 # Complex Numbers

### Complex Numbers

Just as we need negative integers to represent what positive integers can’t, we need imaginary numbers to represent what the real numbers can’t.

Complex numbers represent a number system that combines both real and imaginary numbers. While it may be tempting to ask what is the practical use of complex numbers, their importance is undeniable especially in the field of signal processing.

The usage of complex numbers is an essential part in designing “the poles and zeroes” or simply put, the stability of a system.

“The complex power” of an electrical signal is also an important design consideration in power lines and generators.

We frequently encountered imaginary numbers before, take for example, the solution of this quadratic equation:

$$\\[5pt] \hspace{2em} {x}^{2} \ + \ 1 \ = \ 0$$
$$\\[5pt] \hspace{2em} {x}^{2} \hspace{2.4em} = \ -1$$
$$\\[5pt] \hspace{2em} x \hspace{2.8em} = \ \pm \sqrt{-1}$$

$${\small \sqrt{-1} \ }$$ is an example of an imaginary number. Whenever you met an even n-th root of a negative number, you will need to know the complex number theory to simplify or solve it further.

We will start with the two basic forms to represent complex numbers, the cartesian form and the polar form.

Complex numbers in cartesian form can be shown as below:

$$\hspace{2em} {\large z \ = \ a \ + \ b \textrm{i} }$$

$$\\[5pt] {\small a \ = \ \textrm{Re(} z \textrm{)} }$$
: the real part of a complex number z

$$\\[5pt] {\small b \ = \ \textrm{Im(} z \textrm{)} }$$
: the imaginary part of a complex number z

$${\small \textrm{i} \ = \ \sqrt{-1} }$$

Argand diagram is usually used to show complex number graphically. The x-axis represents the “real part” and the y-axis represents the “imaginary part”.

The Argand diagram representation of the complex number z above can be seen as follows: The cartesian form is especially handy in dealing with addition and subtraction of complex numbers.

The result of addition and subtraction of complex numbers can be found by adding or subtracting each of the real parts and each of the imaginary parts separately.

Example:

$$\hspace{2em} {z}_{1} \ = \ a \ + \ b \textrm{i}$$
$$\hspace{2em} {z}_{2} \ = \ c \ + \ d \textrm{i}$$

$$\hspace{2em} {z}_{1} \ + \ {z}_{2} \ = \ (a + c) \ + \ (b + d) \textrm{i}$$
$$\hspace{2em} {z}_{1} \ – \ {z}_{2} \ = \ (a \ – \ c) \ + \ (b \ – \ d) \textrm{i}$$

The second form of complex numbers is the polar form or the modulus-argument form.

$$\\[7pt] \hspace{2em} {\large z \ = \ r \cos \theta \ + \ \textrm{i} \ r \sin \theta }$$
$$\hspace{3.1em} {\large = \ r \ (\cos \theta \ + \ \textrm{i} \ \sin \theta) }$$

$$\\[5pt] {\small r \ = \ |z| }$$
: the modulus of a complex number z

$$\\[5pt] {\small \theta \ = \ \textrm{arg(} z \textrm{)} }$$
: the principal argument of a complex number z, $$\enspace {\small -\pi \lt \theta \le \pi }$$

The modulus $${\small r }$$ and argument $${\small \theta}$$ can be used to show the sets of points or regions of complex numbers in Argand diagram.

The Argand diagram of a complex number z in its polar form can be seen below: Alternatively, the polar form can also be expressed in euler notation and phase-angle notation.

In euler notation,

$$\\[10pt] \hspace{2.6em} {\large {e}^{\textrm{i} \theta} \ = \ \cos \theta \ + \ \textrm{i} \ \sin \theta }$$
Since, $$\\[7pt] \hspace{1em} {\large z \ = \ r \ (\cos \theta \ + \ \textrm{i} \ \sin \theta) }$$
then $$\hspace{1.5em} {\large z \ = \ r \ {e}^{\textrm{i} \theta} }$$

In phase-angle notation,

$$\\[10pt] \hspace{2.6em} {\large \angle \theta \ = \ {e}^{\textrm{i} \theta} }$$
Since, $$\\[7pt] \hspace{1em} {\large z \ = \ r \ {e}^{\textrm{i} \theta} }$$
then $$\hspace{1.5em} {\large z \ = \ r \ \angle \theta }$$

The euler and phase-angle notation are merely a simpler way of writing the polar form.

The polar form, especially the euler notation and phase-angle notation are useful in multiplication and division of complex numbers.

The result of multiplication of complex numbers can be found by multiplying the moduli and adding the arguments.

The result of division of complex numbers can be found by dividing the moduli and subtracting the arguments.

Example:

$$\hspace{2em} {z}_{1} \ = \ {r}_{1} \ {e}^{\textrm{i} {\theta}_{1}}$$
$$\hspace{2em} {z}_{2} \ = \ {r}_{2} \ {e}^{\textrm{i} {\theta}_{2}}$$

$$\\[10pt] \hspace{2em} {z}_{1} \times {z}_{2} \ = \ ({r}_{1} \times {r}_{2}) \ {e}^{\textrm{i} ({\theta}_{1} \ + \ {\theta}_{2})}$$
$$\hspace{2em} {\large \frac{{z}_{1}}{{z}_{2}}} \hspace{1.9em} = \ {\large \frac{{r}_{1}}{{r}_{2}}} \ {e}^{\textrm{i} ({\theta}_{1} \ – \ {\theta}_{2})}$$

By comparing the complex numbers in both cartesian & polar form, they can be converted from one form to another.

To find the cartesian form from the polar form, $$\enspace z \ = \ a \ + \ b \textrm{i}$$

$$\\[5pt] \enspace a \ = \ r \ \cos \theta$$
$$\enspace b \ = \ r \ \sin \theta$$

To find the polar form from the cartesian form, $$\enspace z \ = \ r \ (\cos \theta \ + \ \textrm{i} \sin \theta )$$

$$\\[7pt] \enspace r \ = \ \sqrt{{a}^{2} \ + \ {b}^{2}}$$
$$\enspace \theta \ = \ {\tan}^{-1} \Big({\large \frac{b}{a} } \Big)$$

The complex conjugate of a complex number has the same real part and opposite sign for its imaginary part. That is, to say, if $$z \ = \ a \ + \ b \textrm{i} \$$ then the complex conjugate is $${z}^{*} \ = \ a \ – \ b \textrm{i}$$.

One such example of usage is when dealing with polynomial equations with real coefficients (non-complex number coefficients).

If a polynomial equation has a complex number z as one of its roots, then the complex conjugate $${\small {z}^{*} }$$ is also a root of the polynomial equation.

To find the n-th power of a complex number, De Moivre’s Theorem can be used. It can be shown as follows:

If $$\enspace z \ = \ r \ {e}^{\textrm{i} \theta}$$

then $$\enspace {z}^{n} \ = \ {r}^{n} \ {e}^{ \textrm{i} (n\theta) }$$

I have put together some of the questions I received in the comment section below. You can try these questions also to further your understanding on this topic.

To check your answer, you can look through the solutions that I have posted either in Youtube videos or Instagram posts.

You can subscribe, like or follow my youtube channel and IG account. I will keep updating my IG daily post, preferably.

Furthermore, you can find some examples and more practices below! =).

Try some of the examples below and if you need any help, just look at the solution I have written. Cheers ! =) .
$$\\[1pt]$$

EXAMPLE:

$${\small 1.\enspace}$$ 9709/03/SP/17 – Specimen Paper 2017 Pure Maths 3 No 9
$$\\[1pt]$$
The complex number $${\small \ 3 \ − \ \textrm{i} \ }$$ is denoted by u. Its complex conjugate is denoted by u*.
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{i}).\hspace{0.7em}}$$ On an Argand diagram with origin O, show the points A, B and C representing the complex numbers u, u* and u*u respectively. What type of quadrilateral is OABC?
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{ii}).\hspace{0.7em}}$$ Showing your working and without using a calculator, express $${\small \ {\large\frac{u}{u*}} }$$ in the form of x + iy, where x and y are real.
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{iii}).\hspace{0.5em}}$$ By considering the argument of $${\small \ {\large\frac{u}{u*}} }$$, prove that
$${\small\hspace{3em} \ {\tan}^{-1} \big(\frac{3}{4}\big) \ = \ 2 {\tan}^{-1} \big(\frac{1}{3}\big) }$$

$$\\[1pt]$$
$${\small 2.\enspace}$$ 9709/03/SP/20 – Specimen Paper 2020 Pure Maths 3 No 6
$$\\[1pt]$$
The complex number $${\small \ 1 \ + \ 3\textrm{i} \ }$$ and $${\small \ 4 \ + \ 2\textrm{i} \ }$$ are denoted by u and v respectively.
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{a}).\hspace{0.8em}}$$ Find $${\small \ {\large \frac{u}{v}} \ }$$ in the form of x + iy, where x and y are real.
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{b}).\hspace{0.8em}}$$ State the argument of $${\small \ \large { \frac{u}{v} }}$$.
$$\\[1pt]$$
In an Argand diagram, with origin O, the points A, B and C represent the complex numbers u, v and u – v respectively.
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{c}).\hspace{0.8em}}$$ State fully the geometrical relationship between OC and BA.
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{d}).\hspace{0.8em}}$$ Show that angle $${\small AOB = \frac{1}{4} \pi \small }$$ radians.

$$\\[1pt]$$
$${\small 3.\enspace}$$ The complex number w is given by $${\small w = {\large -\frac {1}{2}} \ + \ {\large \textrm{i}\frac {\sqrt{3}}{2}} }$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(i\right).\hspace{0.8em}}$$ Find the modulus and argument of w.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(ii\right).\hspace{0.8em}}$$ The complex number z has a modulus R and argument $${\small \theta}$$, where $$\ {\small – {\large \frac{1}{3}}\pi \lt \theta \lt {\large \frac{1}{3}}\pi }$$. State the modulus and argument of wz and the modulus and argument of $${\small {\large \frac{z}{w}} }$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(iii\right).\hspace{0.8em}}$$ Hence explain why, in Argand diagram, the points representing z, wz and $${\small {\large \frac{z}{w}} }$$ are the vertices of an equilateral triangle.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(iv\right).\hspace{0.8em}}$$ In an Argand diagram, the vertices of an equilateral triangle lie on a circle with centre at the origin. One of the vertices represents the complex number 4 + 2i. Find the complex numbers represented by the other two vertices. Give your answers in the form x + iy, where x and y are real and exact.

$$\\[1pt]$$
$${\small 4.\enspace}$$ The complex number -2 + i is denoted by u.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(i\right).\hspace{0.8em}}$$ Given that u is a root of the equation $${\small {x}^{3} \ – \ 11x \ – \ k \ = \ 0}$$, where k is real, find the value of k.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(ii\right).\hspace{0.8em}}$$ Write down the other complex root of this equation.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(iii\right).\hspace{0.8em}}$$ Find the modulus and argument of u.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(iv\right).\hspace{0.8em}}$$ Sketch an Argand diagram showing the point representing u. Shade the region whose points represent the complex numbers z satisfying both the inequalities
$${\small \quad |z| \ \lt \ |z \ – \ 2| \ }$$ and $$\ {\small 0 \ \lt \ \textrm{arg}(z \ – \ u) \ \lt \ {\large \frac{1}{4} }\pi }$$.

$$\\[1pt]$$
$${\small 5.\enspace}$$ The complex number w is defined by $${\small w = -1 + \textrm{i} }$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(i\right).\hspace{0.8em}}$$ Find the modulus and argument of $${\small {w}^{2} }$$ and $${\small {w}^{3}}$$, showing your working.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(ii\right).\hspace{0.8em}}$$ The points in an Argand diagram representing $${\small w }$$ and $${\small {w}^{2}}$$ are the ends of a diameter of a circle. Find the equation of the circle, giving your answer in the form $${\small |z \ – \ (a \ + \ b\textrm{i})| \ = \ k}$$.

$$\\[1pt]$$
$${\small 6.\enspace (a).\hspace{0.8em} }$$ Without using a calculator, solve the equation
$$\\[1pt]$$
$${\small \quad 3w \ + \ 2\textrm{i}{w}^{*} \ = \ 17 \ + \ 8\textrm{i}, }$$
$$\\[1pt]$$
where $${\small {w}^{*} }$$ denotes the complex conjugate of $${\small w }$$. Give your answer in the form $${\small (a \ + \ b\textrm{i}). }$$
$$\\[1pt]$$
$${\small \hspace{1.3em} (b).\hspace{0.8em} }$$ In an Argand diagram, the loci
$$\\[1pt]$$
$${\small \quad \textrm{arg}(z \ – \ 2\textrm{i}) \ = \ {\large \frac{1}{6} }\pi \ }$$ and $$\ {\small |z \ – \ 3| \ = \ |z \ – \ 3\textrm{i}| }$$
$$\\[1pt]$$
intersect at the point P. Express the complex number represented by P in the form of $${\small r {e}^{\textrm{i}\theta}}$$, giving the exact value of $${\small \theta }$$ and the value of r correct to 3 significant figures.

$$\\[1pt]$$
$${\small 7.\enspace}$$ The complex number z is defined by $${\small \ z \ = \ {\large \frac{9\sqrt{3} \ + \ 9\textrm{i}}{\sqrt{3} \ – \ \textrm{i}} } }$$. Find, showing all your working,
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(i\right).\hspace{0.8em}}$$ an expression for z in the form $${\small r {e}^{\textrm{i}\theta}}$$, where $${\small r \ \gt \ 0 \ }$$ and $$\ {\small -\pi \lt \theta \le \pi }$$,
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(ii\right).\hspace{0.8em}}$$ the two square roots of z, giving your answers in the form $${\small r {e}^{\textrm{i}\theta}}$$, where $${\small r \ \gt \ 0 \ }$$ and $$\ {\small -\pi \lt \theta \le \pi }$$.

$$\\[1pt]$$
$${\small 8.\enspace (a).\hspace{0.8em} }$$ Showing all working and without using a calculator, solve the equation
$$\\[1pt]$$
$$\quad {\small (1 \ + \ \textrm{i}){z}^{2} \ – \ (4 \ + \ 3\textrm{i})z \ + \ 5 \ + \ \textrm{i} \ = \ 0}$$.
$$\\[1pt]$$
Give your answers in the form x + iy, where x and y are real.
$$\\[1pt]$$
$${\small \hspace{1.3em} (b).\hspace{0.8em} }$$ The complex number u is given by
$$\quad {\small u \ = \ -1 \ – \ \textrm{i} }$$.
$$\\[1pt]$$
On a sketch of an Argand diagram show the point representing u. Shade the region whose points represent complex numbers satisfying the inequalities
$${\small \ |z| \ \lt \ |z \ – \ 2\textrm{i}| }$$ and $${\small {\large \frac{1}{4} }\pi \ \lt \ \textrm{arg}(z \ – \ u) \ \lt \ {\large \frac{1}{2} }\pi }$$.

$$\\[1pt]$$

PRACTICE MAKES PERFECT!

$${\small 1.\enspace(a).\hspace{0.8em}}$$ Showing all necessary working, express the complex number $${\small {\large \frac{2 \ + \ 3\textrm{i}}{1 \ – \ 2\textrm{i}} } }$$ in the form $${\small r {e}^{\textrm{i}\theta}}$$, where $${\small r \ \gt \ 0 \ }$$ and $$\ {\small -\pi \lt \theta \le \pi }$$. Give the values of r and $${\small \theta}$$ correct to 3 significant figures.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(b\right).\hspace{0.8em}}$$ On an Argand diagram sketch the locus of points representing complex numbers z satisfying the equation $${\small |z \ – \ 3 \ + \ 2\textrm{i}| \ = \ 1 }$$. Find the least value of |z| for points on this locus, giving your answer in an exact form.
$$\\[1pt]$$

$${\small 2.\enspace(a) \ (i).\hspace{0.5em}}$$ Without using a calculator, express the complex number $${\small {\large \frac{2 \ + \ 6\textrm{i}}{1 \ – \ 2\textrm{i}} } }$$ in the form x + iy, where x and y are real.
$$\\[1pt]$$
$${\small\hspace{2.4em}\left(ii\right).\hspace{0.5em}}$$ Hence, without using a calculator, express $${\small {\large \frac{2 \ + \ 6\textrm{i}}{1 \ – \ 2\textrm{i}} } }$$ in the form $${\small r(\cos \theta \ + \ \textrm{i} \sin \theta)}$$, where $${\small r \ \gt \ 0 \ }$$ and $$\ {\small -\pi \lt \theta \le \pi }$$, giving the exact values of r and $${\small \theta}$$.
$$\\[1pt]$$
$${\small\hspace{1.5em}\left(b\right).\hspace{0.8em}}$$ On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying both the inequalities $${\small |z \ – \ 3 \textrm{i}| \ \le \ 1 }$$ and Re $${\small z \ \le \ 0}$$, where Re z denotes the real part of z. Find the greatest value of arg z for points in this region, giving your answer in radians correct to 2 decimal places.

$${\small 3. \enspace}$$ The complex number $${\small 1 \ − \ (\sqrt{3})\textrm{i} }$$ is denoted by u.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(i\right).\hspace{0.8em}}$$ Find the modulus and argument of u.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(ii\right).\hspace{0.8em}}$$ Show that $${\small \ {u}^{3} \ + \ 8 \ = \ 0}$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(iii\right).\hspace{0.8em}}$$ On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying both the inequalities $${\small |z \ – \ u| \ \le \ 2 }$$ and Re $${\small z \ \ge \ 2}$$, where Re z denotes the real part of z.

$${\small 4. \enspace}$$ The polynomial $${\small {z}^{4} \ + \ 3{z}^{2} \ + \ 6z \ + \ 10 }$$ is denoted by p(z). The complex number $${\small -1 \ + \ \textrm{i} }$$ is denoted by u.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(i\right).\hspace{0.8em}}$$ Showing all your working, verify that u is a root of the equation p(z) $${\small = \ 0}$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(ii\right).\hspace{0.8em}}$$ Find the other three roots of the equation p(z) $${\small = \ 0}$$.

$${\small 5.\enspace(a).\hspace{0.8em}}$$ Showing all necessary working, solve the equation $${\small \textrm{i}{z}^{2} \ + \ 2z \ − \ 3\textrm{i} \ = \ 0 }$$, giving your answers in the form x + iy, where x and y are real and exact.
$$\\[1pt]$$
$${\small \quad(b) \ (i).\hspace{0.8em}}$$ On a sketch of an Argand diagram, show the locus representing complex numbers satisfying the equation $${\small |z| \ = \ |z \ − \ 4 \ − \ 3\textrm{i}| }$$.
$$\\[1pt]$$
$${\small \hspace{2em} (ii).\hspace{0.7em}}$$ Find the complex number represented by the point on the locus where |z| is least. Find the modulus and argument of this complex number, giving the argument correct to 2 decimal places.

$${\small 6. \enspace}$$ Sketch, on an Argand diagram, the locus of the points representing the complex number z such that $${\small |\textrm{i}z \ – \ 2| \ = \ |2 \ – \ z|}$$. Hence find the least value of $${\small |z \ + \ 2|}$$.

$${\small 7. \enspace}$$ Solve the equation $${\small \textrm{i}{z}^{4} \ = \ -81 }$$, expressing the roots in the form $${\small r {e}^{\textrm{i}\theta}}$$, where $${\small r \ \gt \ 0 \ }$$ and $$\ {\small -\pi \lt \theta \le \pi }$$.

$${\small 8. \enspace}$$ In an Argand diagram, the point P represents the complex number z such that:
$$\\[1pt]$$
$${\small |z \ – \ 2 \ + \ 4\textrm{i}| \le 4 \ }$$ and $${\small \ {\large \frac{\pi}{4} } \le \textrm{arg}(z \ + \ 2\textrm{i}) \lt 0 }$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(i\right).\hspace{0.8em}}$$ Sketch the locus of P.
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(ii\right).\hspace{0.8em}}$$ Hence, find the exact range of values of $${\small |z \ + \ 2|}$$.

$${\small 9. \enspace}$$ In an Argand diagram, the point P represents the complex number z. Draw on a single clearly labelled diagram to show the locus of P when z satisfies
$$\\[1pt]$$
$${\small\hspace{1.2em}\left(i\right).\hspace{0.8em}}$$ $${\small |z \ – \ 3 \ – \ 4\textrm{i}| = 5 \ }$$
$${\small\hspace{1.2em}\left(ii\right).\hspace{0.8em}}$$ $${\small |z \ – \ 2 \ + \ 3\textrm{i}| = |z \ – \ 10 \ – \ 3\textrm{i}| \ }$$
$$\\[1pt]$$
Hence find the greatest and least possible values of |z| when z satisfies both
$$\\[1pt]$$
$${\small |z – 3 – 4\textrm{i}| \le 5 }$$ and $${\small |z – 2 + 3\textrm{i}| \ge |z – 10 – 3\textrm{i}| }$$

$${\small 10.\enspace}$$ Given that $${\small {z}^{*} \ = \ \frac{{(2 \ – \ 2\textrm{i})}^{3}}{{(-1 \ + \ \sqrt{3}\textrm{i})}^{4}}}$$, find the exact value of |z| and arg(z). Hence, state the smallest positive integer n such that $${\small {z}^{n}}$$ is a purely real number.

As always, if you have any particular questions to discuss, leave it in the comment section below. Cheers =) .

1. Hello I’d like to ask a question:
(a)On a sketch of an Argand diagram, shade the region whose points represent complex
numbers z satisfying the inequalities 􏰈z − 2 − 2i􏰈 ≤ 1 and arg􏰆z − 4i􏰇 ≥ −1π.
and
(b)Find the least value of Im z for points in this region, giving your answer in an exact form.

I know how to do (a) but may I know how do i solve (b)? Thank you in advance

2. Hello, I have a problem I’m trying to solve:

Find the locus of points in the Argand diagram for which the imaginary part of z + 1/z is zero.

so far I have got x(x^2 + y^2) + x = 0

therefore I could get x^2 + y^2 = -1, but I dont think this is right?

1. Hi Nick,
Maybe there is some slight error in your calculation.

Here’s how I do this question:
Let $$\hspace{1em} z \ = \ x \ + \ y \textrm{i}$$

$$\\[5pt]$$ Then,
$$\hspace{2em} {\large \frac{1}{z} \ = \frac{1}{\ x \ + \ y \textrm{i}} }$$

Rationalize the denominator by multiplying it with the complex conjugate,
$$\\[15pt] {\large \frac{1}{z} \ = \ \frac{1}{ x \ + \ y \textrm{i}} \times \frac{ x \ – \ y \textrm{i}}{ x \ – \ y \textrm{i}}}$$
$$\hspace{1.5em} {\large = \ \frac{ x \ – \ y \textrm{i}}{ x^2 \ + \ y^2} }$$

Therefore,
$$\hspace{1em} {\large z \ + \frac{1}{z} \ = \ x + iy + \frac{ x \ – \ y \textrm{i}}{ x^2 \ + \ y^2} }$$

Make the denominator the same,

$$\\[15pt] {\large z \ + \frac{1}{z} \ = \ \frac{ (x^2 \ + \ y^2)(x \ + \ iy) \ + \ x \ – \ y \textrm{i}}{ x^2 \ + \ y^2} }$$
$$\hspace{4em} {\large = \ \frac{ x^3 \ + \ xy^2 \ + \ i(x^2y \ + \ y^3 \ – \ y)}{ x^2 + y^2} }$$

Since the imaginary part of the above expression is zero, then

$$\\[15pt] \hspace{1em} {\large \frac{ x^2y \ + \ y^3 \ – \ y}{ x^2 \ + \ y^2} \ = \ 0 }$$
$$\\[15pt]\hspace{1em} x^2y \ + \ y^3 \ – \ y \ = \ 0$$
$$\\[15pt] \hspace{1em} y(x^2 \ + \ y^2 \ – \ 1) \ = \ 0$$
$$\hspace{1em} y \ = \ 0 \$$ or $$\ x^2 \ + \ y^2 \ = \ 1 \$$

The locus of points in the Argand diagram are the x-axis and a unit circle at the origin.

Cheers,
Mr Will