Argand Diagram, Modulus, Argument, and Loci

Overview

This branch develops the geometric side of complex numbers: Argand diagrams, modulus, argument, distance interpretation, and loci. It is the main reference when a complex-number question is really asking about points, directions, distances, regions, or geometric constraints.

The Argand Diagram

A complex number can be represented geometrically in a plane called the Argand diagram.

If

$$z=x+iy,$$

then $z$ is represented by the point

$$(x,y).$$

The horizontal axis is the real axis, and the vertical axis is the imaginary axis.

So:

• the real part $\mathrm{Re}(z)=x$ gives the horizontal coordinate;
• the imaginary part $\mathrm{Im}(z)=y$ gives the vertical coordinate.

Equivalently, $z=x+iy$ can be represented by the position vector from the origin to the point $(x,y)$.

Caption: In the Argand diagram, $z=x+iy$ is represented by the point $(x,y)$ and by the position vector from the origin.

This representation gives complex numbers a geometric meaning.

For example, if

$$z=3+2i,$$

then $z$ is represented by the point

$$(3,2).$$

If

$$w=-1+4i,$$

then $w$ is represented by the point

$$(-1,4).$$

The Argand diagram also makes several important operations easier to understand.

Conjugation as reflection

If

$$z=x+iy,$$

then

$$z^{\ast }=x-iy.$$

In the Argand diagram, $z$ corresponds to the point $(x,y)$, while $z^{\ast }$ corresponds to the point $(x,-y)$.

So conjugation reflects a point in the real axis.

Modulus as distance

The modulus $|z|$ measures the distance from the origin to the point representing $z$.

If

$$z=x+iy,$$

then

$$|z|=\sqrt{x^2+y^2}.$$

So the modulus is not just an algebraic expression. It is a geometric distance.

Argument as direction

The argument $\arg z$ measures the direction of $z$ from the positive real axis.

It is the angle made by the position vector of $z$ with the positive real axis.

So the argument is a geometric angle, not a length. It can be viewed as the general angle of the position vector representing $z$.

Difference as distance between two points

One of the most important geometric interpretations is this:

$$|z_1-z_2|$$

is the distance between the points representing $z_1$ and $z_2$ in the Argand diagram.

This idea is essential for loci. For example, a condition such as

$$|z-a|=r$$

means that the point representing $z$ is always a fixed distance $r$ from the fixed point representing $a$.

Therefore, the locus is a circle with centre at $a$ and radius $r$.

Why the Argand diagram matters

The Argand diagram is not just a way to draw complex numbers. It is the bridge between algebra and geometry.

It allows us to reinterpret:

• real part as horizontal position;
• imaginary part as vertical position;
• conjugation as reflection;
• modulus as distance;
• argument as direction;
• equations involving $z$ as geometric loci.

The main skill is to translate between the two languages:

$$\text{complex algebra}⟷\text{plane geometry}.$$

This translation becomes especially important when studying loci in the complex plane.

Modulus and Argument

The Argand diagram allows us to describe a complex number by two geometric quantities:

• its modulus, which tells us how far it is from the origin;
• its argument, which tells us its direction from the origin.

Together, modulus and argument give a geometric way to understand the position of a complex number.

Modulus: distance from the origin

Let

$$z=x+iy,$$

where $x,y\in \mathbb{R}$.

The modulus of $z$ is defined by

$$|z|=\sqrt{x^2+y^2}.$$

Geometrically, $z=x+iy$ is represented by the point

$$(x,y)$$

in the Argand diagram. Therefore, $|z|$ is the distance from the origin to the point $(x,y)$.

This formula comes directly from Pythagoras:

$$|z|=\sqrt{x^2+y^2}.$$

For example, if

$$z=3-4i,$$

then

$$|z|=\sqrt{3^2+(-4{)}^2}.$$

So

$$|z|=\sqrt{9+16}=5.$$

The modulus is always a non-negative real number:

$$|z|\ge 0.$$

Also,

$$|z|=0$$

only when

$$z=0.$$

Argument: direction from the origin

The argument of a non-zero complex number $z$ is the angle made by the position vector of $z$ with the positive real axis. It is the general angle of the point representing $z$ in the Argand diagram.

It is denoted by

$$\arg z.$$

The angle is measured from the positive real axis:

• anticlockwise angles are positive;
• clockwise angles are negative.

For example, if a point lies in the first quadrant, its argument is positive and between

$$0$$

and

$$\frac{\pi }{2}.$$

If a point lies in the fourth quadrant, its principal argument is negative and between

$$-\frac{\pi }{2}$$

and

$$0.$$

The argument of $0$ is undefined, because the origin has no direction from itself. So

$$\arg 0$$

is undefined.

Principal argument

The argument of a complex number is not unique.

For example, the angles

$$\frac{\pi }{4}$$

and

$$\frac{\pi }{4}+2\pi$$

point in the same direction.

More generally, if $\theta$ is an argument of $z$, then

$$\theta +2k\pi$$

is also an argument of $z$ for any integer $k$.

To avoid ambiguity, we usually choose the principal argument, which lies in the range

$$-\pi <\theta \le \pi .$$

So when a question asks for

$$\arg z,$$

it usually expects the principal argument unless otherwise stated.

Finding the argument from Cartesian form

Suppose

$$z=x+iy.$$

To find $\arg z$, do not blindly write

$${\tan }^{-1}\left(\frac{y}{x}\right).$$

This can give the wrong quadrant.

A safer method is:

1. locate the point $(x,y)$ in the Argand diagram;
2. find the basic angle

$$\alpha ={\tan }^{-1}\left|\frac{y}{x}\right|;$$

3. adjust the answer according to the quadrant.

For example, if

$$z=1+i,$$

then the point $(1,1)$ lies in the first quadrant. The basic angle is

$$\alpha ={\tan }^{-1}(1)=\frac{\pi }{4}.$$

Therefore,

$$\arg (1+i)=\frac{\pi }{4}.$$

If

$$z=-1+i,$$

then the point $(-1,1)$ lies in the second quadrant. The basic angle is still

$$\alpha =\frac{\pi }{4},$$

but the actual argument is

$$\pi -\frac{\pi }{4}=\frac{3\pi }{4}.$$

So

$$\arg (-1+i)=\frac{3\pi }{4}.$$

If

$$z=1-i,$$

then the point $(1,-1)$ lies in the fourth quadrant. The principal argument is

$$-\frac{\pi }{4}.$$

So

$$\arg (1-i)=-\frac{\pi }{4}.$$

The quadrant is therefore essential when finding the argument.

Modulus versus argument

The modulus and argument answer different geometric questions.

The modulus answers:

$$\text{How far is the point from the origin?}$$

The argument answers:

$$\text{In which direction is the point from the origin?}$$

So for a complex number

$$z=x+iy,$$

the modulus gives its distance from the origin, while the argument gives its direction from the positive real axis.

Distance between two complex numbers

A very important interpretation is that

$$|z_1-z_2|$$

is the distance between the points representing $z_1$ and $z_2$ in the Argand diagram.

To see why, let

$$z_1=x_1+i{y}_1$$

and

$$z_2=x_2+i{y}_2.$$

Then

$$z_1-z_2=(x_1-x_2)+i(y_1-y_2).$$

So

$$|z_1-z_2|=\sqrt{(x_1-x_2{)}^2+(y_1-y_2{)}^2}.$$

This is exactly the distance between the points

$$(x_1,y_1)$$

and

$$(x_2,y_2).$$

Therefore,

$$|z_1-z_2|$$

should be read geometrically as the distance between two points in the Argand diagram.

Bridge to loci

This distance interpretation is one of the most important ideas for complex loci.

For example,

$$|z-a|$$

means the distance between the point representing $z$ and the fixed point representing $a$.

Therefore, an equation such as

$$|z-a|=r$$

means:

$$\text{the point }z\text{ is always distance }r\text{ from the fixed point }a.$$

Geometrically, this describes a circle with centre at $a$ and radius $r$.

So many locus problems become much easier once modulus is interpreted as distance.

Caption: The modulus is the distance from the origin and the argument is the directed angle from the positive real axis.

Loci in the Complex Plane

What a locus means

In the Argand diagram, a complex number

$$z=x+iy$$

is represented by a point.

When $z$ is allowed to vary, the point representing $z$ can move. A condition on $z$ restricts where this point is allowed to move.

The set of all possible positions of the point representing $z$ is called a locus.

For locus problems, it is useful to name the moving point:

• let $P$ be the point representing the variable complex number $z$;
• let $A$ be the point representing a fixed complex number $a$;
• let $B$ be the point representing a fixed complex number $b$.

Then expressions involving $z$ can be translated into ordinary geometry.

The main translation

The key idea is that modulus means distance.

If $P$ represents $z$ and $A$ represents $a$, then

$$|z-a|$$

means the distance from $P$ to $A$.

So

$$|z-a|=PA.$$

Similarly,

$$|z-b|=PB.$$

Arguments also have a geometric meaning. The expression

$$\arg (z-a)$$

means the direction of the line from $A$ to $P$, measured from the positive real-axis direction.

Therefore:

• modulus statements become distance statements;
• equal-modulus statements become equal-distance statements;
• argument statements become direction or angle statements.

This translation is the foundation of complex loci.

Circle loci

Consider the condition

$$|z-a|=r,$$

where $a$ is fixed and $r>0$.

Let $A$ be the point representing $a$, and let $P$ be the point representing $z$.

Since

$$|z-a|=PA,$$

the condition says

$$PA=r.$$

So $P$ must always be a fixed distance $r$ from the fixed point $A$.

Therefore,

$$|z-a|=r$$

represents a circle with centre $A$ and radius $r$.

If

$$a=\alpha +i\beta ,$$

then the centre is the point

$$(\alpha ,\beta ).$$

For example,

$$|z-(3+i)|=2$$

represents a circle with centre

$$(3,1)$$

and radius

$$2.$$

Inequality versions

The inequality signs describe regions.

The condition

$$|z-a|<r$$

means that $P$ is less than distance $r$ from $A$. So it represents the interior of the circle, excluding the boundary.

The condition

$$|z-a|\le r$$

represents the interior of the circle together with the boundary.

The condition

$$|z-a|>r$$

represents the exterior of the circle, excluding the boundary.

The condition

$$|z-a|\ge r$$

represents the exterior of the circle together with the boundary.

So:

• strict inequalities exclude the boundary;
• non-strict inequalities include the boundary.

Perpendicular bisector loci

Now consider the condition

$$|z-a|=|z-b|.$$

Let $A$ represent $a$, $B$ represent $b$, and $P$ represent $z$.

Since

$$|z-a|=PA$$

and

$$|z-b|=PB,$$

the condition says

$$PA=PB.$$

So $P$ is equidistant from the two fixed points $A$ and $B$.

The set of all points equidistant from two fixed points is the perpendicular bisector of the line segment joining them.

Therefore,

$$|z-a|=|z-b|$$

represents the perpendicular bisector of the segment joining the points representing $a$ and $b$.

For example,

$$|z-1|=|z-(3+2i)|$$

means that $P$ is equidistant from

$$(1,0)$$

and

$$(3,2).$$

So the locus is the perpendicular bisector of the line segment joining those two points.

Inequality versions

The condition

$$|z-a|<|z-b|$$

means that $P$ is closer to $A$ than to $B$.

So the region is the side of the perpendicular bisector containing $A$.

Similarly,

$$|z-a|>|z-b|$$

means that $P$ is closer to $B$ than to $A$.

So the region is the side of the perpendicular bisector containing $B$.

Again, strict inequalities exclude the boundary, while non-strict inequalities include it.

Argument loci

Now consider the condition

$$\arg (z-a)=\alpha .$$

Let $A$ be the point representing $a$, and let $P$ be the point representing $z$.

The expression

$$z-a$$

represents the displacement from $A$ to $P$.

Therefore,

$$\arg (z-a)=\alpha$$

means that the line from $A$ to $P$ makes angle $\alpha$ with the positive real-axis direction.

So the locus is a half-line starting from $A$ and making angle $\alpha$ with the positive real-axis direction.

However, the point $A$ itself is excluded, because at $z=a$,

$$z-a=0,$$

and

$$\arg 0$$

is undefined.

For example,

$$\arg (z-(1+i))=\frac{\pi }{4}$$

represents a half-line starting from

$$(1,1)$$

making angle

$$\frac{\pi }{4}$$

with the positive real-axis direction, excluding the starting point.

Argument inequalities describe angular regions between rays. When sketching them, pay careful attention to whether the boundary rays are included or excluded.

Summary of standard locus patterns

Complex condition	Geometric meaning	Locus
$	z-a	=r$
$	z-a	<r$
$	z-a	\le r$
$	z-a	=
$	z-a	<
$\arg (z-a)=\alpha$	direction from $A$ to $P$ is $\alpha$	half-line from $A$, excluding $A$

Workflow for solving locus questions

For a locus question, avoid memorising templates blindly. Use the following workflow.

1. Identify the variable point $P$ representing $z$.

2. Identify the fixed points, such as the points representing $a$, $b$, or $c$.

3. Translate modulus expressions into distances.

For example,

$$|z-a|$$

means

$$PA.$$

4. Translate argument expressions into directions or angles.

For example,

$$\arg (z-a)$$

means the direction from $A$ to $P$.

5. Decide whether the condition gives a boundary or a region.

Usually:

• equality gives a curve, line, or ray;
• inequality gives a region.

6. Decide whether the boundary is included.

Usually:

• $<$ and $>$ exclude the boundary;
• $\le$ and $\ge$ include the boundary;
• for argument loci, the starting point is excluded because $\arg 0$ is undefined.

7. Sketch or describe the locus clearly.

A good description should state the type of locus, its fixed point or centre, its radius or direction, and whether the boundary is included.

Inequalities and regions

Many locus questions involve inequalities. The key point is that inequalities usually describe regions, not just curves.

For example,

$$|z-(2+i)|<3$$

means that $P$ is less than distance $3$ from the point $(2,1)$.

So the locus is the interior of the circle centred at $(2,1)$ with radius $3$, excluding the boundary.

Similarly,

$$|z-a|<|z-b|$$

means that $P$ is closer to $A$ than to $B$.

So the locus is one side of the perpendicular bisector of $AB$.

To decide which side to shade, use the meaning of the inequality rather than algebra alone.

Optimization on loci

Some questions ask for the greatest or least value of an expression such as

$$|z-c|$$

subject to a locus condition.

The expression

$$|z-c|$$

means the distance from the moving point $P$ to the fixed point $C$ representing $c$.

So these are geometric optimization problems.

The usual strategy is:

1. convert the condition on $z$ into a geometric locus;
2. interpret the expression to be optimized as a distance or angle;
3. use the diagram to find the greatest or least possible value.

For example, suppose $P$ lies on a circle with centre $A$ and radius $r$, and we want the greatest or least value of

$$|z-c|.$$

Let $C$ be the point representing $c$.

Then

$$|z-c|=PC.$$

So the question asks for the greatest or least possible distance from $C$ to a point $P$ on the circle.

These extreme distances occur along the line joining $C$ to the centre $A$.

If

$$AC=d,$$

then the greatest possible value is

$$d+r.$$

The least possible value is usually

$$|d-r|.$$

This geometric approach is often much simpler than converting everything into Cartesian equations.

Main idea

The main skill in complex loci is translation:

$$\text{complex condition}⟶\text{geometric statement}.$$

Once the condition has been translated, the problem usually becomes ordinary plane geometry.

The most important translations are:

$$|z-a|=\text{distance from }z\text{ to }a,$$

and

$$\arg (z-a)=\text{direction from }a\text{ to }z.$$

Related

• Complex Numbers
• Content Map
• Complex Number Algebra and Roots
• Polar Form, de Moivre, and Roots