Video: Representing Complex Numbers on an Argand Diagram

If the number 𝑍 = 8 + 𝑖 is represented on an Argand diagram by the point 𝐴, determine the Cartesian coordinates of that point.

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Video Transcript

If the number 𝑍 equals eight plus 𝑖 is represented on an Argand diagram by the point 𝐴, determine the Cartesian coordinates of that point.

To answer this question, we absolutely could go ahead and plot the complex number 𝑍 on the Argand diagram and then read the information from there. But that’s quite a long-winded way to go about answering this question. Instead, we remind ourselves of the definition of the Argand diagram. We know that a complex number of the form π‘₯ plus 𝑦𝑖 can be represented by a point whose Cartesian coordinates are π‘₯, 𝑦. The real part is the π‘₯-coordinate. And the imaginary part is the 𝑦-coordinate.

The real part of our complex number is eight. And we can figure the imaginary part as the coefficient of 𝑖. So in this case, the imaginary part of 𝑍 is one. This means the Cartesian coordinates of the point that represents the complex number 𝑍 on the Argand plane are eight, one. And what about complex conjugate pairs? How might they appear on the Argand diagram?

Let’s have a look at the point that represents the complex number eight plus 𝑖 on the Argand diagram. We saw that it’s represented by a point whose Cartesian coordinates are eight, one. We can find the complex conjugate of 𝑍 by changing the sign of the imaginary part. And so the conjugate of eight plus 𝑖 is eight minus 𝑖. We therefore represent the conjugate of 𝑍 on our Argand diagram by the point whose Cartesian coordinates are eight, negative one. We can see that the point is a reflection in the real axis. And in fact, this is true for all complex numbers and their complex conjugate.

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