### Video Transcript

Given that π§ equals π root two
over one minus π, write π§ in exponential form.

To answer this question, we have
two options. We could divide these two complex
numbers in algebraic form. And to do this, we need to multiply
both the numerator and denominator by the conjugate of the denominator then
distribute and simplify as far as possible. Iβm sure youβll agree thatβs rather
a lengthy process. Instead, weβll choose to write
these complex numbers in exponential form. So weβll need to calculate their
moduli and arguments.

π root two is a purely imaginary
number. On an Argand diagram, itβs
represented by the point whose Cartesian coordinates are zero, root two. Its modulus is the length of the
line segment that joins this point to the origin. So itβs root two. And since the argument is measured
in the counterclockwise direction from the positive real axis, we can see that the
argument of this complex number is equivalent to 90 degrees. Thatβs π by two radians. And in exponential form, we can say
that this is the same as root two π to the π by two π.

The complex number one minus π is
a little more tricky. Its real part is positive and its
imaginary part is negative. So it lies in the fourth
quadrant. Now, its modulus is independent of
this fact. We simply use the formula the
square root of the sum of the square of the real and imaginary parts. So thatβs the square root of one
squared plus negative one squared which once again is the square root of two.

We do need to be a little bit more
careful with the argument. Since itβs in the fourth quadrant,
we can use the formula thatβs unique to complex numbers that are plotted in the
first and fourth quadrants. Thatβs arctan of π over π, arctan
of the imaginary part divided by the real part. So in this case, thatβs the arctan
of negative one over one which is negative π by four. We expected a negative value for
the argument as this time weβre measuring in a clockwise direction. And so, we can rewrite our fraction
as root two π to the π by two π over root two π to the negative π by four
π. And now, we can easily divide.

To divide complex numbers in
exponential form, we divide their moduli and subtract their arguments. Root two divided by root two is one
and π by two minus negative π by four is three π by four. In exponential form then, π§ is
equal to π to the three π by four π.