Argand Diagram, Modulus, Argument, and Loci

Overview

This branch first develops the 9758-core geometry of Argand diagrams, modulus, argument, and transformations. Its later complex-loci section is retained as clearly labelled enrichment beyond the syllabus.

Syllabus boundary

Argand representation, modulus, argument, and the effects of conjugation, negation, addition, subtraction, and multiplication by $i$ are core. Complex loci are explicitly excluded from 9758 and begin only at the enrichment heading below.

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 $(x,y)$.

How to read the figure. Read the horizontal coordinate as $\mathrm{Re}(z)=x$ and the vertical coordinate as $\mathrm{Im}(z)=y$. The arrow from the origin is the position vector of the same point, not a second object.

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.

Core transformations in the Argand diagram

Let $z=x+iy$ and let the fixed complex number $a=u+iv$.

• Negation: $-z=-x-iy$ maps $(x,y)$ to $(-x,-y)$, a half-turn about the origin.
• Addition: $z+a=(x+u)+i(y+v)$ translates every point by the vector $(u,v)$.
• Subtraction: $z_2-z_1$ represents the displacement vector from the point $z_1$ to the point $z_2$.
• Multiplication by $i$: $iz=-y+ix$ maps $(x,y)$ to $(-y,x)$, a $90^{\circ }$ anticlockwise rotation about the origin.

How to read the figure. Compare the coordinates before and after multiplication: $(x,y)\mapsto (-y,x)$. The equal radial lengths show that the modulus is unchanged; the directed arc shows the anticlockwise quarter-turn. If arguments are used, normalise the result to the chosen principal range.

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.

Useful core identities are

$$|z^{\ast }|=|z|,|z_1{z}_2|=|z_1||z_2|,\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}(z_2\ne 0).$$

They say that conjugation preserves distance from the origin, multiplication multiplies distances, and division divides distances.

Argument as direction

The argument measures the direction of $z$ from the positive real axis. In this wiki, $\arg z$ denotes the principal argument in $(-\pi ,\pi ]$, while all arguments are $\arg z+2k\pi$ for $k\in \mathbb{Z}$.

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 distance idea also explains the enrichment topic of complex loci. For example, outside the current 9758 core, 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 $r$. This locus interpretation is retained later as enrichment, not as required Core 9758 content.

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 in enrichment work.

The main skill is to translate between the two languages:

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

This translation becomes especially important in the optional enrichment section on 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 $k$.

This wiki defines $\arg z$ to be the principal argument, which lies in the range

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

Thus

$$\arg z,$$

always denotes the principal value here. Other arguments must be written explicitly as $\arg z+2k\pi$.

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.

Handle points on the axes directly rather than using $y/x$:

$$\begin{array}{cc}x=0,y>0& \arg z=\pi /2\\ x=0,y<0& \arg z=-\pi /2\\ y=0,x>0& \arg z=0\\ y=0,x<0& \arg z=\pi \end{array}$$

The negative real axis uses $\pi$, not $-\pi$, because the chosen range is $(-\pi ,\pi ]$. For $z=0$, $\arg z$ is undefined.

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.

Enrichment bridge to loci

Scope preview

The distance interpretation below prepares for complex loci, which are outside the current 9758 core. Learn it as enrichment unless your teacher explicitly assigns it.

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.

How to read the figure. The radial length is $|z|=\sqrt{x^2+y^2}$; the directed angle from the positive real axis is the principal value $\arg z$. The dashed projections recover the Cartesian components $x$ and $y$.

Enrichment — Loci in the Complex Plane (beyond 9758)

Scope

Complex loci are explicitly excluded from the 9758 syllabus. The material below is optional enrichment and must not be treated as required examination content.

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 $r$.

The assumption $r>0$ matters: if $r=0$, the locus is the single point $z=a$; if $r<0$, there is no solution because a distance cannot be negative.

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, with $a\ne b$,

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

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 $b$.

If $a=b$, both distances are identical for every $z$. Therefore equality, $\le$, and $\ge$ hold throughout the complex plane, while the strict inequalities $<$ and $>$ have no solutions.

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

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

This is an open ray beginning at $a$: the point $z=a$ is excluded because $\arg 0$ is undefined. When an argument inequality is used, interpret it in the stated principal range and check whether the interval crosses the branch cut at the negative real axis; a wrapped interval may need to be described as two angular sectors.

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
$|z-a|=r$, $r>0$	distance from $A$ is $r$	circle with centre $A$ and radius $r$
$|z-a|<r$, $r>0$	distance from $A$ is less than $r$	interior; boundary excluded
$|z-a|\le r$, $r\ge 0$	distance from $A$ is at most $r$	interior plus boundary
$|z-a|=|z-b|$, $a\ne b$	equal distance from $A$ and $B$	perpendicular bisector of $AB$
$|z-a|<|z-b|$, $a\ne b$	closer to $A$ than to $B$	open side of the bisector containing $A$
$\arg (z-a)=\alpha$, $-\pi <\alpha \le \pi$	direction from $A$ to $P$ is $\alpha$	open ray 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

How to read the figure. Shading represents every point satisfying the inequality, not an area to calculate. In $|z-a|<r$, the modulus $|z-a|$ is the distance from the moving point $z$ to the fixed centre $a$, so the shaded set is the interior of the radius-$r$ circle. A strict sign such as $<$ excludes the boundary; replacing it by $\le$ includes the same boundary. For equal-distance comparisons, the boundary is the perpendicular bisector of the two fixed points.

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

$$d+r.$$

For a point $P$ moving on the full circle of centre $A$ and radius $r$, the least possible value is exactly

$$|d-r|.$$

Thus the exact maximum and minimum are $d+r$ and $|d-r|$. These results need not apply unchanged to a disc, arc, or locus restricted by another condition.

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