Geometry of complex numbers

This chapter explores the geometric interpretation of complex numbers within the Argand plane, a foundational skill for advanced topics in complex analysis. Mastery of these concepts, including distance, locus, and transformations, is crucial for solving a wide array of examination problems requiring geometric reasoning.

Chapter Contents

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| Topic |

|---|-------| | 1 | Distance in Argand plane | | 2 | Locus in complex plane | | 3 | Circle and line interpretation | | 4 | Simple transformations in complex plane |

We begin with Distance in Argand plane.

Part 1: Distance in Argand plane

Distance in Argand Plane

Overview

Distance in the Argand plane is encoded by the modulus of a complex number. Once a complex number is viewed as a point, the quantity $|z_1-z_2|$ becomes ordinary Euclidean distance. In exam problems, this idea is used for geometry, loci, triangle inequalities, midpoint arguments, and minimum-distance interpretation. ---

Learning Objectives

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Core Distance Formula

This is the most important distance formula in complex geometry. ---

Coordinate Form

So the distance formula in the Argand plane is exactly the usual Euclidean distance formula. ---

Geometric Meaning of Standard Modulus Conditions

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Midpoint in Complex Form

This is useful in symmetry and perpendicular-bisector questions. ---

Triangle Inequality

This is just the triangle inequality in geometric language. This is useful in minimum-distance arguments. ::: ---

Minimal Distance Interpretation

In many such questions, drawing the Argand picture is more efficient than algebra. ---

Minimal Worked Examples

Example 1 Find the distance between $2+3i$ and $-1+i$. $\qquad |(2+3i)-(-1+i)| = |3+2i| = \sqrt{3^2+2^2}=\sqrt{13}$ --- Example 2 Find the midpoint of the points represented by $1+i$ and $5-3i$. $\qquad \dfrac{(1+i)+(5-3i)}{2} = \dfrac{6-2i}{2}=3-i$ So the midpoint is $\qquad 3-i$. ---

Standard Patterns

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Common Mistakes

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CMI Strategy

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Practice Questions

:::question type="MCQ" question="The distance between the points represented by $1+i$ and $4+5i$ is" options=["$5$","$\sqrt{13}$","$7$","$\sqrt{17}$"] answer="A" hint="Use $|z_1-z_2|$." solution="The distance is $\qquad |(4+5i)-(1+i)|=|3+4i|=\sqrt{3^2+4^2}=5$ So the correct option is $\boxed{A}$." ::: :::question type="NAT" question="If $z=3-4i$, then the distance of $z$ from the origin is" answer="5" hint="Use modulus." solution="The distance from the origin is $\qquad |z|=|3-4i|=\sqrt{3^2+(-4)^2}=\sqrt{9+16}=5$ Hence the answer is $\boxed{5}$." ::: :::question type="MSQ" question="Which of the following are true?" options=["$|z|$ is the distance from the origin","$|z_1-z_2|$ is the distance between $z_1$ and $z_2$","$\dfrac{z_1+z_2}{2}$ represents the midpoint","$|z_1-z_2|=|z_1|-|z_2|$ always"] answer="A,B,C" hint="Only one statement is false." solution="1. True.

True.

True.

False in general.

Hence the correct answer is $\boxed{A,B,C}$." ::: :::question type="SUB" question="Find the locus of points $z$ satisfying $|z-(1+i)|=|z-(3-i)|$ and interpret it geometrically." answer="Perpendicular bisector of the segment joining $(1,1)$ and $(3,-1)$; equation $x-y=2$" hint="Use equidistance from two fixed points." solution="Let $z=x+iy$. Then $\qquad |z-(1+i)|=|z-(3-i)|$ means the point $(x,y)$ is equidistant from $(1,1)$ and $(3,-1)$. So the locus is the perpendicular bisector of the segment joining these points. Midpoint: $\qquad (2,0)$ Slope of the segment: $\qquad \dfrac{-1-1}{3-1}=-1$ So the perpendicular bisector has slope $1$. Equation through $(2,0)$: $\qquad y=x-2$ Hence the Cartesian equation is $\qquad x-y=2$ So the locus is the perpendicular bisector of the segment joining the two fixed points, with equation $\boxed{x-y=2}$." ::: ---

Summary

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Part 2: Locus in complex plane

Locus in Complex Plane

Overview

In the complex plane, a complex number $z=x+iy$ is treated as a point $(x,y)$. A locus problem asks for the set of all points satisfying a given condition. In CMI-style questions, the real task is to convert the condition into a geometric meaning: distance, angle, perpendicular bisector, circle, line, or ray. ---

Learning Objectives

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Core Idea

This is the most important fact in locus problems. ---

Standard Loci from Modulus

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Ratio of Distances

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Argument-Based Loci

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Coordinate Method

This is especially useful for obtaining Cartesian equations of circles and lines. ---

Minimal Worked Examples

Example 1 Find the locus of $\qquad |z-(2+i)|=3$ This is the circle with centre $(2,1)$ and radius $3$. --- Example 2 Find the locus of $\qquad |z-1|=|z+1|$ Let $z=x+iy$. Then $\qquad |(x-1)+iy| = |(x+1)+iy|$ Squaring, $\qquad (x-1)^2+y^2 = (x+1)^2+y^2$ So $\qquad x=0$ Hence the locus is the imaginary axis, which is the perpendicular bisector of the segment joining $1$ and $-1$. ---

Common Patterns

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Common Mistakes

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CMI Strategy

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Practice Questions

:::question type="MCQ" question="The locus of points satisfying $|z-2i|=3$ is" options=["a line","a circle of centre $(0,2)$ and radius $3$","a circle of centre $(2,0)$ and radius $3$","a ray"] answer="B" hint="Read $|z-a|=r$ as a distance condition." solution="The condition $|z-2i|=3$ means the distance of $z$ from the point $2i=(0,2)$ is $3$. Hence the locus is a circle with centre $(0,2)$ and radius $3$. Therefore the correct option is $\boxed{B}$." ::: :::question type="NAT" question="If the locus of $z=x+iy$ is given by $|z-1|=|z+1|$, find the value of $x$." answer="0" hint="The point is equidistant from $1$ and $-1$." solution="The condition $|z-1|=|z+1|$ means the point $z$ is equidistant from the points $1$ and $-1$. Hence it lies on the perpendicular bisector of the segment joining them, which is the imaginary axis. Therefore $\boxed{x=0}$." ::: :::question type="MSQ" question="Which of the following statements are true?" options=["$|z-a|=r$ represents a circle","$|z-a|=|z-b|$ represents the perpendicular bisector of the segment joining $a$ and $b$","$\left|\dfrac{z-a}{z-b}\right|=1$ represents the set of points equidistant from $a$ and $b$","$\arg(z-a)=\theta$ always represents a full line"] answer="A,B,C" hint="Check the geometric meaning of modulus and argument." solution="1. True. Fixed distance from a point gives a circle.

True. Equal distance from two fixed points gives the perpendicular bisector.

True, because it is equivalent to $|z-a|=|z-b|$.

False. $\arg(z-a)=\theta$ usually gives a ray, not a full line.

Hence the correct answer is $\boxed{A,B,C}$." ::: :::question type="SUB" question="Find the locus of points $z$ satisfying $|z-1|=|z-i|$." answer="The perpendicular bisector of the segment joining $(1,0)$ and $(0,1)$, namely $x=y$" hint="Equidistance from two fixed points gives a perpendicular bisector." solution="Let $z=x+iy$. The condition is $\qquad |z-1|=|z-i|$ So $\qquad |(x-1)+iy| = |x+i(y-1)|$ Squaring both sides gives $\qquad (x-1)^2+y^2 = x^2+(y-1)^2$ Expanding, $\qquad x^2-2x+1+y^2 = x^2+y^2-2y+1$ Hence $\qquad -2x = -2y$ So $\qquad x=y$ Therefore the locus is the line $\boxed{x=y}$, which is the perpendicular bisector of the segment joining $(1,0)$ and $(0,1)$." ::: ---

Summary

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Part 3: Circle and line interpretation

Circle and Line Interpretation

Overview

In the geometry of complex numbers, equations in $z$ often represent familiar geometric objects in the Argand plane. The most important skill is to translate an algebraic condition such as $|z-a|=r$ or $|z-a|=|z-b|$ into a geometric picture. In exam problems, this topic is less about raw calculation and more about correct interpretation. ---

Learning Objectives

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Basic Setup

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Circle Interpretation

This is the most important circle interpretation in complex geometry. Reason: $|z-a|$ is the distance between the point $z$ and the fixed point $a$. ---

Interior and Exterior of a Circle

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Line Interpretation from Equal Distances

This is one of the most tested locus forms. ---

Horizontal and Vertical Lines

More generally:

• $\qquad \operatorname{Re}(az+b)=c$

• $\qquad \operatorname{Im}(az+b)=c$

represent straight lines, provided $a\ne0$. ::: ---

Rays and Directed Angle Conditions

This is not a full line; it is a half-line. ---

Constant Angle and Circle Arc

This is the complex-number version of the fixed-angle locus in geometry. ---

General Circle Equation in Complex Form

In most exam problems, however, the form $|z-a|=r$ is much easier and more useful. ---

Minimal Worked Examples

Example 1 Interpret $\qquad |z-(2-i)|=3$ This is the circle with center $\qquad 2-i$ and radius $\qquad 3$ --- Example 2 Interpret $\qquad |z-1|=|z+1|$ The fixed points are $1$ and $-1$ on the real axis. So the locus is the perpendicular bisector of the segment joining them, namely the imaginary axis. Thus the locus is $\qquad \operatorname{Re}(z)=0$ ---

Standard Patterns

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Common Mistakes

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CMI Strategy

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Practice Questions

:::question type="MCQ" question="The locus of $z$ satisfying $|z-(1+2i)|=4$ is" options=["a line","a circle","a parabola","a ray"] answer="B" hint="Interpret modulus as distance." solution="$|z-(1+2i)|$ is the distance of the point $z$ from the fixed point $1+2i$. So the locus is the circle centered at $1+2i$ with radius $4$. Hence the correct option is $\boxed{B}$." ::: :::question type="NAT" question="The locus $|z-2|=|z+2|$ is the line $\operatorname{Re}(z)=\underline{\qquad}$." answer="0" hint="Think perpendicular bisector." solution="The points $2$ and $-2$ lie on the real axis. The set of points equidistant from them is the perpendicular bisector of the segment joining them, namely the imaginary axis. So the equation is $\operatorname{Re}(z)=0$. Hence the answer is $\boxed{0}$." ::: :::question type="MSQ" question="Which of the following statements are true?" options=["$|z-a|=r$ represents a circle","$|z-a|=|z-b|$ represents a perpendicular bisector","$\operatorname{Im}(z)=k$ is a vertical line","$\arg(z-a)=\theta$ represents a ray"] answer="A,B,D" hint="Interpret each condition geometrically." solution="1. True.

True.

False, because $\operatorname{Im}(z)=k$ is a horizontal line.

True.

Hence the correct answer is $\boxed{A,B,D}$." ::: :::question type="SUB" question="Interpret geometrically the locus $|z-(1+i)|=|z-(3-i)|$ and find its Cartesian equation." answer="Perpendicular bisector of the segment joining $(1,1)$ and $(3,-1)$; equation $x-y=1$" hint="Square the distances or use midpoint-slope geometry." solution="The fixed points are $(1,1)$ and $(3,-1)$. The locus of points equidistant from them is the perpendicular bisector of the segment joining them. Midpoint: $\qquad \left(\dfrac{1+3}{2},\dfrac{1+(-1)}{2}\right)=(2,0)$ Slope of the segment: $\qquad \dfrac{-1-1}{3-1}=-1$ So the perpendicular bisector has slope $1$. Equation through $(2,0)$: $\qquad y=x-2$ So the Cartesian form is $\qquad x-y=2$ Therefore the locus is the perpendicular bisector of the segment joining the two fixed points, and its equation is $\boxed{x-y=2}$." ::: ---

Summary

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Part 4: Simple transformations in complex plane

Simple Transformations in Complex Plane

Overview

Complex numbers give a very efficient language for planar transformations. Multiplication by a complex number rotates and scales, while addition translates. In CMI-style problems, the key is to read expressions such as $z+a$, $\lambda z$, and $e^{i\theta}z$ geometrically rather than algebraically. ---

Learning Objectives

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Core Transformations

If $a=u+iv$, then each point moves by $(u,v)$. --- --- ---

Polar Form View

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Combined Transformations

So if $\qquad \alpha = re^{i\theta}$ then the map is:

scaling by $r$,

rotation by $\theta$.

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Rotation About Another Point

This is one of the most important formulas in geometry-of-complex-numbers problems. ---

Reflection Notes

These are useful, but translation, rotation, and scaling appear much more often in elementary problems. ---

Minimal Worked Examples

Example 1 Describe the transformation $\qquad z \mapsto z+(2-3i)$ This is translation by the vector $(2,-3)$. --- Example 2 Describe the transformation $\qquad z \mapsto iz$ Since $\qquad i = e^{i\pi/2}$, this is rotation by $90^\circ$ anticlockwise about the origin. --- Example 3 Describe the transformation $\qquad z \mapsto (1+i)z$ Now $\qquad |1+i|=\sqrt{2}$ and $\qquad \arg(1+i)=\dfrac{\pi}{4}$ So the map is:

• scaling by factor $\sqrt{2}$,
  • rotation by $45^\circ$ anticlockwise about the origin.
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  Images of Sets

  💡 How to Track a Locus

  To find the image of a set under a transformation:
  

  • understand what the transformation does geometrically,

  
  

  • transform the centre, radius, direction, or endpoints,

  
  

  • write the new geometric object directly.

  For example:
  • translation sends circles to circles,
  • rotation sends lines and circles to lines and circles,
  • scaling changes radius and distance from the origin.
  ::: ---

  Common Patterns

  📐 Patterns to Recognize

  • translation by a fixed complex number,

  
  

  • multiplication by $i$,

  
  

  • multiplication by $re^{i\theta}$,

  
  

  • rotation about a point,

  
  

  • image of a circle or line under a simple transformation.

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  Common Mistakes

  ⚠️ Avoid These Errors

  • ❌ thinking $z\mapsto iz$ changes modulus,

  ✅ it only rotates

  • ❌ forgetting that multiplication acts about the origin,

  ✅ use $w-a=e^{i\theta}(z-a)$ for rotation about another point

  • ❌ ignoring the modulus of the multiplier,

  ✅ $|\alpha|$ gives scaling factor

  • ❌ confusing translation with multiplication,

  ✅ addition translates, multiplication rotates/scales
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  CMI Strategy

  💡 How to Solve Fast

  • First rewrite the multiplier in polar form if needed.

  
  

  • Separate translation from multiplication.

  
  

  • Decide whether the map is acting about the origin or another point.

  
  

  • Describe the transformation geometrically before doing algebra.

  
  

  • Only then compute images of specific points or loci.

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  Practice Questions

  :::question type="MCQ" question="The transformation $z\mapsto iz$ represents" options=["translation by $i$","rotation by $90^\circ$ anticlockwise about the origin","reflection in the real axis","scaling by factor $2$"] answer="B" hint="Write $i=e^{i\pi/2}$." solution="Since $i=e^{i\pi/2}$, multiplication by $i$ rotates every point by $90^\circ$ anticlockwise about the origin. Hence the correct option is $\boxed{B}$." ::: :::question type="NAT" question="Under the map $z\mapsto z+(3-2i)$, the origin moves to which point? Enter the answer in the form $a+bi$." answer="3-2i" hint="Add the translation vector to $0$." solution="The image of the origin is $\qquad 0+(3-2i)=3-2i$ Hence the answer is $\boxed{3-2i}$." ::: :::question type="MSQ" question="Which of the following statements are true?" options=["$z\mapsto z+a$ is a translation","$z\mapsto e^{i\theta}z$ is a rotation about the origin","$z\mapsto \lambda z$ with real $\lambda>0$ is a scaling about the origin","$z\mapsto \overline{z}$ is reflection in the real axis"] answer="A,B,C,D" hint="Interpret each map geometrically." solution="1. True. Addition by a fixed complex number translates every point.
• True. Multiplication by $e^{i\theta}$ rotates about the origin.
• True. Positive real multiplication scales distances from the origin.
• True. Complex conjugation reflects in the real axis.
Hence the correct answer is $\boxed{A,B,C,D}$." ::: :::question type="SUB" question="Describe the transformation $z\mapsto (1+i)z$ geometrically." answer="Rotation by $45^\circ$ anticlockwise about the origin followed by scaling by factor $\sqrt{2}$" hint="Write $1+i$ in polar form." solution="We write $\qquad 1+i = \sqrt{2}\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right)=\sqrt{2}\,e^{i\pi/4}$ Hence the map $\qquad z \mapsto (1+i)z$ multiplies each modulus by $\sqrt{2}$ and increases each argument by $\dfrac{\pi}{4}$. Therefore it is a rotation by $45^\circ$ anticlockwise about the origin together with scaling by factor $\boxed{\sqrt{2}}$." ::: ---

Summary

❗ Key Takeaways for CMI

• Addition translates.




• Multiplication by a positive real scales.




• Multiplication by $e^{i\theta}$ rotates about the origin.




• Multiplication by a general nonzero complex number combines scaling and rotation.




• Rotation about another point uses $w=a+e^{i\theta}(z-a)$.




• Geometry should be read directly from the complex expression.

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Chapter Summary

❗ Geometry of complex numbers — Key Points

The geometric interpretation of $|z_1 - z_2|$ as the distance between points $z_1$ and $z_2$ is fundamental for defining loci in the Argand plane.
Standard loci, including circles ($|z-c|=r$ or $z\bar{z} + \alpha z + \bar{\alpha}\bar{z} + k = 0$), lines ($|z-a|=|z-b|$ or $\operatorname{Re}(\alpha z) = k$), rays ($\operatorname{arg}(z-a)=\theta$), and arcs ($\operatorname{arg}\left(\frac{z-a}{z-b}\right)=\theta$), are critical for problem-solving.
Complex number operations have clear geometric meanings: addition as vector addition, and multiplication by $re^{i\theta}$ as rotation by $\theta$ and scaling by $r$.
Simple transformations such as translation ($z \mapsto z+c$), rotation ($z \mapsto ze^{i\theta}$), reflection ($z \mapsto \bar{z}$), and inversion ($z \mapsto 1/z$) can be effectively analyzed using complex numbers.
Geometric conditions like collinearity, perpendicularity, and properties of triangles and quadrilaterals can be elegantly expressed and proven using complex number identities.
The triangle inequality, $|z_1 + z_2| \le |z_1| + |z_2|$ and its variations, provides powerful tools for bounding distances and magnitudes in geometric contexts.

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Chapter Review Questions

:::question type="MCQ" question="What is the locus of $z$ if $|z-1| + |z+1| = 4$?" options=["A circle centered at the origin", "An ellipse with foci at $(1,0)$ and $(-1,0)$", "A hyperbola", "A line segment"] answer="An ellipse with foci at $(1,0)$ and $(-1,0)$" hint="Recall the definition of an ellipse based on distances from foci." solution="The equation $|z-1| + |z+1| = 4$ represents the set of all points $z$ such that the sum of its distances from two fixed points $1$ and $-1$ is constant and equal to $4$. This is the definition of an ellipse with foci at $F_1 = 1$ (i.e., $(1,0)$) and $F_2 = -1$ (i.e., $(-1,0)$). The constant sum of distances is $2a = 4$, so $a=2$. The distance between foci is $2c = |1 - (-1)| = 2$, so $c=1$."
:::

:::question type="NAT" question="Let $z$ be a complex number such that $|z-3|=1$. If $w = \frac{1}{z}$, what is the maximum value of $|w|$?" answer="0.5" hint="Interpret $|z-3|=1$ geometrically and relate it to $|w|$. To maximize $|w|$, minimize $|z|$." solution="The equation $|z-3|=1$ represents a circle centered at $C=(3,0)$ with radius $R=1$. We want to find the maximum value of $|w| = \left|\frac{1}{z}\right| = \frac{1}{|z|}$. This maximum occurs when $|z|$ is minimized. The minimum value of $|z|$ for points on the circle is the distance from the origin to the closest point on the circle. This point is $z = 3-1 = 2$. So, $\min|z|=2$. Thus, the maximum value of $|w|$ is $\frac{1}{\min|z|} = \frac{1}{2} = 0.5$."
:::

:::question type="MCQ" question="If $z_1, z_2, z_3$ are the vertices of an equilateral triangle in the Argand plane, which of the following relations holds true?" options=["$z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_2z_3 + z_3z_1$", "$z_1 + z_2 + z_3 = 0$", "$|z_1-z_2|^2 = |z_2-z_3|^2 + |z_3-z_1|^2$", "$z_1z_2z_3 = 1$"] answer="$z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_2z_3 + z_3z_1$" hint="Consider the conditions for an equilateral triangle involving cube roots of unity, or the rotational properties of its sides." solution="For an equilateral triangle, the condition $(z_1 + \omega z_2 + \omega^2 z_3)(z_1 + \omega^2 z_2 + \omega z_3) = 0$ holds, where $\omega = e^{i2\pi/3}$ is a primitive cube root of unity. Expanding this product, and using the properties $\omega^3=1$ and $1+\omega+\omega^2=0$ (which implies $\omega+\omega^2=-1$), we get:
$z_1^2 + z_2^2 + z_3^2 + (\omega^2+\omega)z_1z_2 + (\omega^2+\omega)z_2z_3 + (\omega^2+\omega)z_3z_1 = 0$
$z_1^2 + z_2^2 + z_3^2 - z_1z_2 - z_2z_3 - z_3z_1 = 0$
Therefore, $z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_2z_3 + z_3z_1$."
:::

:::question type="NAT" question="A square in the Argand plane has vertices at $0, 1, 1+i, i$. If this square is transformed by the mapping $w = (1+i)z$, what is the area of the transformed region?" answer="2" hint="Consider how scaling and rotation by a complex number affect the area of a geometric figure." solution="The original square has side length $1$ and an area of $1^2 = 1$. The transformation $w = (1+i)z$ involves multiplication by the complex number $k = 1+i$. This transformation scales the lengths of all figures by a factor of $|k|$ and rotates them by $\operatorname{arg}(k)$. The area of the transformed region is scaled by a factor of $|k|^2$.
Here, $|k| = |1+i| = \sqrt{1^2+1^2} = \sqrt{2}$.
Therefore, the area scaling factor is $|k|^2 = (\sqrt{2})^2 = 2$.
The area of the transformed region is $2 \times (\text{original area}) = 2 \times 1 = 2$."
:::

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What's Next?

💡 Continue Your CMI Journey

Building on the geometric intuition developed here, the next chapters will delve deeper into the algebraic properties of complex numbers, specifically exploring roots of unity and complex polynomials, where geometric symmetry plays a crucial role. Furthermore, the interplay between complex numbers and trigonometry will become more apparent as we utilize Euler's formula and De Moivre's theorem to simplify trigonometric identities and solve equations. A strong grasp of the Argand plane is also foundational for understanding complex functions and transformations, which are explored in advanced topics.