Find the value of 𝑧 bar given 𝑧 on the Argand diagram below.
We begin by recalling that an Argand diagram is a way of representing a complex number of the form 𝑥 plus 𝑦𝑖, where 𝑥 and 𝑦 are real numbers. We can see from the coordinate axes that 𝑧 has Cartesian coordinates negative three, five. This means that the complex number 𝑧 is equal to negative three plus five 𝑖. The 𝑥-coordinate is the real part and the 𝑦-coordinate the complex part.
We are asked to find the value of 𝑧 bar, which is also sometimes written as 𝑧 star. This is known as the complex conjugate. If 𝑧 is equal to 𝑥 plus 𝑦𝑖, then 𝑧 bar, the complex conjugate, is equal to 𝑥 minus 𝑦𝑖. To find the complex conjugate of any complex number, we simply change the sign of the imaginary part. In this question, as 𝑧 is equal to negative three plus five 𝑖, then 𝑧 bar, the complex conjugate, is equal to negative three minus five 𝑖.
It is worth noting that this would have Cartesian coordinates negative three, negative five as shown on the axes. This leads us to a general rule that we can use to find the complex conjugate on an Argand diagram. The complex conjugate can be found by reflecting the point 𝑧 in the 𝑥-axis or the line 𝑦 is equal to zero. This is because the 𝑥-axis is the real axis and the real part of our complex number is not changing.