 𝑧 sub one has coordinates two, three. This means that the complex number 𝑧 sub one is equal to two plus three 𝑖. The point 𝑧 sub two has coordinates negative four, negative three. This means that 𝑧 sub two is equal to the complex number negative four minus three 𝑖.
We are asked in this question to find the value of the complex number 𝑧 sub one plus 𝑧 sub two. This means that we need to add two plus three 𝑖 and negative four minus three 𝑖. Adding the real parts two and negative four gives us negative two. Adding the imaginary parts three 𝑖 and negative three 𝑖 gives us zero 𝑖 or just zero. 𝑧 sub one plus 𝑧 sub two is therefore equal to negative two. This corresponds to the point on the Argand diagram with coordinates negative two, zero. Even though 𝑧 sub one and 𝑧 sub two were complex numbers with both real and imaginary parts, 𝑧 sub one plus 𝑧 sub two is a real number.