### Video Transcript

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

To answer this question, we
absolutely could go ahead and plot the complex number π on the Argand diagram
and then read the information from there. But thatβs 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 π¦π can be represented by a point whose Cartesian
coordinates are π₯, π¦. The real part is the
π₯-coordinate. And the imaginary part is the
π¦-coordinate.

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

Letβs have a look at the point
that represents the complex number eight plus π on the Argand diagram. We saw that itβs represented by
a point whose Cartesian coordinates are eight, one. We can find the complex
conjugate of π by changing the sign of the imaginary part. And so the conjugate of eight
plus π is eight minus π. We therefore represent the
conjugate of π 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.